# Theory of Reality

## A Projective Seven-Layer Framework for Hidden State, Time, Quantum Observation, Computation, and Testable Physics

**Version:** 1.97
**Date:** 26 September 2026
**Originating framework and research direction:** Anthony Michael Collado  
**Computational formalization:** Reality Bit Engine 0.20.0; Rope Tomography Lab 0.14.0; RRLS Studio 0.27 audited extension 2; Hydrogen Matrix Lab 0.8.0; Element Projection Lab 0.15.0; Fixed Grid String Lab 0.22.0; Matter Harmony Lab 1.5.0; Born Outcome Frontier Audit 0.2.0; Wallbreaker Audit 0.1.0; P3-R1 Hardware Runner 1.0.0; Dimensional Chess C1 1.0.0; Dimensional Chess C2 1.0.0; Dimensional Chess C3 1.0.0; Dimensional Chess C4 1.0.0; Dimensional Chess C5 1.0.0; Dimensional Chess C6 1.0.0; Dimensional Chess C7 1.0.0; Dimensional Chess C8 1.0.0
**Status:** Revisable mathematical framework and executable research program; not an experimentally confirmed seven-dimensional theory of nature

**Revision 1.97 paired-reference identifiability.** Appendix CT proves that a zero-lifetime reference identifies a sample's material-delay kernel by a Laplace-transform ratio only when the reference and sample have the same detector and optical-path kernels. If the sample has an uncalibrated path mismatch, the ratio identifies the convolution of that mismatch with material response. An exact counterexample makes a genuine 0.8 ns material delay observationally identical to a zero-lifetime sample with a 0.8 ns sample-only path delay. Exact rational checks, 200,000-path Monte Carlo controls and four new tests pass. A separately justified 0.05 ns bound on path-mean mismatch would bound the inferred material mean to a 0.10 ns-wide interval; the bound is an assumption, not a measurement made here. No extra dimension or measured afterglow follows.

**Revision 1.96 optical-tail ambiguity.** Appendix CS proves an exact all-distance equivalence between a causal exponential material delay and a causal exponential excess-light-path distribution in one calibrated scalar time-resolved optical channel. The frozen 24 synthetic checks, two independent quadratures and a pure-Python audit agree; a calibrated direct-path trace differs. Thus the suggested multi-distance shadow test can check propagation and the presence of a tail but **cannot by itself locate its physical cause**, even with arbitrarily many noiseless distances. A separately validated intervention that changes only one mechanism is required. No extra spatial coordinate or measured result follows. The unpersisted 1.95 local paper and its symmetric-first-trial failure are retained intact in the lab's historical folder.

**Revision 1.95 orientation and shadow identifiability.** Appendix CR gives an exact unlabelled 4D mirror pair with opposite orientation but identical projection under every accessible 3D illumination experiment that factors through the stated projection map. Sixteen frozen synthetic lamp images, an exact rational distance-symmetry audit, and an independently integrated all-direction contact-shadow control pass. The earlier labelled-only symmetric prototype is preserved as a counterexample to a stronger chirality claim. In ideal vacuum a removed blocker changes a screen after ordinary light-travel delay, with no persistent shadow substance. A hypothetical instrument that couples to the fourth spatial coordinate would separate the example; no such physical instrument or extra spatial dimension is established.

**Revision 1.94 local continuum comparator.** Appendix CQ freezes a local 2D nonrelativistic charged-field continuum PDE with no discrete spatial scale, alongside the existing nonlinear gauge lattice. At one off-site point, gauge-invariant density and plaquette magnetic field differ at every tested finite N=8/16/32 and the differences shrink approximately fourfold under refinement. Independent analytic and time-integrator audits pass. This supplies a conditional experiment between two declared equations, not a physical preference: neither dimensionless observer has a validated instrument/unit map, and available collider and gamma-ray data observe different physical quantities. The engineered nonlinear interpolation countermodel from revision 1.93 remains possible when the continuum class is unrestricted.

**Revision 1.93 nonlinear sampled-field ambiguity.** Appendix CP constructs an exact nonlinear continuous-field representation of the *finite* coupled lattice dynamics at its designated sample nodes, using trigonometric interpolation and an explicitly nonlocal, grid-dependent evolution rule. The identity is a written mathematical conjugacy, not physical evidence of a continuum or a lattice. Four frozen synthetic runs and an independently assembled force law check the implementation. This corrects any suggestion that merely adding nonlinear sampled observations would identify a physical grid against an unrestricted continuum alternative. An independently restricted, physically motivated comparator and measured held-out predictions remain necessary.

**Revision 1.92 dispersion identifiability.** Appendix CO derives one calibration-plus-held-out linear-mode contrast that distinguishes the declared nearest-neighbor lattice dispersion from a restricted local isotropic nondispersive continuum equation, then constructs a continuous spatial-shift operator with exactly the same linear frequencies at all modes. Numerical frequency agreement cannot identify a fundamental spatial grid while that continuous alternative remains allowed. Frozen synthetic controls and independent off-grid audits are recorded in Fixed Grid String Lab 0.17.0; no measured physics or extra dimensions are claimed.

**Revision 1.91 rough-field counterexample.** Appendix CN adds a frozen cusp-profile control at the same grids as a smooth control. At t=0.25, the matter self-error ratio drops from 3.6579 to 1.7657 although norm, energy and Gauss checks pass; an independently assembled vectorized replay agrees. Smooth-field near-four ratios must not be generalized to arbitrary rough initial states. This remains synthetic computation, not evidence of a fundamental lattice or higher dimension.

**Revision 1.90 longer unforced gauge–matter check.** Appendix CM uses a frozen source-free extension of the existing smooth interacting case to t=1 on three grids. Matter, electric and gauge links show finite self-consistency ratios 3.22–4.18 across the longer check; Gauss, norm and energy drifts remain small, and independent vectorized evolution agrees. This is one numerical example, not a general continuum convergence proof or measured evidence.

**Revision 1.89 forced-solution location correction.** Appendix CL adds one exact driven coupled trajectory. The first frozen trial failed its spatial gate because the prescribed current was evaluated at sites rather than link midpoints. After preserving that negative result and freezing a corrected protocol, all five evolving components converge at approximately second order across grids 8/16/32; independent differentiation verifies the driving sources. These artificially driven equations validate a numerical method, not physical extra dimensions or unforced general convergence.

**Revision 1.88 three-grid evolving coupled check.** Appendix CK reports a frozen short-time (t=0.03) spatial self-convergence test of interacting matter, electric and magnetic gauge links on grids 6/12/24. Five comparison ratios range 3.8156–4.1557 after staggered-link alignment; independent vectorized RHS and RK4 replay agree. This is finite synthetic evidence consistent with second-order convergence of this example, not a general proof, measured physics or a discovered underlying grid. Revision 1.87 and the historical 1.86 frequency error remain preserved.

**Revision 1.87 interacting consistency and frequency correction.** Appendix CJ corrects the 1.86 vacuum dispersion frequency: the actual model has wave speed `sqrt(kappa beta)`, omitted from the prior table, whose values are only for an auxiliary unit-speed mode. The frozen nonzero-current and nonzero-magnetic-field RHS consistency check converges at second order for N=12,24,48, including matter, electric links and gauge links; an independent Fourier eigenspectrum and numerical derivative audit agrees. This verifies the spatial equations at one smooth state, not interacting trajectory convergence or physical evidence. Original version 1.86 and negative results are retained.

**Revision 1.86 magnetic plaquette benchmark.** Appendix CI and Fixed Grid String Lab 0.11.0 extend the self-consistent gauge–matter system to a periodic two-dimensional noncompact U(1) lattice with electric links and quadratic magnetic plaquette energy. Three frozen coupled cases preserve the written charge, Gauss, gauge-covariance and Hamiltonian identities numerically: maximum norm drift 2.39e-12, energy drift 1.20e-11, Gauss residual 7.18e-13 and gauge-related trajectory error 1.01e-12. A separate component-loop implementation agrees within 1.76e-13. Removing matter feedback breaks Gauss and energy accounting; removing magnetic feedback breaks energy while leaving the omitted curl force divergence-free, proving that Gauss preservation alone is insufficient. One linear vacuum magnetic mode converges toward its continuum dispersion at second order, with error ratios 3.97384 and 3.99345. All 142 package tests pass. These are written conditional mathematics and finite synthetic computation—not interacting continuum validation, QED, measured evidence, a physical lattice/tick or evidence for extra dimensions. Complete revision 1.85 is preserved.

**Revision 1.85 self-consistent field–matter benchmark.** Appendix CH and Fixed Grid String Lab 0.10.0 add a dimensionless one-dimensional periodic Hamiltonian in which a complex matter wave drives canonical electric-link variables and the changing link phase acts back on the matter. The written finite ODE conserves norm and total Hamiltonian energy and preserves a neutral-background Gauss constraint. In three frozen synthetic cases, numerical norm drift is at most 2.91e-12, energy drift 8.77e-12, Gauss residual 9.29e-13 and gauge-related trajectory error 1.95e-12; an independent scalar-loop RK4 implementation agrees within 2.18e-13. Removing field backreaction breaks Gauss preservation and energy conservation, so prescribed-field covariance is not mislabeled self-consistency. All 130 package tests pass. This is conditional mathematics and finite synthetic computation, not full 3D Maxwell theory, QED, spatial continuum validation, measured evidence, a derived fine-structure constant, a fundamental grid/tick or evidence for extra dimensions. Complete revision 1.84 is preserved.

**Revision 1.84 coverage and electromagnetic foundation.** Appendix CG and Fixed Grid String Lab 0.9.0 establish a 20-sector physical coverage ledger and a prescribed-field U(1) wave module. A declared nonrelativistic matter action yields a gauge-covariant lattice Hamiltonian, with an explicit scalar-potential term for time-dependent phase changes and a locally conserved probability/charge current. Eight production gates, all 120 package unit tests, 37 independent numerical checks and nine independent failure controls pass. The grid error for one frozen smooth mode decreases by factors 3.9826 and 3.9956 when the spacing is halved. A counterexample preserves gauge covariance while changing the potential and predictions, showing that symmetry does not determine a unique Hamiltonian. Maxwell equations are derived in writing from a separately postulated field action; self-consistent field evolution remains open. These are conditional mathematics and finite synthetic verification, not a derivation of all physics, the fine-structure constant or physical extra dimensions. Complete revision 1.83 and all earlier corrections are preserved.

**Revision 1.83 synthesis and checked wave formulation.** Appendix CF and Fixed Grid String Lab 0.8.0 define the Wave–Observer Formulation: finite Fourier representation, specified mode dynamics and interaction, and an explicit measurement map. A written invertible transformation connects positive-stiffness conservative classical oscillators to complex rotating modes; quantum amplitudes and a scoped Markovian density-matrix channel retain their different physical assumptions. Counterexamples show that nonlinear squaring generates omitted harmonics, zero modes can drift, equal initial densities can produce opposite futures when phases differ, and zero coherent dipole does not imply zero emission. Nine production gates, 112 package unit tests and 29 independent numerical audit checks pass. The formulation supplies an executable architecture, not a newly derived universal Hamiltonian, measured spacetime grid, physical fourth/fifth dimension, or new experimental evidence. The complete 1.82 paper and prior engine records are preserved.

**Revision 1.82 addition and correction.** Appendix CE and Fixed Grid String Lab 0.7.0 separate exact 4D slice reconstruction from projection. A 137-voxel discrete four-ball reconstructs exactly from five ordered, coordinate-labelled 3D slices. In contrast, the disjoint even- and odd-parity subsets of a `2^4` hypercube have identical line-sum projections along all four coordinate axes, so even four orthogonal shadows do not identify every 4D set. The revision also corrects the unqualified claim that one scalar can only trace a curve: a scalar can enumerate a finite catalog and a continuous Peano curve can be space-filling; the rank/area limitation requires a `C1` or locally Lipschitz generator. Nine production gates, all 103 package tests and 61 independent checks pass. These are written mathematics, finite computation and synthetic visualization—not measured evidence for a physical fourth or fifth dimension. Revision 1.81 and all earlier corrections remain in force.

**Revision 1.81 correction and evidence gate.** Appendix CD and Fixed Grid String Lab 0.6.0 audit 14 supplied claims about exceptional mathematics. The unqualified claim that every dimension except four has exactly one smooth structure is retracted: the exceptional theorem concerns Euclidean spaces, where `R^4` has uncountably many smoothings; oriented smooth homotopy 7-spheres supply 28 classes and refute the broader sentence. The claimed prime-count crossover below `10^316` is also corrected to a published interval near `1.398 x 10^316`. Banach–Tarski does not break physical conservation because its pieces are nonmeasurable. The 2022 experiment excludes a specified real-valued standard quantum formalism, while a peer-reviewed 2026 real-variable construction changes the composition rule and reproduces standard finite-dimensional predictions; the broad claim that complex numbers are uniquely necessary in nature is therefore model-class dependent. Zero of 14 catalog entries passes the new physical-dimension promotion gate. All six production gates, all 85 package tests and 21 independent checks pass. This is written mathematics, literature scope correction and finite computation—not new measured evidence or proof of a fourth physical dimension. Revision 1.80 and all earlier corrections remain in force.

**Revision 1.80 correction and addition.** Appendix CC and Fixed Grid String Lab 0.5.0 correct the release-0.4 acoustic load from `rho_particle V g` to the buoyancy-adjusted `(rho_particle-rho_air)Vg`, preserving the former result as history. The polystyrene threshold changes from 1010.85847797 Pa to 1010.27875288 Pa per array, only 0.05735%. Equal-radius calculations for 12 declared condensed materials show acoustic contrast tightly clustered near 5/6, so threshold is dominated by effective density: 720.396 Pa for lithium, 985.047 Pa for water, 4331.773 Pa for gold and 4686.466 Pa for osmium. A separate density-only screen covers 85 sourced condensed elements but is explicitly approximate. Free gas parcels are not ranked as levitated objects because they mix with air unless bounded; bubbles and membranes require interface and resonance physics. All five new production gates, all 70 package tests and 86 independent scalar checks pass. These are conditional acoustics, sourced property compilation and finite synthetic computation—not measured validation, a universal chemical law, mass reduction, or evidence for a physical grid or extra dimension. Revision 1.79 and all earlier corrections remain in force.

**Revision 1.79 addition.** Appendix CB and Fixed Grid String Lab 0.4.0 calculate whether a declared multichannel ultrasonic field can support a small object. In the frozen linear-acoustic, ideal-monopole and Gor'kov small-sphere model, two opposed 8 by 8 arrays at 40 kHz give a stable equilibrium for a 0.549778714 mg sphere. At 3000 Pa peak coherent-focus pressure per array, equilibrium is 86.5744778 micrometres below the pressure node; the computed support threshold is 1010.85847797 Pa. All six production gates, 53 package tests and 24 independently implemented scalar checks pass. The decisive correction is interpretive: the acoustic field supplies an external force equal to `mg`; it does not change mass or gravity, and the source receives the equal and opposite reaction. No measured pressure map, force curve or hardware validation was used. The calculation is conditional mathematics plus finite synthetic computation, not evidence for a physical grid, a fundamental tick, extra dimensions or a new ontology of matter. Revision 1.78 and all earlier corrections remain unchanged.

**Revision 1.78 addition.** Appendix CA and Fixed Grid String Lab 0.3.0 execute the frozen hydrogen 1S→2P electric-dipole baseline. The declared l=0/l=1 radial finite-difference Hamiltonians converge toward the exact infinite-mass nonrelativistic gap, radial matrix element and oscillator strength. At the finest grid the oscillator strength is 0.416548743989 versus 0.416196717980, a 0.08458% relative error; all three tested errors decrease at approximately second order. Exact angular rules allow the z-polarized 1S→2P m=0 transition and reject the declared 1S→2S, 1S→3D and m=1 controls. All 32 unit tests and 58 independent sparse-eigensolver/quadrature checks pass. No parameter was fitted and no measured wavelength was used for validation. The Coulomb Hamiltonian and electric-dipole operator are imported standard quantum assumptions, not derived string dynamics or evidence of extra dimensions. Revision 1.77's corrections and all earlier boundaries remain unchanged.

**Revision 1.77 correction and foundation test.** Appendix BZ and Fixed Grid String Lab 0.2.0 correct the minimum-hidden-memory statement in §4.2: exact minimality uses the maximum actual number of future-equivalence classes across all original observation fibers, not a loose upper bound or one chosen fiber. An exact stationary four-cycle also shows why §4.1's one-step conditional need not be an autonomous Markov law. The new foundation contract separates mathematical representation, fixed physical model, measurement likelihood, implementation and evidence. A frozen radial Coulomb finite-difference model has a written exact half-line spectrum; 32 finite-matrix eigenvalues, 23 unit tests and 102 independently implemented audit checks pass. Exact exponential ticks are sampled continuous-time evolution of the same matrix, so this agreement cannot identify a fundamental clock. No measured validation or parameter fitting was performed. The Coulomb interaction and quantum rules are imported, not derived from higher-dimensional strings. Earlier versions, negative results, sources and mathematical corrections remain intact.

**Revision 1.76 addition.** Appendix BY and Element Projection Lab 0.15.0 add a coherent crystal diffraction observer for all 92 available phases. An independent implementation verifies 56,656 reciprocal reflections, 5,962 powder shells, 3,993 analytic selection-rule cases and 132 explicit finite-cluster sums. NIST SRM640g silicon reproduces 11 published calculated peak positions within rounding; this checks consistency with a certified lattice parameter rather than supplying 11 new measurements. An exact homometric pair proves that different point arrangements can produce identical intensities for every scattering vector. All 272 unit tests and 48,786 DOM/SVG interaction checks pass. The model uses identical point scatterers; calibrated intensities, electronic color prediction and physical extra dimensions are not established. Earlier reference data, negative results and mathematical corrections remain unchanged.

**Revision 1.75 addition.** Appendix BX and Element Projection Lab 0.14.0 expand the element reference table to 118 typed property records, 1,544 populated values, 90 complete selected crystal cells and 30 visible optical-response datasets. Graphite and a separately labeled 300 GPa DFT hydrogen phase are additional options; unmeasured or uningested data remain explicit. Independent raw-CIF audits check all 25 added phase records, including the prediction; independent scalar optics reproduce 7,619 reflectances and 38 colors. All 259 unit tests and 14,105 view checks pass. The original 67 geometries, 11 optical records and previous mathematical/physical corrections remain unchanged. These are external reference inputs and checked computation, not new measurements or geometric predictions of color.

**Revision 1.74 addition.** Appendix BW and Element Projection Lab 0.13.0 add a crystal periodic table: all 118 identities, 67 complete sourced primitive cells and 11 measured-input sunlight-color reconstructions. Primitive cells are repeating units, not minimum stable isolated clusters; missing bases and optical constants remain unassigned. Au, Ag and Cu have identical normalized FCC topology but distinct visible reflectances, so topology alone does not determine color. Their atomic lattice spacings also fail the scale condition for ordinary first-order visible Bragg diffraction. The interactive 4D-shadow view is an explicit w=0 embedding and projection, separate from physical inference and optical calculation. All 218 tests pass; independent replay checks 67 cells, 4,411 reflectances and 22 colors. The result is compiled reference geometry and computation using external measured optical response, not an ab-initio color prediction or new dimensional evidence. Earlier sources, results and the full 1.73 paper are retained.

**Revision 1.73 addition.** Appendix BV and Element Projection Lab 0.12.0 implement an explicit-loss synthetic detector response and frozen injected-signal closure test. At unit strength, 500 expected truth events give 396.970157 reconstructed and 103.029843 missed events. Correct-response mean-data fitting recovers strength 1, while underestimating efficiency by 20% recovers 1.25 with an identical fitted spectrum and likelihood. This is an exact source-yield/efficiency non-identifiability result, independently checked over 6,000 synthetic histories and 18,000 fits. All 192 unit tests and 1,231,728 independent numerical checks pass. Gaussian resolution, equal weights, efficiency and background are toy assumptions, not ATLAS calibration; the original excluded benchmark, published negative results and real-data recast block remain unchanged. The contribution is written conditional mathematics and finite synthetic computation, not new physical evidence or a unique identification of hidden particles or dimensions.

**Revision 1.72 addition.** Appendix BU and Element Projection Lab 0.11.0 add an offline, four-scenario contour-relative parameter explorer. A declared computational viewport contains 12,099 nodes per observed contour; independent source-decimal rational arithmetic verifies all 48,396 grid classifications and 2,916 additional probe classifications. Nine nonzero horizontal segments at five table/k levels are missed by the coarse regular grid but retained by off-grid tests; a repeated vertex is not mistaken for an ambiguous interval. The engine distinguishes above/below contour from missing k coverage, ambiguous geometry, boundary, and the nonperturbative region. The viewport is not a certified likelihood scan rectangle, no joint confidence is calculated, and above-contour points are not confirmation. The unchanged benchmark remains excluded in all four individual observed scenarios. No physics parameter or prior negative result is retuned; the full previous paper is preserved.

**Revision 1.71 addition.** Appendix BT and Element Projection Lab 0.10.0 ingest the full user-supplied HEPData YAML archive: 24 contour tables, 1,978 coordinate pairs, and submission metadata. At the unchanged `k=500 GeV`, `M5=6000 GeV` benchmark, observed limits are approximately 7.063338 TeV (dielectron with mass thresholds), 9.815289 TeV (diphoton with thresholds), 8.735511 TeV (dielectron without thresholds), and 10.896422 TeV (diphoton without thresholds). Each observed scenario excludes the benchmark under its own published 95% CL result; no combined confidence level or independent experimental replication is claimed. All 24 slices are unique at this k, and an independent exact-rational implementation agrees through 349 checks, with maximum boundary discrepancy 1.82e-12 GeV. The twenty expected-limit curves are retained only as diagnostics. The archive contains no spectra, response matrices or likelihood objects, so it closes the published-contour lookup gap, not the full detector-recast gap. This is retrospective computation from external measured limits, not a new measurement, a fit, dimensional discovery, or universal exclusion. Earlier results and the complete 1.70 paper are preserved.

**Revision 1.70 addition.** Appendix BS and Element Projection Lab 0.9.0 apply the first machine-readable HEPData values supplied by the user to the revision-1.69 benchmark. HEPData Table 1 is the observed 95% CL dielectron exclusion contour with mass thresholds. Its adjacent vertices `(k,M5)=(489.1,6973.0)` and `(502.3,7082.4)` GeV bracket `k=500 GeV`; local linear interpolation gives `M5_limit=7063.337879 GeV`. The compiled `M5=6000 GeV` point is 1063.337879 GeV below the boundary and is therefore excluded by this measured result. Under ATLAS's approximate `sigma proportional to M5^-3` scaling, the point's signal strength is about 1.63146 times the boundary value. Ten new production tests and an independent eight-check 50-digit Decimal replay pass. This corrects the benchmark's interpretation: it remains useful for validating template construction but is not experimentally viable in this channel/scenario. The supplied record index inventories 24 contour tables but advertises no spectra, response matrices, likelihood, covariance, or classifier artifact, so it enables a complete published-limit audit once the archive bytes are supplied, not an exact detector-level recast. This is external measured evidence against one parameter point plus finite arithmetic--not positive fifth-dimensional evidence, universal exclusion, or theological identification.

**Revision 1.69 addition.** Appendix BR and Element Projection Lab 0.8.0 compile the primary spin-2 clockwork/linear-dilaton equations at `k=500 GeV` and `M5=6 TeV`, a benchmark explicitly generated by ATLAS for analytic-template validation. The Planck relation gives `R=0.0206137688164 GeV^-1`, `kR=10.3068844082`, and `1/R=48.5112649176 GeV`. There are 102 modes below the common 5 TeV cap, from 502.347831 to 4973.346835 GeV. In the paper's massless-SM approximation, the maximum `Gamma_SM/m` is `1.64244e-4` and the maximum `Gamma_SM/Delta_next` is `0.0169231`; all compiled modes lie below the paper's 28.6083 TeV resolved-tower condition. The massless channel ratios are `BR(gamma gamma)=12/283` and `BR(e+e-)=6/283`. Eleven new production tests pass, and an independent implementation performs 2,454 comparisons over every mode. This closes the numerical mass/coupling/SM-partial-width gap for one truth-level benchmark, not the hadronic production, detector response, likelihood, nuisance, or classifier gaps. HEPData again presented a persistent human-verification challenge; that is an access result, not non-publication. The benchmark is not treated as a fit or viable point, and this is conditional mathematics plus finite computation--not a measured excess, exact ATLAS recast, physical fifth-dimension evidence, universal grid/tick, or theological identification.

**Revision 1.68 addition.** Appendix BQ and Element Projection Lab 0.7.0 execute the recast-readiness test demanded by revision 1.67. The exact ATLAS forward equation shows that a truth-level resonance tower does not determine reconstructed bin counts without per-mode production, branching fractions, acceptance/efficiency, and detector transfer. A frozen 13-component contract finds only three components ready and blocks the detector-level replay rather than imputing the other ten. The exact counterexample keeps truth weights and efficiencies fixed while two admissible column-stochastic detector responses change the reconstructed spectrum from `[8,3,0]` to `[4.75,4.75,1.5]`, with the same total count of 11. Nine new production tests and an independent 15-check scalar replay pass. The official HEPData endpoint could not be materialized through the available web client because of an access challenge; that is explicitly not reclassified as non-publication. This is written mathematics and finite synthetic computation, not a new measurement, successful ATLAS recast, positive evidence for another dimension, universal exclusion, or theological identification.

**Revision 1.67 addition.** Appendix BP and Element Projection Lab 0.6.0 execute a frozen retrospective evidence audit on the exact signed significances printed in ATLAS's 139 fb^-1 periodic dielectron/diphoton search. At the preselected periodicity-focused `R<50%` row, the dielectron value is +1.5 standard deviations and the diphoton value is -0.2. Across all four reported threshold variants, one-sided normal-tail probabilities with an arbitrary-dependence Bonferroni correction are 0.267229 and 1.0 for the most positive dielectron and diphoton values. The paper's model-dependent search also reports no significant deviation. The necessary cross-channel evidence gate therefore fails. Eight new production tests and a twelve-check independent standard-library replay pass. The equally important correction is procedural: because the publication and its null conclusion were already public, this is not a blind preregistration or a reproduction of the ATLAS likelihood. It is measured-input retrospective auditing plus finite computation, not a new measurement, discovery of another dimension, universal exclusion, or theological identification. A separate latest-source check found that the RRLS package no longer matched its own documented integer-validation and independent-verifier contract. Audited extension 2 restores strict integer inputs, records the collision search bound, and reconstructs collision certificates without calling the production solver; all 20 regression tests, five legacy checks, 16,250 exhaustive integer cases, and 600 generated rational cases pass. That is a software correction, not physical evidence.

**Revision 1.66 addition.** Appendix BO and Element Projection Lab 0.5.0 impose a physical-dimension evidence gate and close one invalid route to such evidence. For the declared real five-coordinate skew-linear motion, every continuous linear three-vector observation is exactly a constant plus at most two ordinary sine/cosine pairs; two conventional temporal oscillators therefore reproduce the alleged 5D projection for all times. In 512 random trials the two realizations agree over 512,512 vector samples to maximum component error 5.33e-15; an independent scalar implementation checks 299,919 components with maximum error 3.55e-15. A simple flat-circle Kaluza--Klein relation is retained only as a model-specific example: two labelled masses merely calibrate its two parameters, while later masses become predictions. Current ATLAS periodic-spectrum, CMS high-mass diphoton, and Eot-Wash short-range-gravity searches report null results and exclusions rather than positive evidence. Thus the present animations and optical oscillators are representations, not evidence for another spatial dimension. The next admissible experiment is a preregistered public-data search for a correlated tower, including frozen masses, widths, spin/coupling/channel relations, background, multiplicity correction, and reserved validation. This is written mathematics and finite synthetic computation plus cited external measured limits--not a new measurement, dimensional discovery, or theological identification.

**Revision 1.65 addition.** Appendix BN and Element Projection Lab 0.4.0 freeze the revision-1.64 Au/Ag response comparison under identity `EPL-AU-AG-COLOR-2026-09-20-v1`. The manifest fixes the four coefficient vectors as hexadecimal IEEE-754 values, source table and executable hashes, train/holdout split, equations, wavelength domain, scoring rules, boundary warnings, headline results, and prohibited changes. The evaluation module contains no optimizer and verifies every locked artifact before scoring; future refitting, bound changes, and model selection are forbidden under this identity. Eight new tests pass, and a separate implementation verifies the manifest plus four artifact hashes and replays all 64 predictions with maximum discrepancy 4.44e-16. This is an anti-leakage and reproducibility result, not new measured evidence or an upgrade from phenomenological response to unique band, relativistic, higher-dimensional, or theological mechanism.

**Revision 1.64 addition.** Appendix BM and Element Projection Lab 0.3.0 execute the first frozen real-data model-class test behind the gold/silver color observer. For each metal, eleven Johnson--Christy wavelengths fit either a three-parameter damped free-electron response or a six-parameter response with one additional interband oscillator; five fixed wavelengths remain untouched until scoring. Gold holdout reflectance MAE falls from 0.154691 to 0.019718, an 87.253% reduction. Silver falls from 0.006840 to 0.002727, a smaller 0.004112 absolute gain because its visible response is already nearly free-electron-like. Nine new tests pass, and an independent scalar implementation replays 64 predictions and 218 checks with maximum aggregate-metric discrepancy 2.16e-16. The correction is equally important: an oscillator ablation is not a relativistic/nonrelativistic comparison, the richer model has three extra fitted parameters, and two fits hit declared parameter bounds. This supports a wavelength-dependent interband channel within the tested classes, not a unique band Hamiltonian, physical extra dimension, or theological cause.

**Revision 1.63 addition.** Appendix BL and Element Projection Lab 0.2.0 add the missing condensed-matter optical observer for bulk gold and silver without assigning color to one isolated atom. Johnson--Christy's room-temperature measured complex refractive indices are passed through the exact normal-incidence Fresnel map and a declared finite CIE observer. At 450.9 nm the calculated reflectances are 0.408220 for Au and 0.980347 for Ag; at 659.5 nm they are 0.962585 and 0.990566. Gold's mean red-minus-blue reflectance is 0.537373, versus 0.013155 for silver. All 19 package tests pass, and an implementation importing no production optics code replays 802 dense reflectance values with maximum discrepancy 5.55e-16. The decisive correction is that the all-element atom atlas cannot predict bulk color: lattice-periodic bands, occupations, optical transition matrix elements, damping, and relativistic terms intervene. This is a measured-input forward calculation, not an ab-initio color prediction, measured work by this project, proof of a fourth/fifth dimension, or theological identification.

**Revision 1.62 addition.** Appendix BK and Element Projection Lab 0.1.0 compile all 118 neutral elements into one typed comparison atlas. The exact correction is that an isolated atom has no unique rigid classical surface: the released whole-atom channel is an explicitly rotationally averaged one-electron density and is therefore spherical, while the separate oriented channel displays a declared canonical differentiating-subshell basis. Closed-subshell sphericity follows exactly from the spherical-harmonic addition theorem; oriented \(s,p,d,f\) basis densities have allowed even angular ranks through \(2l\). Reference configurations, shell populations, radial-node counts, and periodic-table positions are separated from a Slater-screened hydrogenic radial proxy; configurations for atomic numbers 109--118 are marked predicted. All 11 production tests and 13 independent checks pass over every element, including electron counts, capacities, grid uniqueness, hydrogenic normalization, and the addition theorem. These are written mathematics, compiled reference inputs, and finite proxy computation--not measured 3D densities for every atom, many-electron ab initio solutions, rigid electron paths, fourth/fifth-dimensional causes, or a theological identification.

**Revision 1.61 addition.** Appendix BJ and Hydrogen Matrix Lab 0.8.0 replace the undefined goal “display everything” with a typed forward-observer contract. One declared ideal coherent `1s + 2p_z` state now generates synchronized position density, conserved probability flow, electric charge/current, expectation dipole, far-zone coherent electromagnetic radiation, Poynting flux, and an operator-filtered harmonic catalog. Eight frozen checks pass, including charge continuity, transversality, `|B|=|E|/c`, angular power integration, and the parity-silent `1s + 2s` control; a separate implementation passes 12 checks. The decisive exact boundary is not a fit: cycle-averaged classical expectation-dipole power is exactly one half of the initial quantum E1 energy-loss rate for the equal packet's half-excited population. Therefore a classical projected wave is not the complete spontaneous-photon channel. The compiler marks quantized-field/vacuum statistics and spin magnetization as blocked and higher-dimensional cause as non-identified. No new measurement, complete QED engine, physical fourth/fifth dimension, or theological cause is claimed; all earlier rejections and fixed-grid boundaries remain unchanged.

**Revision 1.60 addition.** Appendix BI and Hydrogen Matrix Lab 0.7.0 audit golden-ratio, golden-angle, Fibonacci, icosahedral, and proposed “golden alignment” claims with exact algebra before accepting numerical matches. The exact positive result is structural: the Fibonacci companion matrix has dominant eigenvalue `phi`, declared icosahedral coordinates contain `phi`, and the `E8` identity is `2 cos(pi/5)=phi`. The exact negative result is equally important: every ideal Coulomb bound-state energy gap is rational in Hartree, so no ratio of two nonzero gaps can equal irrational `phi`. An exhaustive frozen `n<=40` scan checks 303,809 distinct pairs; the closest ratio differs from `phi` by `9.76537e-6` and remains a coincidence. Likewise, a finite angular grid step `k/N` cannot equal the irrational golden-turn fraction `1/phi^2`; Fibonacci grids are periodic approximations only. All 51 package tests and nine independent checks pass. External `E8`, quasicrystal, and phyllotaxis measurements are retained as evidence for their declared systems, not transferred to hydrogen, hidden dimensions, or theology.

**Revision 1.59 addition.** Appendix BH answers the bounded question of which quantum equation was missing from the hydrogen visualization. Schrödinger evolution was already present; the omitted bridge was the conserved probability-current field, electromagnetic minimal coupling, and the observable matrix element that filters raw energy-gap harmonics. Hydrogen Matrix Lab 0.6.0 implements the field-free current for the existing `1s + 2p_z` packet, proves `partial_t rho + div(j)=0`, recovers the exact dipole matrix element `256/(243 sqrt(2)) a0`, and computes the ideal E1 lifetime `1.59532505 ns`. In an equal `1s + 2p_z + 3s` packet all three density beats exist but only two are visible to the `z` electric dipole; the `1s <-> 3s` harmonic vanishes by parity, as does the one-photon `1s <-> 2s` control. All 42 Hydrogen Matrix tests pass, and an independent quadrature/fourth-order finite-difference audit passes seven frozen checks while retaining the larger full-interior Coulomb-cusp residual. These are written standard quantum mechanics and finite computation, not a measured atom, a particle trajectory, an extra dimension, or a theological identification.

**Revision 1.58 addition.** Appendix BG turns one member of revision 1.57's non-identifiable interaction family into a frozen prediction. The range is set at the electron reduced Compton wavelength and the coupling is fitted only to the published H--D 1S--2S central residual. Without refitting it predicts a -201.996425 Hz correction to the historical H--D 2S--8S isotope shift. A published bound-state reference with experimental and theory correlations gives an ordinary signed error of -0.700628 combined standard deviations; adding the operator changes it to -0.720112. The operator therefore earns no predictive credit, but the combined uncertainty is 51.326 times the predicted contrast, so this target cannot exclude it either. This is a historical public-data holdout, not a blind prediction. All 164 package tests and an independent direct-radial-quadrature audit pass. No physical dimension, universal grid/tick, string particle, or theological cause is inferred.

**Revision 1.57 addition.** Appendix BF replaces the unspecified extra-dimensional hydrogen correction with one explicit finite-range central proxy, \(\delta V/h=g e^{-r/\lambda}/r\), but first evaluates the published H--D 1S--2S correction ladder. A 2022 QED/nuclear-structure expression supplied with the muonic-deuterium radius \(2.12763(78)\) fm predicts \(670994336.042632\) kHz versus the direct measured \(670994334.606(15)\) kHz: a +1.436632 kHz error, only +0.309753 combined standard deviations. The ordinary ladder reduces the crude reduced-mass-only error by a factor of 8837.3. The operator is therefore not required. Moreover, all 61 ranges from 1 fm through 10 nm can be paired with a coupling that fits the same central residual exactly, while the endpoint pairs disagree by 121.784 Hz on a prospective 1S--3S isotope correction. This is an exact one-equation/two-parameter non-identifiability result, not evidence for a fifth dimension. All 154 package tests and a separate adaptive-quadrature audit pass. The published values are measured/theory inputs; no new spectroscopy or theological identification is claimed.

**Revision 1.56 addition.** Appendix BE replaces the hydrogen feature-count exercise with a frozen transition holdout. NIST H-alpha and H-beta alone calibrate a one-parameter Rydberg-form baseline and a two-parameter phenomenological `Delta2 + Delta4` candidate after a declared standard-air-to-vacuum conversion. On untouched H-gamma and H-delta, the added term increases vacuum-wavelength RMSE from 6.030008 to 6.948567 ppm, a 15.2331% deterioration, and is rejected as an improvement. A written counterexample adds an arbitrary term that vanishes on both training lines but changes every later prediction; one coefficient forces H-gamma exactly and still misses H-delta by 7.1389 ppm. Thus perfect calibration does not identify a continuation, a quantum generator, or a physical dimension. All 143 package tests and a separate 50-digit Decimal replay pass. The NIST values are compiled measured references; the conversion and scores are finite computation; no new spectroscopy, complete QED calculation, or higher-dimensional operator is claimed.

**Revision 1.55 addition.** Appendix BD generalizes the fixed skew-linear harmony engine to arbitrary latent dimension and exposes explicit latent position vectors, tangent vectors, and their calibrated 3D projections. A single three-output instantaneous observer has nullity at least (d-3), while a simultaneous known observer bank identifies the common state exactly iff its vertically stacked matrix has rank (d). The frozen banks reconstruct synthetic states in dimensions 4, 5, 6, and 8 to numerical precision, while an exact 5D counterexample moves the latent state by one unit without changing one observer's output. More observers remove projection ambiguity but do not add intrinsic generator frequencies: the capacity remains (lfloor d/2\rfloor). Thus 4D/5D have two positive modes, 6D/7D three, and 8D/9D four. Eight dimensions first pass the four-Balmer-feature count, but this is only a necessary capacity gate and predicts no wavelength, intensity, selection rule, or physical dimension. All 132 package tests and a separate observer-bank audit pass. These are written mathematics and finite synthetic computation, not measured higher-dimensional strings.

**Revision 1.54 addition.** Appendix BC subjects the new five-coordinate harmony observer to an external hydrogen spectroscopy gate. The NIST H I strong-line table supplies four resolved visible Balmer features at representative air wavelengths 410.17400, 434.04620, 486.13615, and 656.28518 nm. A fixed real 5-by-5 skew generator with a linear observation has at most two distinct positive temporal frequencies; any further linear exposure can only attenuate or erase them. Therefore the current model cannot reproduce all four features as one linear temporal spectrum, even though two free frequencies can be calibrated to any selected pair. In the three frozen ratio families, the 3:2 case matches two of four features at 1% only after calibration, while the exact capacity deficit remains two. Seven new tests and an independent Decimal replay pass. This is a useful rejection of the present linear observation model, not of every higher-dimensional theory: adding state transitions, nonlinear mixing, time variation, interactions, or more state dimensions creates different model classes that require fresh predictions. No mismatch is assigned to a hidden string, a fifth dimension, or a theological cause.

**Revision 1.53 addition.** Appendix BB replaces the undefined phrase “five-dimensional harmony” with a complete classification for fixed norm-preserving linear motion.  Every real 5-by-5 skew generator has two nonnegative rotation frequencies and one zero direction, so every linear 3D projection is a constant plus at most two elliptical sine/cosine harmonies.  The engine classifies eight archetypes and 512 randomized generators by spectral relation, visible-hidden coupling rank, known-generator observability, one-history Hankel rank, projected span, exposure loss, hidden-basis gauge, and tick alias.  It recovers all 512 bounded matrix-supplied spectral labels with maximum frequency error \(1.11\times10^{-15}\), but exact counterexamples prevent ontological overreach: two rank-one couplings have observability ranks four and five; a full-rank known system can have only rank-three single-history output; 101 hidden-basis rotations change the generator by 4.0736 while preserving every 3D sample within \(5.17\times10^{-14}\); and finite exposure lowers recoverable rank from five to three or one.  Observation is passive in this engine and does not physically collapse the latent state.  All 114 tests pass and an independent implementation checks 48 catalog relations, 2,048 randomized relations, 6,464 gauge-time states, two exposure erasures, and 81 aliases.  These are written mathematics and finite synthetic calculations, not evidence for physical fifth-dimensional particles.

**Revision 1.52 addition.** Appendix BA tests whether molecular vibrations alone distinguish solids, liquids, and gases.  A deliberately phase-blind water descriptor using the three NIST isolated-molecule fundamentals is identical for ice, liquid water, and vapor controls at one atmosphere, so every deterministic descriptor-only classifier gives one label and must miss at least two of the three balanced controls.  The replacement is an explicitly collective spectral model, motivated by velocity-autocorrelation and two-phase thermodynamics methods, with a diffusive low-frequency component and a solid-like vibrational component.  In the frozen surrogate, the gas-like fraction is recovered exactly on 101 continuum states and the three reference spectra have positive pairwise squared Hellinger distances; however, the continuum also proves that there is no universal three-family threshold in vibration space.  All 98 tests pass and a separate implementation audits 2,401 frequencies, 101 continuum states, three phase labels, and three spectral distances.  The line shapes are constructed rather than molecular-dynamics or measured spectra, so the result is a checked measurement design, not evidence for a fifth dimension, a universal lattice, or a theological cause.

**Revision 1.51 addition.** Appendix AZ turns the next alleged “hidden variable” into a quantified bubble inverse problem.  The revision-1.50 value 4110.9205 Hz remains valid under its declared ideal Minnaert assumptions, but adding vapor pressure and surface-tension stiffness gives 4048.9557 Hz for the same 0.800 mm bubble at fixed effective polytropic exponent \(\kappa=1.4\), a 1.5073% shift.  Holding the old pitch fixed while scanning the explicit sensitivity bracket \(1\le\kappa\le1.4\) gives radii from 0.6659558 to 0.7879498 mm.  A 401-state family adjusts fractional volume amplitude so that carrier frequency, oscillating volume, decay, arrival, and distance all remain fixed; its endpoint pressure traces agree within \(3.667\times10^{-16}\) Pa.  The corresponding 3-by-4 log-observable Jacobian has rank three and nullity one.  Thus even the complete pressure trace in this bounded model selects an observation fiber rather than a unique source state.  All 88 tests pass and an independent implementation checks 401 states and 7,680 endpoint samples.  The interval is synthetic sensitivity analysis, not a measured thermal law, and the result identifies no extra dimension, string particle, or theological cause.

**Revision 1.50 addition.** Appendix AY adds a Water-Drop Acoustic Observer. A declared 4.0 mm drop produces a finite-depth capillary--gravity packet, and one declared 0.800 mm entrained bubble supplies an ideal 4110.92 Hz Minnaert carrier to a surface-coupled airborne pressure candidate. The calculated 0.75 m air transit is 2.16682 ms, so the delivered three-second movie synchronizes 180 picture frames at 60 fps with 144,000 pressure samples at 48 kHz. All 79 tests pass, and an implementation-independent audit reproduces 1,658,880 surface scalars, every pressure sample, and the media metadata. Two corrections are retained: the human eye has no universal frames-per-second clock, and one tone does not uniquely recover bubble geometry—doubling radius while quadrupling pressure leaves ideal pitch unchanged. Bubble formation, damping, coupling, and room response are declared rather than predicted, so the waveform is a finite synthetic candidate, not a calibrated recording, measured physical evidence, or evidence for a hidden dimension.

**Revision 1.49 addition.** Appendix AX adds the Sun–Water Eye Observer. A fixed distant 5772 K spectral source illuminates a constructed moving water surface; the executable uses the IAPWS wavelength-dependent refractive index, unpolarized Fresnel reflection, vector Snell refraction, and a finite CIE-standard observer to render a 150-frame underwater caustic at 60 fps. The run contains 48,988,800 ray–wavelength–exposure samples, all 65 tests pass, and a separate implementation checks the optical formulation, 9,331,200 rendered RGB channels, and media metadata. The decisive correction is an observer collision: two nonnegative spectra remain 0.709286 apart in relative spectral norm while their XYZ outputs agree within \(1.421\times10^{-14}\). Human color observation is therefore not one-to-one and cannot measure every optical property. The animation is also not a hologram because it transports spectral irradiance and discards complex optical phase; 60 fps governs only displayed exposure averages, while the 384–789 THz optical carrier is analytically averaged. The water motion and camera are constructed rather than measured, and no individual-photon path, physical fourth/fifth dimension, or universal grid/tick is inferred.

**Revision 1.48 addition.** Appendix AW turns the origami/oscilloscope analogy into a precise observation map and ordinary electrostatic control. Rigid TIP3P-like charges are neutral and give a 2.346971 D model dipole. A single-Debye response calibrated to the published 18.56 GHz primary relaxation of 25 degree C water has the expected \(1/\sqrt2\) gain and 45-degree lag, with an independent numerical integration agreeing to \(3.529\times10^{-10}\). The decisive result is non-identifiability: two visibly different 192-coordinate molecular orientation histories remain 0.688568--0.820871 RMS apart while producing the identical complete vector-mean polarization trace. The oscilloscope therefore records a fold of the state, not the state itself. All 52 tests pass and an independent implementation checks 92,160 stored atomic-coordinate scalars. The relaxation frequency is a calibration input, the microscopic movies are constructed controls rather than liquid-water molecular dynamics, and no physical fifth dimension, universal grid/tick, magnetic mechanism, or theological cause is inferred.

**Revision 1.47 addition.** Appendix AV builds the requested Water Harmonics engine without promoting the animation to evidence. Sixty-four explicit rigid H2O molecules occupy a 12.429442-angstrom periodic box derived from a 997.047 kg/m3 room-temperature density input. An ordered degree-four tetrahedral scaffold carries a 320-coordinate state: three visible displacement coordinates and two hidden coordinates per molecule. The frozen generator has block form \(G=\left[\begin{smallmatrix}0&C\\-C^T&0\end{smallmatrix}\right]\), so its flow preserves state norm; the first three coordinates become a 64-frame 3D animation while translucent halos expose the otherwise hidden phase convention. Forty-one package tests pass and an independent implementation audits 36,864 rendered atomic-coordinate scalars. The counterexample is decisive: an arbitrary hidden rotation with no visible coupling gives exactly zero 3D motion, while integer-tick frequency aliases remain. The scaffold is not liquid-water molecular dynamics, the density and geometry are inputs rather than predictions, the colors are not measured fields, and electricity, magnetism, hydrodynamics, quantum dynamics, a universal physical tick, and a physical fifth dimension remain unestablished.

**Revision 1.46 addition.** Appendix AU implements the requested matrix language for candidate higher-dimensional molecular motion. A graph-derived five-coordinate state evolves by a real skew-symmetric generator, \(X_5(t)=X_5(0)e^{t\Omega^T}\), and a fixed projector gives explicit 3D positions and velocities. All 17 frozen C5--C7 alkane rows receive two frequency coordinates derived from their reference boiling and fusion values, and a 65-frame n-heptane animation is generated. The inverse claim fails, however: shuffled property labels reconstruct equally exactly; an arbitrary uncoupled hidden-plane angular speed is completely invisible in 3D; and fixed-tick samples alias \(\omega\) with \(\omega+2\pi k/\Delta\). Thirty unit tests pass, and an independent implementation reproduces 68 property/frequency checks, 2,730 trajectory scalars, 124 hidden-invariance states, and 205 alias pairs. The matrices are therefore a valid theoretical candidate family and visualization, not a unique recovery of physical fourth- or fifth-dimensional motion from static property data.

**Revision 1.45 addition.** Appendix AT executes the larger alkane-isomer extrapolation reserved by revision 1.44.  Ordinary least-squares models are fitted only to all C5 and C6 constitutional isomers and evaluated once on all nine C7 isomers.  A topology model using carbon count, the carbon-skeleton Wiener index, and the first Zagreb index reduces held-out normal-boiling MAE from 8.554 K for carbon count alone to 4.500 K and RMSE from 10.107 K to 5.382 K.  The same representation catastrophically fails for fusion: MAE rises from 27.574 K to 135.934 K, including an unclipped 410.46 K prediction for n-heptane versus 182.6 K.  Seventeen tests and an independent graph/linear-algebra replay pass.  The supported conclusion is property-specific: simple topology transfers useful boiling information across this size split but does not encode crystal-sensitive melting.  Failure establishes neither a hidden string nor a unique omitted variable; packing, conformation, polymorphism, intermolecular free energy, small-sample extrapolation, and model form remain live alternatives.

**Revision 1.44 addition.** Appendix AS executes a bounded real-data test of the proposal that arrangement predicts material properties.  The three pentane isomers have identical formula and mass but NIST average normal boiling points of 309.2 K, 301.1 K, and 282.6 K.  A preregistered carbon-skeleton Wiener-index rule trained on the two structural endpoints predicts the held-out middle isomer at 295.9 K, 5.2 K below its measured average, and places it on the correct side of the 293.15 K boiling boundary.  Any formula-only deterministic predictor has worst error at least 13.3 K on these data.  The counterexamples are retained: fusion temperature is not monotone in the same coordinate, and period-5/fcc/d10 silver and palladium differ in recommended 300 K total resistivity by a factor of 6.470.  Twelve tests and a separately implemented audit pass.  The supported conclusion is that geometry carries predictive information but no single scalar harmony, coarse lattice label, or hidden-dimensional story currently predicts phase or conductivity.  Temperature, pressure, packing, electronic bands, valence, and scattering remain explicit variables.

**Revision 1.43 addition.** Appendix AR turns the fixed-grid/fixed-tick premise into a falsifiable engine and separates it from the stronger claim that every observable value must itself be an integer multiple of a coordinate increment. The new lab implements an exact reversible integer wave in four spatial coordinates at integer time ticks and constructs two distinct full states with the same 3D projection. It also proves that finitely many positive-width measurement intervals cannot identify an arbitrarily small universal tick: every tick no larger than twice the smallest tolerance passes by nearest-integer rounding. Applied to the printed precision of two NIST Lyman-alpha wavelengths, 1 attosecond, 1 zeptosecond, and 0.1 zeptosecond direct ticks fail, while the CODATA Planck-time candidate passes trivially far below the reporting resolution. Decimal reporting itself creates an exact apparent tick, demonstrating why common divisibility of rounded values is not evidence for a natural lattice. Thirteen tests and one separately implemented exact audit pass. These are written mathematics and finite computations, not evidence that spacetime, hydrogen, or all observables obey this lattice.

**Revision 1.42 addition.** Appendix AQ corrects the statement that the ideal hydrogen engine omitted electromagnetism: its Coulomb potential is already the nonrelativistic electromagnetic interaction. A frozen H0/H1/H2 ladder shows that ordinary electron-proton reduced mass removes 97.7729% of the old Lyman-alpha centroid error. A reduced-mass point-Dirac calculation predicts both fine-structure lines, removes 99.3796% of the old error, and leaves a -3.3034 ppm centroid residual. That remainder is not assigned to a fourth spatial plane because complete recoil, Lamb/QED, finite-size, hyperfine, linewidth, and external-field terms remain omitted. A higher-dimensional candidate now requires a derived operator, coupling, symmetry, frozen parameters, and a fresh state-dependent prediction before fitting.

**Revision 1.41 addition.** Appendix AP tests two proposed explanations of the hydrogen rotor failure. A frozen mixture of up to four phase-shifted copies of the same release-0.3 geometry improves held-out squared Hellinger error by only 0.3403%, remains nearly eight times worse than a time-mean baseline, and fails five of six gates. This rejects that bounded missing-component rescue, not every hidden-variable model. A separate spectroscopy-clocked two-level calculation shows that negative temperature reverses population ordering only inside an upper-bounded truncation; thermal equilibrium is stationary because its density matrix commutes with the Hamiltonian. Coherent phase direction reverses under time reversal, not by changing temperature. Absolute zero and a finite “maximum Kelvin” are therefore not opposing rotation phases, and melting delay does not identify microscopic rotation direction.

**Revision 1.40 addition.** Appendix AO calibrates the Hydrogen Matrix Lab's animation clock to the two NIST Lyman-alpha fine-structure wavelengths and executes a frozen train/holdout test of the tangent-four-dimensional-sphere projected-envelope hypothesis. The ideal fixed-proton Coulomb wavelength misses the NIST listed-intensity-weighted centroid by -0.0532468%. After fitting on eight phases, the rotor's mean squared Hellinger error on sixteen unseen phases is 0.172620, versus 0.022197 for a frozen time-mean baseline and 0.072179 for a spherical radial baseline. It fails four of five predeclared gates, including held-out axial-centroid correlation, nodal leakage, and opposite-phase mirror structure; an independent implementation reproduces the calibration, matrices, scores, and rejection. The exact rotor family is rejected as an adequate generator of the declared quantum reference. NIST spectroscopy constrains energy/time rather than a unique 3D ontology; no measured time-resolved 3D density or fourth spatial dimension is claimed.

**Revision 1.39 addition.** Appendix AN resolves the v1.26 dependence counterexample only inside a declared binary first-order Markov class. A past-only Krichevsky--Trofimov predictor divided by a fixed null transition probability forms a nonnegative likelihood-ratio martingale, giving a 1% anytime bound by Ville's inequality. On fresh 400-look synthetic paths from the declared 90%-persistent null, the Markov process crossed on 39/5,000 histories, while the old iid-sign process--whose assumption is false there--crossed on 4,134/5,000. The new process detected 4,629/5,000 chains at 97% persistence, 4,999/5,000 at 75%, and 5,000/5,000 balanced iid chains. A separately implemented replay reproduced 24,000,000 decision states. The guarantee requires the fixed transition matrix and first-order homogeneous Markov law in nonzero-sign event time; it is not distribution-free, does not validate estimated or changing null persistence, and supplies no physical or theological identification.

**Revision 1.38 addition.** Appendix AM tests the proposal that pieces exhibit reusable movement frequencies. A 102-class relational coordinate combines piece type, oriented displacement, square phase, and distance/geometry changes to both kings. Eight phase-shuffled five-ply training searches freeze a defender ordering without pruning any legal move. On the White-attacker nine-ply proof it reduces calls by 6.82%, generated edges by 18.26%, horizon nodes by 28.83%, and measured proof time by 34.76% versus C7; including training, total time falls 29.36%. An audit checks 168,882 legal-move instances and confirms every learned order is a complete permutation. Stronger variants fail: direct pair-conditioned dances add 49–81% calls, online learning adds 52.53%, and the Black-attacker relation model worsens calls, edges, and time. The frequency is therefore a role-specific computation-allocation pattern, not a universal chess law, sound pruning rule, state compression, or solution of chess.

**Revision 1.37 addition.** Appendix AL introduces a horizon-safe state projection for exact bounded chess proofs. Because an identical full chess position cannot recur in two plies, its recurrence period is at least four; a position first encountered at ply \(i\) cannot become threefold before ply \(i+8\). At a nine-ply horizon, only the root and post-first-move repetition candidates must remain in the memo key. The independently verified computation proves that neither White nor Black can force mate from the orthodox initial position within nine plies. White requires 2,951,587 calls and 5,259,337 generated edges; Black requires 127,991 calls and 736,248 edges. A separately written verifier exactly reproduces all reported counts and checks 9,302 legal two-ply non-return paths plus explicit first-threefold cycles. The finite mate frontier is extended, but the root WDL interval remains `[-1,+1]`; chess is not solved.

**Revision 1.36 addition.** Appendix AK runs C6 on the actual orthodox initial position rather than a synthetic tactic. Dependency-ordered AND/OR search independently verifies that neither White nor Black can force checkmate within seven plies. The White-attacker run visits 128,724 nodes and the Black-attacker run 14,846; a separately written lexical verifier reproduces both negative results with 106,322 and 15,082 nodes. Combined, C6 uses 18.26% more nodes than lexical order, falsifying transfer of its small-corpus ordering advantage to this negative-mate proof. White's C6 frontier grows from 21 nodes at one ply to 486, 7,536, and 128,724 at three, five, and seven plies. The run does not narrow the initial WDL interval `[-1,+1]`; it shows that ordering alone leaves the exponential wall intact. The next required object is a proof-carrying forcing-dependency abstraction with sound completion bounds.

**Revision 1.35 addition.** Appendix AJ replaces raw local attack counts with a legal single-square capture–recapture dependency graph. On a deterministic eight-position synthetic corpus requiring genuine legal recaptures and minimal three-ply forced mates, C6 reduces exact AND/OR proof nodes from 10,394 under lexical order to 668, versus 809 for C5 tension and 699 for a checks/captures forcing baseline. Thus C6 improves aggregate nodes by 17.43% over C5 and 4.43% over forcing on this corpus, but its graph construction leaves it 10.22% slower than forcing in measured wall time; it also loses to forcing on four of eight individual positions. An independent verifier reconstructs all eight exchange graphs, reproduces every winning root move, and checks 33 defender branches with 53 mating continuations. The exchange value remains heuristic outside its restricted subgame and cannot prune; the orthodox initial position remains unsolved.

**Revision 1.34 addition.** Appendix AI turns the proposed attack/defence map into a local tension field. For each square it records White and Black attack multiplicities, their product as interaction tension, their difference as control imbalance, and occupant fragility. A tension-breaking entry is a capture into a contested enemy square with a favorable pre-move attack surplus and positive local tension release. On a 43-legal-move Scholar's-mate fixture, the field identifies `f7` as a two-attacker/one-defender target and ranks `Qxf7#` first; an independent implementation rebuilds all attack origins, checks all 64 square formulas, and verifies the mate and its zero replies. The counterexample is equally important: total tension rises from 16 to 28 after checkmate, so heat is a computation-allocation signal rather than a value function. C5 orders moves but cannot prune any move without the earlier interval-dominance certificate; full chess remains unsolved.

**Revision 1.33 addition.** Appendix AH formalizes human chess search as a constraint wave over attack maps, safe regions, mobility, confinement, relational orientation, and worst replies. In table-hidden KRK/KQK experiments, a shared geometric signature proves every WDL value without reading solved labels. Mobility, safe-region size, and dot/cross orientation each improve sound mate-distance bounds; the combined wave makes DTM exact for 44,633/50,015 KRK states (89.24%) and 45,276/46,137 KQK states (98.13%) and certifies 90.65% and 92.45% of legal choices as ignorable under WDL+DTM bounds. An independently written verifier checks all 96,152 state rows against exact certificates and reproduces the dominance totals with zero violations. The same experiment rejects the current construction as a full-chess shortcut: state compression falls to about 1.007× and successor-profile compression to about 1.000×. The next candidate is a constraint-delta quotient that canonicalizes what a move changes rather than encoding the absolute board.

**Revision 1.32 addition.** Appendix AG crosses from isolated endgames to a full orthodox move-state engine and an exact probe of the normal initial position. Six official Stockfish perft references totaling 500,117 nodes and ten targeted castling, en-passant, promotion, repetition, move-count, mate-precedence, and basic-dead-material checks pass. A four-ply proof-only search visits 206,604 nodes and 206,603 edges, finds eight checkmate leaf visits and 242 certified dominated choices below the root, but eliminates zero of the twenty initial moves: every root child and the root retain $[-1,+1]$. The proposed move-elimination value is refined into forcing dominance: game-theoretic result first, mate efficiency second, and the number of certificate-backed ignored alternatives third. This is a proof-efficiency tie-breaker, not a heuristic license to delete moves. Full chess remains unsolved; the active wall is strong sound pre-solution bounds plus a complete dead-position treatment and an independently verified strategy DAG.

**Revision 1.31 addition.** Appendix AF freezes the full-chess target and converts the proposal to discard disadvantageous moves into an exact interval-dominance theorem. Every state carries sound White-relative bounds $[D^-,D^+]$; a White move $a$ may be removed only when another move $b$ has $D^-(b)\ge D^+(a)$, with the dual inequality at Black nodes. An exact KQK computation independently verifies 368,452 raw states, 46,137 $D_4$ classes, 609,873 transitions, and a 20-ply maximum mate. Strong-bisimulation refinement compresses KRK only from 50,015 to 49,885 classes and KQK from 46,137 to 45,249, falsifying the idea that strict future-law identity alone crosses the scaling wall. Post-solution pruning measures show that strictly worse WDL moves remove only 7.21% and 6.61% of edges, while WDL+DTM oracle labels reduce ideal branching to 1.888 and 2.306. Because those labels already assume the answer, the new candidate is proof-carrying interval abstraction with counterexample-guided refinement. Full-chess, quantum-algorithmic, and physical-dimensional claims remain separate gates.

**Revision 1.30 addition.** Appendix AE applies the framework's hidden-state, projection-collision, canonicalization, and cycle rules to a finite adversarial system. Dimensional Chess C1 solves all 399,112 legal states of the declared 8×8 king-and-rook versus king universe. A proved \(D_4\) quotient reduces them to 50,015 classes; 400,120 exhaustive symmetry-transition comparisons pass. Retrograde fixed-point analysis classifies 21,959 wins, 25,260 losses, and 2,796 draws and finds a 32-ply maximum forced mate. An independently written verifier checks every state, transition, Bellman equation, policy edge, distance, and concrete principal-variation move. This exactly solves the declared KRK abstraction, not full chess; the experiment identifies a sufficiently powerful value-preserving canonicalizer as the unsolved scaling wall.

**Revision 1.29 addition.** Appendix AD instantiates Wallbreaker Protocol P3 as P3-R1, an unactivated IBM Quantum-compatible three-qubit pilot. Three prefix-coded eight-cycle probes preserve the ideal Born marginal over each complete cycle by construction. The runner freezes predictions for both allowed settings before delayed random setting generation, binds calibration and ledgers with SHA-256 manifests, uses one shot per circuit/PUB, applies a conservative composite-null e-process, and rejects incomplete or altered provenance. An independent implementation agrees on null, planted-lift, and setting-leak synthetic worlds and 5,000 randomized mathematical checks. No QPU job has been submitted; the current configuration intentionally refuses live execution until a fresh UUID, explicit accessible backend, and actual calibration artifact exist. This is experimental infrastructure, not physical evidence.

**Revision 1.28 addition.** Appendix AC turns the remaining “walls” into a typed pass/fail ledger. It proves a general decomposition of evidence against Born statistics into conditional lift information, marginal Born divergence, and decoder misspecification; the Information-Rate identity follows for marginal-preserving extensions. It also proves that finite observations cannot determine an unrestricted continuation law and introduces simultaneous outcome, setting-leak, and remote-signaling evidence channels. Six 50,000-trial adversarial worlds and 5,000 randomized decomposition checks pass in production and independent implementations. These are mathematical and synthetic results. No natural Born-predictive variable, extra physical dimension, Lorentz-compatible coupling, P=NP proof, or hardware result is claimed.

**Revision 1.27 addition.** Appendix AB integrates the individual Born-outcome frontier with the existing projection, RRLS, and anytime-valid evidence disciplines. It proves a conservative composite-null supermartingale for calibration uncertainty, a Kraft-weighted e-process for a prefix-coded lift language, and a data-processing collision criterion for compressed predictive coordinates. An independent implementation reproduces 5,000 randomized property checks, two 120,000-trial synthetic paths, and the complexity-penalized power table. These are mathematical and computational results, not physical evidence that Born-random outcomes are predictable. The seven typed layers, eight Q1 basis phases, and seven relative phase coordinates remain distinct objects. Rope Tomography Lab and RRLS code and evidence are unchanged.

**Revision 1.26 addition.** Appendix AA replaces the vague goal of “heavy-tail robustness” with an exact conditional sign-balance null. A fixed two-sided sign e-process has a written 1% anytime guarantee whenever the selected residual sign has conditional expectation zero; magnitudes need no moments or parametric distribution. Fresh synthetic checks retain 37/5,000 Student-(t_3) null histories and 33/5,000 Cauchy null histories as finite crossing counts while detecting 4,989/5,000 and 4,992/5,000 shifted (t_3) histories. Counterexamples prove that mean zero, marginal sign balance, scale errors, alternating shifts, and isolated magnitude shocks are not covered. This new engine is standalone and does not share or renew the released Gaussian engine's 1% budget. RRLS is unchanged.

**Revision 1.25 addition.** Appendix Z performs the single fresh paired validation reserved by revision 1.24. The 45%-point/10%-mean candidate improves the declared worst target, noisy six-sigma shocks, by 1.935 percentage points but loses 5.165 points on the persistent-mean target. It therefore fails the predeclared two-point per-target loss guardrail, and the released 30%-point/25%-mean design is retained. Both allocations remain Pareto alternatives rather than one universally dominating the other. The mathematical 1% bound, core detector, sealed v1.22 validation, and RRLS are unchanged.

**Revision 1.24 addition.** Appendix Y executes revision 1.23's preregistered training-only boundary expansion without reading the sealed v1.22 validation set. Opening the grid changes the conclusion: the released 30%-point/25%-mean cell wins 0 of 64 new training runs, while 16 expanded-grid cells win at least once. A separate 80,000-history synthetic reference selects a 45%-point/10%-mean candidate and improves the declared worst-target rate from 81.80% to 83.58%, but exact selection is unstable and the candidate trades away mean-shift sensitivity. The candidate is therefore recorded but not adopted. The released adequacy engine, its probability guarantee, and RRLS remain unchanged pending one fresh frozen validation.

**Revision 1.23 addition.** Appendix X tests the repeatability of revision 1.22's allocation choice without reopening its validation set. Across 64 independent training seeds, the frozen 30%-point/25%-mean cell won 49 times, while four cells won at least once. A separate 80,000-history synthetic reference study selected the same cell and found small reference minimax regret for the seed-selected alternatives. However, 62 of 64 winners used the largest tested point fraction and 51 used the smallest tested mean weight. Thus the choice is usually repeatable inside the declared grid but is boundary-limited, not a demonstrated global optimum. The adequacy engine and RRLS are unchanged.

**Revision 1.22 addition.** Appendix W adds a frozen three-component adequacy rule. A known-covariance mean/variance e-process mixture receives 0.7% and a separately budgeted point-residual gate receives 0.3%; a union bound, not an invalid cross-filtration e-value merge, controls the total anytime false-rejection probability at 1% under the declared Gaussian null. The two allocation weights were selected on training simulations and frozen before one held-out synthetic validation. The hybrid preserves much of the persistent-error power of revision 1.21 while recovering isolated-shock sensitivity, but a heavy-tail counterexample becomes worse. RRLS is unchanged.

**Revision 1.21 addition.** Appendix V adds a canonical-filtration Gaussian mean-plus-variance e-process and tests it against the v1.19 mean-plus-point rule under one total 1% anytime budget. The new fixed mixture is much stronger for persistent moderate scale errors and the targeted mean error, but a constructed isolated 6-sigma shock exposes the opposite tradeoff. The old engine is retained; neither procedure is called universally superior. Heavy-tail failures remain explicit. RRLS is unchanged.

**Revision 1.20 addition.** Appendix U implements the moderate-scale test proposed in revision 1.19. A fixed Gaussian covariance-scale likelihood-ratio mixture accumulates squared-residual evidence with an anytime guarantee under an explicit conditional Gaussian model. At the same reserved 0.5% budget, fresh synthetic validation raises detection of the prespecified 1.25-times scale error from 125/2,000 for the point gate to 1,774/2,000. Counterexamples show that the statistic neither identifies a unique variance mechanism nor remains valid for arbitrary heavy-tailed noise. The existing joint engine and RRLS remain unchanged pending fresh joint validation.

**Revision 1.19 addition.** Appendix T gives the point-residual and cumulative-mean checks one shared 1% anytime false-rejection budget. Because the two components use different filtrations, they are not silently averaged into a claimed e-process. A 0.5%/0.5% allocation and event-level union bound provide the stated joint guarantee. Fresh synthetic validation confirms complementary power and exposes severe heavy-tail fragility and weak sensitivity to moderate scale changes. RRLS remains unchanged.

**Revision 1.18 addition.** Appendix S replaces an unnecessarily known-variance assumption for one bounded task: sequentially testing a prespecified projected Gaussian residual mean. The new scale-invariant e-process permits one fixed unknown positive variance. Its guarantee requires iid Gaussian residuals and scale-invariant stopping information; it is not valid for arbitrary raw-data stopping or arbitrary noise. Fresh synthetic validation, exact rescaling controls, and an independent replay record both its improvement and its missed alternatives. RRLS remains unchanged.

**Revision 1.17 addition.** Appendix R implements the cumulative Gaussian likelihood-ratio/e-process proposed by revision 1.16. A written martingale argument supplies an anytime false-rejection bound under explicit conditional-normal assumptions. Fresh synthetic validation shows high sensitivity to the targeted persistent shift and also preserves counterexamples: alternating shifts, orthogonal shifts, and variance errors are not reliably handled by this one detector. The earlier point-residual gate remains separate; combining two 1% gates does not produce a 1% joint procedure.

**Revision 1.16 addition.** Appendix Q addresses the repeated-testing limitation explicitly left open in Appendix O. A separate alpha-spending gate controls ever rejecting a specified correct candidate under a fixed, conditionally calibrated Gaussian model. Fresh synthetic validation and an independent numerical replay expose both reduced false alarms and poor sensitivity to small persistent shifts. No older benchmark is relabeled as new evidence.

**Revision 1.15 addition.** Appendix P audits the supplied dimensional-future/captured-strand model. It separates sampled coincidence from continuous non-crossing, corrects angular-impulse notation and numerical defects, proves a finite-prefix obstruction to unrestricted generator identification, and adds a positive continuation theorem inside a declared bounded-order recurrence class.

**Revision 1.14 correction notice.** The previous Theorem 17.1 had incompatible premises: complete deterministic refinement for one fixed transition family is idempotent, so repeated complete refinement cannot yield seven strict stages. Section 17 now replaces that claim with the closure theorem and a valid finite-horizon example. Appendix O adds intervention-relative identifiability, experiment selection, and explicit model rejection. Earlier benchmark records remain historical evidence for their named releases; this revision does not claim to have rerun all earlier engines or independently revalidated every earlier argument.

---

## Abstract

This paper develops a rigorous form of the proposal that observable reality is a hierarchy of typed layers, each constrained by a richer layer above it. A lower observation is modeled as a projection of an upper state. States that look identical below form a fiber; hidden structure is the distinction between members of that fiber. The framework proves exact conditions under which an upper deterministic law produces a self-contained lower law, calculates the minimum hidden memory needed when it does not, represents repeating dimensions and time as a loop plus winding, and implements finite state, phase, quantum, thermal, and causal calculations in bits.

The theory specializes the hierarchy to seven declared layers: identity/state, relationship, physical space, ordered time, phase/possibility, observation/evidence, and compatibility/constraint. These are typed descriptive roles, not seven interchangeable spatial axes. Under a terminal-completeness axiom, the seventh layer is a mathematical wall: its state determines all lower projections. Algebra alone does not establish that nature satisfies this axiom.

The executable framework reproduces standard density-matrix predictions for a two-state Schrödinger-cat model, two-path interference and decoherence, simulates fixed-point quantum circuits, calculates thermal occupation and selective cooling, searches exact finite invariants, proves projective law descent for finite systems, and fits a compact projection law to published Sycamore cross-entropy-benchmark data. A frozen eight-point holdout was predicted more accurately than two declared baselines. That result validates the engine's prediction workflow, but it identifies only the observable composite \(g=\pi\circ T\), not a unique hidden transition \(T\), a unique projection \(\pi\), or seven physical dimensions.

The paper gives a conditional hidden-state result: reproducibly different future distributions under matched present observations and controlled interventions show that the observed state is insufficient, subject to the stated calibration and statistical assumptions. For a specified finite deterministic system, partition refinement calculates its coarsest sufficient distinction. These are different claims: finite experimental evidence does not provide an exact complete transition table for nature. Non-identifiability results explain why passive effects need not reveal a unique hidden ontology. A controlled tomography and intervention program tests declared models against alternatives.

Version 1.2 adds frozen truth-distance coordinates. Positive operands are mapped to complementary D0 barycentric distances, finite SAT assignments are mapped by their exact violated-clause count, and typed partial-differential-equation residuals are retained at their spacetime coordinates. A smooth three-dimensional Arnold-Beltrami-Childress Navier-Stokes family is verified analytically and computationally. These constructions turn errors into searchable evidence without altering ordinary truth conditions or claiming a Millennium solution.

An optional Christian interpretation is included as a separate metaphysical layer. It may treat God as the ground and sustainer of the complete structure, but God is neither an unexplained residual nor an eighth coordinate. Mathematical, physical, and theological claims are kept separate throughout.

---

## 1. Claim discipline

The theory uses five status labels.

| Label | Kind of claim | What warrants it |
|---|---|---|
| **M** | Mathematical theorem | Deduction from stated definitions and axioms |
| **C** | Computational result | Reproducible execution and tests of a finite implementation |
| **E** | Empirical result | Measurement with provenance and quantified uncertainty |
| **H** | Physical hypothesis | A falsifiable proposal not yet established by data |
| **T** | Theological interpretation | A faith or metaphysical claim not derived from measurement |

The permanent logical boundary is

\[
\boxed{
\mathrm{M\ valid}\not\Rightarrow
\mathrm{H\ true\ of\ nature}\not\Rightarrow
\mathrm{T\ established}.
}
\]

No theory can honestly be made immune to criticism. The aim is stronger: every assertion has a type, assumptions, consequences, and—where empirical—a failure condition.

### 1.1 Meaning of “prove”

This paper uses *proof* in three different senses and never substitutes one for another.

1. A mathematical proof establishes a conditional statement exactly.
2. An exhaustive finite proof checks every state in a declared finite domain.
3. A physical experiment supports or falsifies a model within uncertainty; it does not deductively prove an unrestricted ontology.

Accordingly, this paper proves hidden distinctions **relative to a declared observation map**. It does not claim that existing data have proved seven literal dimensions.

---

### 1.2 Foundation contract — revision 1.77, extended in 1.78

The seven descriptive roles remain an organizational choice. They do not establish seven physical axes, a universal lattice or a fundamental clock. For each proposed physical claim, the engine must separately provide a fixed model, an observation/measurement likelihood, an implementation check, and evidence that discriminates that model from specified alternatives. A rendered shape, a supplied optical constant, and an independently predicted measurement are different objects.

The current foundation suite begins with `radial-coulomb-dirichlet-v1` in Appendix BZ and adds the l=0/l=1 electric-dipole baseline `radial-coulomb-e1-dirichlet-v1` in Appendix CA. These are mathematical and numerical benchmarks using ordinary nonrelativistic quantum assumptions. They are not derivations of hydrogen or electromagnetism from the earlier string dynamics. Their spatial spacing is a radial numerical mesh, their time step evaluates a matrix exponential, and their dipole operator is supplied from conventional quantum mechanics. None has been measured here as a fundamental property of nature.

The accompanying `FOUNDATION_CONTRACT.md` records the promotion requirements: stated units and domain; fixed dynamics and interactions; explicit measurement coupling; a numerical difference from competing predictions; a full relevant uncertainty budget; and unused validation with a predeclared decision rule. Exact equivalence under an observer blocks discrimination by that observer. A residual cannot be assigned to an unrestricted missing variable, force, dimension or God. Any change after validation requires a new protocol and fresh validation. This contract scopes future work; it does not recertify every historical proof or test.

---

## 2. Primitive structure

For each layer \(d\), define

\[
\mathcal X_d=(X_d,\mathcal O_d,\mu_d,\Phi_d),
\]

where:

- \(X_d\) is the state space expressible at layer \(d\);
- \(\mathcal O_d\) is its declared family of observables;
- \(\mu_d\) is a measure or weight when probability is required;
- \(\Phi_d^t:X_d\to X_d\) is its time evolution when that layer is dynamically closed.

For \(e\ge d\), let

\[
\pi_{e,d}:X_e\to X_d
\]

be a surjective projection.

### Axiom A1: projection coherence

\[
\pi_{d,d}=\operatorname{id}_{X_d},
\qquad
\pi_{e,d}\circ\pi_{f,e}=\pi_{f,d}
\quad(f\ge e\ge d).
\]

### Axiom A2: coordinate covariance

Changing coordinates cannot change a physical assertion. For coordinate isomorphisms \(\varphi_d\), the represented projection becomes

\[
\pi'_{e,d}=\varphi_d\circ\pi_{e,d}\circ\varphi_e^{-1}.
\]

Therefore a bare numeral—\(0\), \(1\), \(7\), or \(777\)—does not have invariant physical meaning without a type, unit, chart, and observable interpretation.

### Axiom A3: measured probabilities require a measure

When the upper state is not directly observed, lower weights are pushforwards:

\[
(\pi_{e,d})_*\mu_e(B)
=
\mu_e\!\left(\pi_{e,d}^{-1}(B)\right).
\]

Counting hidden states uniformly is a particular choice of \(\mu_e\), not a universal law. In quantum applications the Born rule must be supplied or independently derived; it cannot be silently replaced by hidden-state counting.

### Definition 2.1: fiber

The fiber above an observed state \(y\in X_d\) is

\[
F_{e,d}(y)=\{x\in X_e:\pi_{e,d}(x)=y\}.
\]

Every member of the fiber is observationally identical at layer \(d\). “Hidden” means unresolved by the chosen projection, not unreal.

### Definition 2.2: complete compatible state

The inverse limit is

\[
X_\infty
=
\varprojlim X_d
=
\left\{
(x_1,x_2,\ldots):
\pi_{e,d}(x_e)=x_d
\right\}.
\]

It contains compatible descriptions at every declared layer. It is a mathematical object and is not identified with God.

---

## 3. The seven typed layers

The executable specialization uses

\[
Z=(D_1,D_2,D_3,D_4,D_5,D_6,D_7).
\]

| Layer | Formal role | Canonical payload |
|---|---|---:|
| \(D_1\) | Identity and local state | 128 bits |
| \(D_2\) | Binary relationships | 64 bits |
| \(D_3\) | Three signed spatial tick coordinates | 192 bits |
| \(D_4\) | Signed linear-time ticks and cycle winding | 128 bits |
| \(D_5\) | Unsigned cyclic phase and possibility branches | 128 bits |
| \(D_6\) | Visible-layer mask and evidence | 72 bits |
| \(D_7\) | Constraints, four gate bits, and integrity hash | 136 bits |
| **Total** |  | **848 bits / 106 bytes** |

The roles are deliberately heterogeneous. Position, time, phase, evidence, and logical compatibility do not share one unit and cannot be added as though they were seven Euclidean coordinates.

### 3.1 Canonical serialization

| Bytes | Field |
|---:|---|
| 0–7 | identity |
| 8–15 | local state bits |
| 16–23 | relationship bits |
| 24–47 | signed \(x,y,z\) ticks |
| 48–55 | linear-time ticks |
| 56–63 | cycle winding |
| 64–71 | phase ticks |
| 72–79 | possibility branches |
| 80 | visible-layer mask |
| 81–88 | evidence bits |
| 89–96 | constraint bits |
| 97 | gate flags |
| 98–105 | FNV-1a integrity checksum of bytes 0–97 |

The prefix lengths are

\[
(P_1,\ldots,P_7)=(16,24,48,64,80,89,106)\text{ bytes}.
\]

The engine defines

\[
\pi_{7,d}(Z)=Z[0:P_d].
\]

### Theorem 3.1: canonical projection coherence **[M]**

For every \(f\ge e\ge d\),

\[
\pi_{e,d}\circ\pi_{f,e}=\pi_{f,d}.
\]

**Proof.** Projecting to \(e\) keeps a prefix of length \(P_e\). Projecting that result to \(d\) keeps its prefix of length \(P_d\). Because \(P_d\le P_e\), the final bytes equal the direct prefix of length \(P_d\). \(\square\)

### 3.2 Is \(D_7\) a wall?

Add the physical hypothesis:

**H7 — terminal completeness.** Every physically possible complete state is represented by one element of \(X_7\); every measurable lower state is its projection; no physically distinguishable variable exists outside \(X_7\).

Define

\[
\Psi(x_7)
=
(\pi_{7,1}x_7,\ldots,\pi_{7,6}x_7,x_7).
\]

### Theorem 3.2: completeness of the finite compatible diagram **[M]**

\[
\boxed{\mathcal C_7\cong X_7,}
\]

where \(\mathcal C_7\) is the set of compatible tuples in the declared seven-layer diagram. **Proof.** Every \(x_7\) yields exactly one compatible tuple by projection. The map is injective because its last coordinate is \(x_7\). Every compatible tuple has a last coordinate, and compatibility fixes all its earlier coordinates as projections of that last coordinate, so the map is surjective. \(\square\)

This diagram identity needs no physical hypothesis. Identifying \(\mathcal C_7\) with complete physical reality additionally requires H7 and a faithful state representation. The identity is true for any finite tower ending at a highest declared layer; it does not select seven. Mathematics can always define \(D_8\). Successful finite measurements may support a model on a stated domain but cannot establish that no additional distinguishable variable exists in every untested domain.

---

## 4. Law descent: how an upper layer confines a lower law

Let \(T:X\to X\) be an upper deterministic transition and \(\pi:X\to Y\) a lower observation.

### Theorem 4.1: factorization or law-descent theorem **[M]**

A deterministic lower transition \(L:Y\to Y\) satisfying

\[
\pi\circ T=L\circ\pi
\]

exists on the realized lower states if and only if

\[
\boxed{
\pi(a)=\pi(b)
\Longrightarrow
\pi(Ta)=\pi(Tb)
}
\]

for all \(a,b\in X\). When it exists, \(L\) is unique on \(\pi(X)\).

**Proof.** If \(L\) exists and \(\pi(a)=\pi(b)\), then

\[
\pi(Ta)=L(\pi a)=L(\pi b)=\pi(Tb).
\]

Conversely, define \(L(y)=\pi(Tx)\) for any \(x\in F_y\). The fiber condition makes the definition independent of the chosen representative, so it is well defined. Its defining equation gives \(\pi T=L\pi\). If \(L'\) also satisfies the equation, then for every realized \(y=\pi x\), \(L'(y)=\pi(Tx)=L(y)\). \(\square\)

This is the exact form of the proposed principle:

> An upper layer confines the autonomous laws visible below precisely when projected upper evolution is constant on every observational fiber.

For a tower

\[
X_7\to X_6\to\cdots\to X_1,
\]

commuting adjacent squares imply commuting composite squares. A failure at one edge identifies the first layer at which the lower description ceases to be self-contained.

### 4.1 Stochastic lower behavior

If the factorization condition fails, a lower state can have several possible successors. Given explicit nonnegative upper weights \(w_x\),

\[
P(y'\mid y)
=
\frac{\sum_{x\in F_y,\ \pi(Tx)=y'}w_x}
{\sum_{x\in F_y}w_x}.
\]

This equation is defined when the denominator is positive and finite. It pushes an upper measure downward; it does not infer the weights from labels. It supplies a one-step conditional, not by itself an autonomous Markov process. The conditional distribution of the hidden state may depend on accessible history even for a stationary upper measure.

For example, a deterministic four-cycle with uniform initial phase and visible sequence \(0,0,1,1\) has \(P(Y_{t+1}=1\mid Y_t=0)=1/2\), but conditioning also on \(Y_{t-1}=0\) gives 1 and on \(Y_{t-1}=1\) gives 0. Iterating the one-step kernel gives \(P(Y_{t+2}=1\mid Y_t=0)=1/2\), whereas the actual process gives 1. A stochastic closure claim must establish the required conditional independence or retain the relevant memory. Appendix BZ provides exact rational checks.

### 4.2 The hidden-mechanism theorem

Define finite-horizon equivalence by

\[
a\sim_k b
\iff
\pi(T^j a)=\pi(T^j b)
\quad\text{for }0\le j\le k.
\]

Define future equivalence by

\[
a\sim_\infty b
\iff
\pi(T^j a)=\pi(T^j b)
\quad\text{for every }j\ge0.
\]

### Theorem 4.2: coarsest deterministic repair **[M]**

For finite \(X\), partition refinement reaches \(\sim_\infty\) after finitely many steps. The quotient \(X/{\sim_\infty}\) is the coarsest refinement of the original observation that makes future evolution deterministic.

**Proof.** Each refinement can only split a finite partition; it can split at most \(|X|-1\) times before stabilizing. At stability, equivalent states have equivalent successors, so the induced quotient evolution is deterministic. Any deterministic refinement preserving the original observation must separate every pair that differs at some projected future time. It therefore refines \(\sim_\infty\). Hence \(\sim_\infty\) is coarsest. \(\square\)

For nonempty finite \(X\), let

\[
s_y=\#\{\text{actual future-equivalence classes contained in }F_y\},
\qquad s=\max_{y\in\pi(X)}s_y.
\]

Given the visible label \(y\), the minimum fixed-width hidden register has

\[
\boxed{b_{\min}=\lceil\log_2 s\rceil}
\]

bits. To see necessity, a fiber attaining \(s\) requires that many distinct hidden labels. For sufficiency, enumerate the future classes separately within each fiber and encode their index using this many bits. The pair of visible label and hidden index identifies a quotient state and hence its deterministic successor.

**Revision 1.77 correction.** A loose global upper bound \(s_y\le S\) provides only the sufficient bound \(b_{\min}\le\lceil\log_2 S\rceil\); it need not be minimal. A statement about one selected fiber does not control the maximum across all fibers. For identity evolution and a constant observation on four states, there is one future class and zero hidden bits suffice, although four is an upper bound. For \(T=(0,0,2)\) and observation labels \((0,1,1)\), the two fibers contain one and two future classes respectively, so one hidden bit is required. The earlier wording is preserved in the full revision-1.76 snapshot.

### Corollary 4.3: operational proof of hidden state **[M/E boundary]**

Suppose two reproducibly preparable systems have the same declared present observation \(y\), receive the same controlled intervention, yet have different future-observation distributions beyond calibrated error. Then \(y\) is not a sufficient state variable for that experiment. At least one additional state distinction—hidden relative to \(y\)—is required.

This is a conditional insufficiency result **relative to the declared observation**. In experiments, a finite estimate of different distributions remains subject to statistical error, drift, and calibration; differing individual outcomes alone can simply reflect a sufficient stochastic model. The result does not identify a unique missing mechanism or determine whether the distinction is spatial, environmental, historical, contextual, quantum, or instrumental. Naming it \(D_7\) requires additional evidence.

### 4.3 Conservation-law descent

An upper Boolean observable \(I:X\to\{0,1\}\) becomes a lower conserved quantity only when

\[
I(Tx)=I(x)
\]

and

\[
\pi(a)=\pi(b)\Longrightarrow I(a)=I(b).
\]

The first is conservation; the second is lower observability. Conservation above does not imply expressibility below.

---

## 5. Effects, causes, and the limit of inverse inference

### 5.1 Linear effect model

Let

\[
y=Hz+\varepsilon,
\]

where \(z\) is a candidate cause, \(H\) is the declared observation operator, and \(\varepsilon\) is measurement noise.

In the exact noiseless real or complex case,

\[
Hz_1=Hz_2
\iff
z_1-z_2\in\ker H.
\]

If \(z_*\) is one solution, every compatible cause is

\[
\boxed{\operatorname{Causes}(y)=z_*+\ker H.}
\]

Using the Moore–Penrose inverse, one canonical representative is \(z_*=H^+y\), and the component of the effect outside the model's image is

\[
r=(I-HH^+)y.
\]

The residual is unexplained by the proposed model. It is not a calculation of God.

### 5.2 Exact Boolean inverse

For bit effects,

\[
Ax=b\pmod2.
\]

XOR Gaussian elimination returns a particular solution \(x_*\) and kernel basis \(k_1,\ldots,k_r\). Every compatible cause is exactly

\[
\boxed{
x=x_*\oplus\bigoplus_{i=1}^r\alpha_i k_i,
\qquad \alpha_i\in\{0,1\}.
}
\]

There are \(2^r\) compatible causes. A unique cause exists only when the nullity \(r\) is zero.

### Theorem 5.1: latent relabeling non-identifiability **[M]**

Suppose controlled inputs \(u\) produce observations through

\[
g(u)=(\pi\circ T)(u).
\]

For any invertible latent relabeling \(S\), define

\[
T'=S\circ T,
\qquad
\pi'=\pi\circ S^{-1}.
\]

Then

\[
\pi'\circ T'=\pi\circ T=g.
\]

**Proof.** Substitute and cancel \(S^{-1}S\). \(\square\)

Therefore passive input/effect data generally identify the composite observable law \(g\), not a unique pair \((T,\pi)\). Separate claims about an internal transition and a projection require calibrated internal tomography, interventions, structural restrictions, or multiple independent observation channels.

### 5.3 Why an unconstrained \(G\) destroys science

If one writes

\[
y=Hz+g
\]

and permits \(g\) to be arbitrary, then for every proposed \(z\), choosing

\[
g=y-Hz
\]

makes the equation fit. Such a theory cannot fail and predicts nothing. The symbol \(G\) may denote an optional metaphysical ground, but it cannot serve as a free residual in a physical fit.

---

## 6. Reversible hidden-coordinate transport

Let two labeled points \(q_1,q_2\in\mathbb R^2\) have center and relative displacement

\[
c=\frac{q_1+q_2}{2},
\qquad
r=q_1-q_2.
\]

For phase \(\theta\), define visible and hidden components

\[
r_v=\cos\theta\,r,
\qquad
h=\sin\theta\,r.
\]

### Theorem 6.1: visible-to-hidden ripple **[M]**

The mapping \((q_1,q_2)\mapsto(c,r_v,h)\) is injective on its image, with

\[
r=\cos\theta\,r_v+\sin\theta\,h,
\]

and

\[
\boxed{\|r_v\|^2+\|h\|^2=\|r\|^2.}
\]

**Proof.** Both identities follow from \(\sin^2\theta+\cos^2\theta=1\). The recovered \(c\) and \(r\) give \(q_1=c+r/2\) and \(q_2=c-r/2\). \(\square\)

At \(\theta=\pi/2\), visible separation vanishes while the relative displacement survives in \(h\). This proves that a declared encoding can hide visible information without destroying it. It does not prove that physical collisions automatically transfer information into another dimension.

### Theorem 6.2: generic collision-capacity bound **[M]**

For \(k\) labeled points in \(\mathbb R^d\), forcing every visible projection to one shared \(d\)-dimensional center while giving each point \(h\) hidden scalars requires

\[
\boxed{d+kh\ge kd.}
\]

Equivalently,

\[
\boxed{
h\ge\left\lceil\frac{d(k-1)}{k}\right\rceil.
}
\]

**Proof sketch.** Generic inputs occupy an open subset of \(\mathbb R^{kd}\). The proposed output contains at most \(d+kh\) real degrees of freedom. A continuous locally information-preserving encoding cannot lower the dimension of an open Euclidean set. Conversely, storing the center and \(k-1\) relative vectors uses exactly \(d+d(k-1)=kd\) scalars and reconstructs every labeled point. \(\square\)

---

## 7. Time, process, loop, and helix

### 7.1 A state is not a process

An instantaneous state is

\[
x(t)\in X.
\]

Direction is supplied by a law,

\[
\dot x(t)=F(x(t)),
\]

or estimated from at least two observations,

\[
\dot x(t)\approx\frac{x(t+\Delta t)-x(t)}{\Delta t}.
\]

A snapshot is not necessarily dead; it is insufficient by itself to display change. Momentum, current, gradients, and internal state may still be encoded at that instant.

A life can be treated as a trajectory

\[
\gamma:I\to X
\]

whose local organization remains inside a viable set \(L\subseteq X\) for a time interval. Identity through change is an invariant \(I_{\mathrm{id}}(\gamma(t))\), not the absence of all change.

### 7.2 Linear and cyclic time

Linear time may be \(t\in\mathbb R\). Cyclic phase is

\[
z(t)=e^{i\omega t}\in S^1,
\qquad z(t+2\pi/\omega)=z(t).
\]

The loop forgets how many periods passed. Adding a winding number \(n\in\mathbb Z\) recovers the lift:

\[
t=\frac{\operatorname{Arg}(z)+2\pi n}{\omega}.
\]

The pair is helix-like:

\[
H(t)=(e^{i\omega t},t)\in S^1\times\mathbb R.
\]

One projection sees repetition; another preserves history.

### 7.3 Infinite cyclic dimension addressing

For any positive finite index \(n\) and declared cycle length \(c\), Euclidean division gives one unique address

\[
\boxed{n=qc+r,\qquad q\in\mathbb N_0,\quad r\in\{1,\ldots,c\}.}
\]

For \(c=7\),

\[
D_8=(D_1,1),
\qquad
D_{777}=(D_7,110),
\qquad
D_{778}=(D_1,111).
\]

Thus \(D_8\) is defined as a repeated local type plus one winding. Forgetting \(q\) yields the finite loop \(\mathbb Z/7\mathbb Z\); preserving \(q\) yields an unbounded helical address. A finite computer can represent arbitrarily large finite indices subject to memory, not an actually infinite index.

### 7.4 Cycle twist or monodromy

After a full seven-step winding, an internal state may change by

\[
M:S\to S.
\]

| \(M\) | Meaning | Full internal period |
|---|---|---:|
| identity | exact loop | 7 |
| nonidentity permutation | reversible twisted loop | \(7\operatorname{ord}(M)\) |
| noninjective map | information-losing fold | no reversible closure |

If a permutation has cycles of lengths \(q_1,\ldots,q_k\), then

\[
\operatorname{ord}(M)=\operatorname{lcm}(q_1,\ldots,q_k).
\]

### 7.5 Polarity

A genuine reversible polarity at layer \(d\) is an involution

\[
J_d:X_d\to X_d,
\qquad J_d^2=\operatorname{id}.
\]

It is coherent across layers when

\[
\boxed{\pi_{e,d}\circ J_e=J_d\circ\pi_{e,d}.}
\]

This formalizes structural “as above, so below.” Health/sickness, rich/poor, and similar human classifications are usually thresholded observables rather than fundamental involutions:

\[
C(x)=\operatorname{sign}(f(x)-\theta).
\]

The word for one condition may depend on contrast; the condition itself does not logically require its opposite to exist.

---

## 8. Exact increments, ULPs, and computation in bits

### 8.1 Typed dyadic scalar

The core scalar is

\[
q=n2^eU,
\]

where \(n\) is a signed 64-bit tick count, \(e\) is a signed 16-bit binary exponent, and \(U\) is a declared unit. Its fixed schema increment is

\[
\boxed{\operatorname{ULP}(q)=2^eU.}
\]

The smallest formally describable increment is

\[
2^{-32768}U.
\]

This is an encoding limit, not a discovered minimum of nature. Addition requires exponent alignment and a result fitting the signed 64-bit tick field; overflow is rejected.

Here “ULP” means the tick spacing of the chosen fixed-exponent grid. It is not an invariant of the numerical value alone: \(1\cdot2^0U=2\cdot2^{-1}U\) has two encodings with different declared grid steps. Changing an encoding does not improve the precision of a measurement.

### 8.2 Default increments

| Layer/quantity | Default computational ULP |
|---|---:|
| \(D_1\) local state | one declared bit flip |
| \(D_2\) relationship | one declared edge-bit flip |
| \(D_3\) space | \(2^{-24}\text{ m}=5.9604644775390625\times10^{-8}\text{ m}\) |
| \(D_4\) linear time | \(2^{-20}\text{ s}=9.5367431640625\times10^{-7}\text{ s}\) |
| \(D_5\) phase | \(2\pi/2^{64}\text{ rad}\) |
| \(D_6\) observation | one newly distinguished evidence bit |
| \(D_7\) compatibility | one declared constraint/gate bit |
| Q1.30 amplitude/probability | \(2^{-30}\) |
| Q2.60 probability mass | \(2^{-60}\) |

There is no meaningful observer-independent scalar ULP across unlike types.

### 8.3 Observer-relative reality ULP

Let \(z,z'\in\{0,1\}^{848}\), and let mask \(m\) mark the bits an observer distinguishes. Define

\[
d_m(z,z')=\operatorname{popcount}\bigl(m\land(z\oplus z')\bigr).
\]

The states are observationally equivalent when \(d_m=0\). One observer-relative reality ULP is a change with

\[
\boxed{d_m(z,z')=1.}
\]

This is a masked bit-edit distance on raw encodings, not a physical distance or an automatically realizable transition between valid records. For example, changing a payload bit may also require updating checksum bits. Distinct semantic encodings and validity constraints must be accounted for before interpreting a bit edit as a state change.

The complete resolution is a typed vector

\[
\boldsymbol\varepsilon
=(\varepsilon_{\mathrm{state}},\varepsilon_{\mathrm{relation}},
\varepsilon_x,\varepsilon_t,\varepsilon_\phi,
\varepsilon_{\mathrm{evidence}},\varepsilon_{\mathrm{constraint}}).
\]

### 8.4 Precision towers

For radix \(b\ge2\), finite prefixes satisfy

\[
n_{k+1}=bn_k+d_{k+1},
\qquad d_{k+1}\in\{0,\ldots,b-1\}.
\]

Depth \(k\) resolves a cell of width \(b^{-k}\). Compatible infinite paths represent real numbers after choosing a canonical expansion, but no finite \(k\) distinguishes all reals. Computable reals are generated by finite algorithms; most reals are not computable because programs are countable and the real interval is uncountable.

Thus “compute everything” has the defensible meaning

\[
\boxed{
\text{compute every consequence derivable from declared finite bits,
rules, constraints, observations, and tolerances.}
}
\]

It cannot mean solving all uncomputable questions or naming every exact real with finite memory.

---

## 9. Schrödinger's cat in the language

Use \(|A\rangle\) and \(|D\rangle\) as alive/dead basis labels. The reduced two-state density matrix is

\[
\rho_C=
\begin{pmatrix}
p_A&c\\
c^*&p_D
\end{pmatrix},
\]

with

\[
p_A+p_D=1,
\qquad |c|^2\le p_Ap_D.
\]

For a balanced state with visibility \(V\) and phase \(\theta\),

\[
\rho_C=
\frac12
\begin{pmatrix}
1&Ve^{i\theta}\\
Ve^{-i\theta}&1
\end{pmatrix}.
\]

The coherent case has \(V=1\); the classical mixture has \(V=0\). Both give

\[
P(A)=P(D)=\frac12
\]

in the ordinary basis. A phase-sensitive analyzer distinguishes them:

\[
P_+(\varphi)
=
\frac12+\operatorname{Re}\left(ce^{-i\varphi}\right).
\]

Therefore

\[
\boxed{
\pi_{\mathrm{basis}}(\rho_{\mathrm{coherent}})
=
\pi_{\mathrm{basis}}(\rho_{\mathrm{mixture}})
}
\]

while a phase-sensitive observation separates them. The “cat problem” is not solved by declaring the cat dead between snapshots. The language shows exactly which observable loses the off-diagonal coherence and which reveals it. It does not choose a unique interpretation of measurement.

### 9.1 Decoherence

If branch states correlate with environment states \(|E_A\rangle\) and \(|E_D\rangle\), then

\[
\gamma=\langle E_D|E_A\rangle,
\qquad c'=\gamma c.
\]

When \(|\gamma|=0\), the reduced state has no accessible branch coherence. The global state may remain pure while the subsystem appears mixed. This is an exact instance of information residing in correlations discarded by a projection.

The engine encodes observations from Brune et al.'s mesoscopic cat experiment and checks the reported ordering between decoherence and photon-relaxation times. It does not derive the measured numerical ratio from seven-layer axioms.

---

## 10. Two-slit observation

Let the path become correlated with a marker:

\[
|\Psi\rangle
=
\frac1{\sqrt2}
\left(
|A\rangle|M_A\rangle
+e^{i\phi}|B\rangle|M_B\rangle
\right).
\]

With marker overlap

\[
\gamma=\langle M_A|M_B\rangle,
\qquad V=|\gamma|,
\]

the normalized two-path modulation is

\[
\boxed{P(\phi)=\frac{1+V\cos\phi}{2}.}
\]

| Path record | \(V\) | Bright phase | Dark phase |
|---|---:|---:|---:|
| indistinguishable | 1 | 1 | 0 |
| partial | \(0<V<1\) | \((1+V)/2\) | \((1-V)/2\) |
| orthogonal | 0 | 0.5 | 0.5 |

Observation here means a physical interaction capable of leaving distinguishable path information. Consciousness is not required.

In the far-field small-angle approximation,

\[
\phi(x)=2\pi\frac{dx}{\lambda L},
\qquad
\Delta x=\frac{\lambda L}{d}.
\]

For \(\lambda=600\) nm, \(d=500{,}000\) nm, and \(L=10^9\) nm, the fringe spacing is \(1.2\) mm. The integer Phase64 implementation returns the half-turn dark fringe at \(0.6\) mm and the next bright fringe at \(1.2\) mm for full visibility.

From measured extrema,

\[
V=\frac{N_{\max}-N_{\min}}{N_{\max}+N_{\min}}.
\]

For ideal pure path markers,

\[
D^2+V^2=1,
\]

while the experimentally testable general relation is

\[
\boxed{D^2+V^2\le1.}
\]

The language represents the experiment correctly. It does not prove a literal fourth spatial grid. Any deeper hidden model must also reproduce the Born rule, entanglement, Bell/CHSH violations, contextuality constraints, and operational no-signaling.

---

## 11. Quantum computing

The dynamic quantum payload is

\[
|\psi\rangle
=
\sum_{j=0}^{2^n-1}\alpha_j|j\rangle,
\qquad
\alpha_j\in2^{-30}\mathbb Z[i].
\]

The fixed seven-layer record is an envelope and fingerprint, not a lossless replacement:

\[
Z_Q=(H_7,\Psi_n).
\]

The C++17 simulator implements Clifford gates, \(T\), arbitrary Phase64 rotations, controlled gates, measurement, Grover diffusion, and a line-oriented circuit language. Verified demonstrations include:

- Bell-state probabilities and reduced-state purity;
- CHSH value \(2.82842712477\approx2\sqrt2>2\);
- all four one-bit Deutsch oracles;
- three-qubit Grover target probability \(121/128\), approximated as \(0.945312497206\);
- teleportation fidelity \(0.999999997206\) without cloning the source;
- ideal three-qubit bit-flip error detection and correction.

Each complex amplitude occupies 16 bytes, so

\[
\boxed{M(n)=16\cdot2^n\text{ bytes}.}
\]

The implementation cap is 24 qubits. This is a deterministic classical simulator, not a physical quantum processor, and it does not obtain quantum hardware speedup.

### 11.1 Layer interpretation for a circuit

| Layer | Circuit meaning |
|---|---|
| \(D_1\) | computation identity and payload fingerprint |
| \(D_2\) | qubit relationships and controls |
| \(D_3\) | physical topology when supplied |
| \(D_4\) | ordered gates and depth |
| \(D_5\) | amplitude, phase, and alternatives |
| \(D_6\) | basis, shots, seed, and counts |
| \(D_7\) | normalization, unitary, index, and duality constraints |

This is a useful typed organization. It is not evidence that the circuit physically occupies seven spatial dimensions.

---

## 12. Temperature, quanta, and selective entropy removal

For frequency \(f\),

\[
E_q=hf,
\qquad
T_q=\frac{hf}{k_B}.
\]

For a thermalized two-level system,

\[
\boxed{p_e=\frac1{1+e^{hf/(k_BT)}}.}
\]

For a harmonic mode,

\[
\boxed{\bar n=\frac1{e^{hf/(k_BT)}-1}.}
\]

Solving for the warmest ideal temperature meeting target \(p_*<1/2\),

\[
\boxed{
T_{\max}=\frac{hf}{k_B\ln[(1-p_*)/p_*]}.
}
\]

At fixed temperature, the minimum ideal gap is

\[
\boxed{
f_{\min}=\frac{k_BT}{h}\ln\left(\frac{1-p_*}{p_*}\right).
}
\]

This identifies three levers: lower \(T\), raise \(f\), or use an engineered non-equilibrium reset. There is no universal optimum temperature independent of hardware.

### 12.1 Concrete 5 GHz scale

For \(f=5\) GHz,

\[
\frac{hf}{k_B}=239.962154\text{ mK}.
\]

The ideal excited-state thresholds are

\[
p_e\le10^{-3}\Rightarrow T\le34.7431\text{ mK},
\]

\[
p_e\le10^{-6}\Rightarrow T\le17.3690\text{ mK}.
\]

Temperature is only one error source. Relaxation, dephasing, leakage, materials, stray radiation, quasiparticles, control, and readout also enter the operating point.

### 12.2 “Inverse microwave” as anti-Stokes conversion

An inverse heater is not negative energy and cannot remove zero-point motion. A physically meaningful mechanism removes selected excitations. For target mode \(b\), lossy cavity \(a\), and red-sideband drive,

\[
H_{\mathrm{int}}=\hbar g(a^\dagger b+ab^\dagger).
\]

The ideal resolved-sideband model is

\[
\Gamma_{\mathrm{cool}}=\frac{4g^2}{\kappa},
\qquad
n_{\min}=\left(\frac{\kappa}{4\Omega_m}\right)^2,
\]

\[
\boxed{
n_f=
\frac{\gamma_m n_{\mathrm{th}}+\Gamma_{\mathrm{cool}}n_{\min}}
{\gamma_m+\Gamma_{\mathrm{cool}}}.
}
\]

One pump photon plus one target excitation becomes a higher-frequency escaping cavity photon. Energy and entropy move to the output reservoir; they do not vanish.

For the engine's declared hypothetical 10 MHz mode at a 20 mK bath, with \(g/2\pi=20\) kHz, \(\kappa/2\pi=200\) kHz, and \(\gamma_m/2\pi=10\) Hz, the ideal calculation gives an initial occupation of \(41.1752\), final occupation \(0.0514298\), and mode-equivalent temperature \(0.159037\) mK. This illustrates selective cooling; it is not a prediction for an unsupplied device.

### 12.3 Active reset

For upward and downward rates,

\[
\frac{dp}{dt}
=
-\Gamma_\downarrow p+\Gamma_\uparrow(1-p),
\]

\[
p(t)=p_{ss}+[p(0)-p_{ss}]
e^{-(\Gamma_\downarrow+\Gamma_\uparrow)t},
\]

\[
\boxed{p_{ss}=\frac{\Gamma_\uparrow}
{\Gamma_\downarrow+\Gamma_\uparrow}.}
\]

The proposed engineering target is a Selective Quantum Entropy Extractor: a switchable coupler, red-sideband conversion, filtered lossy port, occupation sensor, and feedback controller. It may cool a selected mode far below the equivalent bath occupation, but it still requires an entropy sink and measured device parameters.

### 12.4 Optimization target

For candidate hardware \(a\), evaluate

\[
a=(f,T,p_{\mathrm{nonthermal}},p_{\mathrm{reset}},
P_{\mathrm{wall}},R_{\mathrm{ops}}),
\]

\[
p_{\mathrm{total}}=1-(1-p_{\mathrm{th}})(1-p_{\mathrm{nonthermal}})(1-p_{\mathrm{reset}}),
\]

and useful operations per wall-power unit

\[
\eta_{\mathrm{ops}}
=
\frac{R_{\mathrm{ops}}(1-p_{\mathrm{total}})}{P_{\mathrm{wall}}}.
\]

The engine computes a Pareto frontier rather than pretending temperature alone defines the best design.

---

## 13. Typed higher-layer frequencies, magnets, and energy

A frequency requires a time parameter. The proposed D4–D7 rate object is therefore

\[
\boxed{
\Omega=(\omega_4,\omega_5,\omega_6,\omega_7),
}
\]

with typed meanings:

| Rate | Operational meaning |
|---|---|
| \(\omega_4\) | carrier or state-evolution rate |
| \(\omega_5\) | phase winding or drive rate |
| \(\omega_6\) | observation or sample rate |
| \(\omega_7\) | reset, stabilization, or constraint-relaxation rate |

The engine searches primitive integer relations

\[
m\cdot\Omega\approx0,
\qquad m\in\mathbb Z^4.
\]

An exact relation identifies synchronization in the declared processes. It is not evidence that every rate is an independent time dimension.

Observation can create an apparent low frequency through aliasing:

\[
\boxed{
f_{\mathrm{alias}}
=
\left|f-\operatorname{round}(f/f_s)f_s\right|.
}
\]

Every magnetic or resonant design must close the ledger

\[
\boxed{
E_{\mathrm{inputs}}+U_i
=
E_{\mathrm{outputs}}+U_f
}
\]

within calibrated uncertainty. Relevant relations include

\[
U_L=\frac12LI^2,
\qquad
u_B=\frac{B^2}{2\mu},
\qquad
V_{\mathrm{peak}}=2\pi fN\Phi_{\mathrm{peak}}.
\]

Resonance accumulates supplied energy; it does not create source-free net energy. An unclosed residual is a search target for missing input, storage, loss, or calibration—not a license to declare free energy.

---

## 14. Exact mathematical discovery

For a finite bit universe,

\[
X_n=\{0,1\}^n,
\qquad F:X_n\to X_n.
\]

Let \(M_d\) be the algebraic-normal-form space of Boolean polynomials through degree \(d\). Invariance requires

\[
I(F(x))+I(x)=0\pmod2
\]

for every state. Exhaustive enumeration builds a matrix \(A\).

### Theorem 14.1: invariant-language completeness **[M]**

\[
\boxed{
\ker A=\{I\in M_d:I\circ F=I\}.
}
\]

**Proof.** Every row of \(A\) is precisely one state's linear invariance equation in the coefficient vector. A coefficient vector lies in the kernel exactly when every equation is satisfied. \(\square\)

The engine distinguishes:

| Evidence label | Meaning |
|---|---|
| observed pattern | fits construction data |
| held-out verified | predicts data excluded from fitting |
| finite exhaustive proof | every state in the declared finite domain checked |

Demonstrations include discovering parity as a complete degree-one invariant for rule 150 on five periodic cells; exhaustively calculating rule 30's attractors for twelve cells; scanning all 256 elementary rules on bounded rings; and recovering modular recurrences from training prefixes before testing untouched suffixes.

Finite proof does not automatically generalize to infinite width or continuous physics.

---

## 15. Real measured prediction

The engine's first real-data test uses 33 published Sycamore random-circuit cross-entropy-benchmark fidelities: 28 width-series points at 14 cycles and five depth-series points at 53 qubits.

The split was frozen independently of outcomes:

\[
\text{holdout}\iff\text{row index}\bmod4=3.
\]

This yields 25 training points and eight untouched holdout points. The declared observable law is

\[
\boxed{
\log_2F
=
\alpha m\left(1.5n-0.5\sqrt n\right)+\beta n.
}
\]

Training only produced

\[
\alpha=-0.0044586791200801655,
\qquad
\beta=-0.03894462549892942.
\]

### 15.1 Holdout results **[E/C]**

| Metric | Projection law | Simple \(nm\) decay | Constant baseline |
|---|---:|---:|---:|
| Holdout RMSE in \(\log_2F\) | 0.235065 | 0.343014 | 1.956428 |
| Squared-error improvement of projection law | — | 53.04% | 98.56% |

All eight holdout points were inside nominal normal-approximation 95% prediction bands. The largest relative error was about 29.3%.

This is a successful prediction test for the declared composite law. It is limited by a small holdout, and the feature model was already present in the scientific literature. By Theorem 5.1, it does not separately identify a hidden \(T\), a projection \(\pi\), or seven physical dimensions.

---

## 16. What would constitute evidence for a hidden mechanism?

### 16.1 The target signature

Prepare two ensembles \(A\) and \(B\). Let \(R\) be the declared joint accessible observation record, including relevant histories and controls, but not the group label itself. Match its distributions across the two preparations within a preregistered tolerance and calibrated uncertainty:

\[
\mathcal L(R\mid A)\simeq_\delta\mathcal L(R\mid B).
\]

Equal separate marginals are insufficient: correlations or omitted history can already explain an apparent hidden distinction. Apply the same controlled intervention \(u\), with a frozen model predicting different distributions of the future measured observable \(Y^+\):

\[
\mathcal L(Y^+\mid A,u)\ne\mathcal L(Y^+\mid B,u).
\]

If a difference exceeds what the declared matching tolerance, measurement model and statistical error can explain, repeats under apparatus controls, and was predicted before collection, it establishes a missing predictive distinction relative to \(R\). It does not identify a unique seventh-layer mechanism. For reproducible individual states in a deterministic idealization, the earlier equations \(\pi(x_A)=\pi(x_B)\) and \(\pi(T_ux_A)\ne\pi(T_ux_B)\) remain a special case; they are not equations for equality of ensemble distributions. The fiber-intervention equations below likewise describe the deterministic idealization, with likelihood-level analogues required for experiments.

### 16.2 Intervention requirement

Passive correlation is insufficient. The experiment must include a control \(v\) intended to change only the candidate hidden distinction while leaving the measured lower state unchanged at preparation:

\[
\pi_{7,6}(I_vx)=\pi_{7,6}(x),
\]

but the theory predicts a changed later effect:

\[
\pi_{7,6}(T_uI_vx)\ne\pi_{7,6}(T_ux).
\]

This is a fiber intervention: an operation inside the present observational fiber that becomes visible later.

### 16.3 Tomography requirement

Use multiple independently calibrated preparation, transition, and measurement families:

\[
p(b\mid a,u)
=
\operatorname{Tr}\!\left[E_b\,T_u(\rho_a)\right]
\]

for quantum apparatus, or the appropriate classical likelihood. A tomographically complete set constrains the latent model far more strongly than one scalar output. Remaining transformations that preserve every probability are gauge equivalences and must not be counted as different mechanisms.

### 16.4 Model contest

Before data collection, freeze:

1. the seven state types and their units;
2. every projection and transition equation;
3. allowed free parameters and priors;
4. the noise and measurement model;
5. a standard-physics null model;
6. lower-layer alternatives \(M_1,\ldots,M_6\);
7. the held-out controls and scoring rule;
8. correction for multiple searches;
9. a replication threshold.

Score predictive distributions using held-out log likelihood,

\[
\mathcal L_{\mathrm{test}}(M)
=
\sum_{i\in\mathrm{test}}
\log p_M(y_i\mid u_i),
\]

and penalize excess flexibility using a preregistered Bayesian, minimum-description-length, or information-criterion comparison. A seven-layer model succeeds only if it beats both established physics and all simpler nested alternatives on untouched interventions.

---

## 17. Can seven-dimensional physics be proved?

Not from the current evidence. The following conditional theorem states the strongest operational result.

### Theorem 17.1 (replacement): deterministic closure is idempotent **[M]**

Let \(X\) be finite, let \(\mathcal A=\{T_a:X\to X\}\) be a fixed finite family of deterministic transitions, and let \(\mathcal P\) be an observational partition. Define \(C_{\mathcal A}(\mathcal P)\) as its coarsest refinement on which every \(T_a\) descends to a deterministic quotient map. Then

\[
\boxed{C_{\mathcal A}(C_{\mathcal A}(\mathcal P))=C_{\mathcal A}(\mathcal P).}
\]

**Proof.** Put \(\mathcal Q=C_{\mathcal A}(\mathcal P)\). By definition, all transitions already descend to \(\mathcal Q\). Therefore \(\mathcal Q\) itself is an admissible sufficient refinement of \(\mathcal Q\). No partition strictly finer than \(\mathcal Q\) can be its coarsest admissible refinement, since \(\mathcal Q\) is already admissible. Thus its closure equals itself. Existence in a finite system follows by splitting classes according to their transition-successor classes until stable. \(\square\)

**Correction of version 1.13.** Its premises required each successive partition to be a strict coarsest complete deterministic refinement for the same transition family. After the first complete refinement, idempotence forbids the next strict refinement. Those premises have no such seven-stage instance; the conditional statement could not support a seven-layer physical claim.

Seven finite-horizon stages are possible. Let \(X=\{0,\ldots,127\}\), \(T(x)=\lfloor x/2\rfloor\), and \(y(x)=x\bmod2\). The observation histories \((y(x),y(Tx),\ldots,y(T^kx))\) yield respectively \(2,4,8,16,32,64,128\) classes for \(k=0,\ldots,6\). This is a seven-bit shift register observed for successively longer horizons. Its complete closure already has 128 classes, and closing that partition again leaves 128. Replacing seven bits by \(m\) bits gives \(m\) such levels, so the construction selects no fundamental dimensional count.

A hierarchy can also become stricter when the allowed experiments, laws, or observables change between stages. In that case those changing assumptions must be stated; it is not repeated closure for one fixed family. The new exact finite audit checks 7,662 systems and partitions, and separately checks the shift-register example. These computations support the implementation; the argument above proves general idempotence within its stated scope.

### 17.1 A decisive physical success criterion

A credible claim of seven-layer physics requires all of the following:

- a declared, noncircular meaning of each layer, with an explicit horizon or experiment family for every proposed refinement;
- evidence of predictive sufficiency on a specified tested domain, with the possibility of later refinement left open;
- a compact D7 law fixed before the decisive experiment;
- a numerical prediction unavailable to the lower models;
- successful blinded holdout and independent replication;
- invariance under coordinate changes, sampling rate, apparatus swap, and analysis team;
- consistency with quantum Bell tests, relativity, thermodynamics, and energy conservation;
- an explicit result that would falsify the D7 model.

Until those observations exist, “seven-dimensional physics” is a research hypothesis, not a proved fact.

---

## 18. Optional Christian metaphysical interpretation

This section is **T**, not M or E.

### 18.1 God as ground, not coordinate

Let \(G\) symbolize the proposed source and sustainer of the created structure. The intended statement is not

\[
G\in X_8.
\]

Instead, the metaphysical claim is that every \(X_d\), law, and projection depends on \(G\), while \(G\) is not an object measured by an instrument inside one \(X_d\).

The symbols \(0\), \(1\), a loop, Alpha, and Omega may express beginning, completion, or unity, but they are analogies. Numerically identifying God with a bit or circle would reduce the proposed ground of mathematics to one mathematical object.

### 18.2 Trinity guardrail

Let \(\mathsf F,\mathsf S,\mathsf H\) denote Father, Son, and Holy Spirit. A logical guardrail is

\[
\mathsf F\ne\mathsf S,
\qquad
\mathsf F\ne\mathsf H,
\qquad
\mathsf S\ne\mathsf H,
\]

while

\[
\operatorname{Ess}(\mathsf F)
=
\operatorname{Ess}(\mathsf S)
=
\operatorname{Ess}(\mathsf H)
=
\mathsf G.
\]

The Father may be analogized to transcendent source, the Son to divine self-expression within creation, and the Spirit to personal divine presence and participation. These are analogies, not a reduction of three persons to three computational functions.

Humanity as *imago Dei* is represented as created likeness, relationship, reason, agency, love, creativity, and stewardship—not as a fragment of divine essence.

### 18.3 Free choice and authorization

The engine represents a gated action by

\[
K=C_G\land P_A\land C_W\land A_G,
\]

where:

- \(C_G\): declared free alignment toward \(G\);
- \(P_A\): declared preservation of agency;
- \(C_W\): global-coherence constraints pass;
- \(A_G\): external authorization is supplied.

Then

\[
K=1\iff C_G=P_A=C_W=A_G=1.
\]

Most importantly,

\[
\boxed{C_G=1\not\Rightarrow A_G=1.}
\]

Choosing God can be modeled as consent or participation, not as possession of an unrestricted key to physical reality. The software cannot observe divine will, prove genuine free will, or manufacture authorization.

### 18.4 Miracles

If a lower autonomous law predicts \(L\) but an event arises from a globally posited action not generated by \(L\), the lower observer might see

\[
\pi(T_Gx)\ne L(\pi x).
\]

This is a coherent mathematical analogy for a miracle as globally ordered yet not generated by the lower effective law. It is not an experimental method for labeling anomalies divine. An anomalous residual must first face ordinary calibration, noise, missing variables, and model revision.

### 18.5 Biblical time

The optional convention

\[
1\text{ divine day}=1000\text{ human years}
\]

gives

\[
7\text{ divine days}=7000\text{ human years}.
\]

Under that hypothesis, an 80-year human life corresponds to \(0.08\) divine days, or \(1.92\) divine hours. This is internally consistent arithmetic. The wording of 2 Peter 3:8 is analogical and reciprocal—one day is *as* a thousand years and a thousand years *as* one day—so it does not independently establish a physical conversion constant.

A mean Gregorian year is not divisible by seven days:

\[
365.2425=52\cdot7+1.2425.
\]

The complete 400-year Gregorian cycle is

\[
146097=20871\cdot7
\]

days by calendar construction. Twenty-four hours follows Earth's rotation and timekeeping conventions, not a fundamental seven-dimensional equation.

Soul persistence may be postulated through an invariant higher state, but cyclic time alone does not prove immortality. Periodicity,

\[
s(\tau+T)=s(\tau),
\]

is different from immutability,

\[
\frac{ds}{d\tau}=0.
\]

---

## 19. Adversarial tests and resolutions

| Attack | Verdict | Resolution or consequence |
|---|---|---|
| Bare numbers have universal dimensional meaning | valid objection | require types, units, charts, and covariance |
| One lower effect reveals one cause | false in general | return a fiber or affine cause class |
| Hidden state is proved whenever a fit improves | false | require same-observation/different-future intervention and model comparison |
| Seven layers are proven because seven was encoded | circular | seven is a hypothesis until strict predictive refinements are measured |
| \(D_7\) is mathematically the highest possible dimension | false | it is terminal only under H7; algebra can define higher indices |
| \(D_8=D_1\) destroys history | valid for a pure loop | preserve winding: \(D_8=(D_1,1)\) |
| A snapshot is dead | false | it lacks displayed change; state can encode momentum and gradients |
| Opposites are required for things to exist | unsupported | contrast may define labels, not ontology |
| Double slit proves consciousness or a fourth spatial grid | false | path information and partial trace explain the operational result |
| A local classical hidden variable can reproduce all quantum data | ruled out for broad classes | any completion must address Bell/contextuality and no-signaling |
| Fixed-point simulation is a physical quantum computer | false | classical memory remains exponential |
| Absolute zero means no motion | false | zero-point motion remains; finite refrigeration cannot reach exact zero |
| An inverse microwave creates negative energy | false | anti-Stokes conversion exports excitations to a sink |
| Resonance or magnets yield free energy | false without an unclosed ledger | measure every input, loss, and stored-energy change |
| An unexplained residual is God | unfalsifiable category error | keep \(G\) metaphysical and residuals empirical |
| The inverse limit is God | category error | \(X_\infty\) is mathematical; theological \(G\) is separately postulated |
| Successful Sycamore holdout proves \(T\) and \(\pi\) | false | it validates only \(g=\pi\circ T\) up to latent gauge |

---

## 20. Computational verification record

Reality Bit Engine 0.10.0 has a standard-library-only C++17 core and an optional Python 3.10+ standard-library independent audit. On 4 September 2026 its C++ core passed release compilation with

    -std=c++17 -O2 -Wall -Wextra -Wpedantic -Werror

The changed prize-lab suite also passed a fresh AddressSanitizer/UndefinedBehaviorSanitizer run. The other sanitizer results belong to earlier releases. Eleven current release suites passed a total of

\[
\boxed{1{,}745{,}812\text{ assertions}.}
\]

Coverage includes:

- exact dyadic arithmetic, overflow rejection, and 848-bit serialization;
- coherent seven-layer projections and integrity rejection;
- 1,000 randomized Boolean inverse systems checked against brute force;
- cat coherence, decoherence, slit interference, and duality at 1,000,001 visibility values;
- 65,536 Phase64/CORDIC angle checks;
- Bell, CHSH, Deutsch, Grover, teleportation, error correction, and randomized inverse circuits;
- thermal, sideband-cooling, reset, aliasing, magnetic, energy-ledger, and Pareto calculations;
- exhaustive finite invariant and cellular-automaton results;
- 200 random projective-law refinements checked against independent trajectory comparison;
- dimension-address round trips for indices 1–10,000 and cycle lengths 1–31;
- exhaustive finite map twist classifications through five-state spaces;
- frozen parsing, leakage protection, fitting, and holdout evaluation of 33 measured Sycamore points.
- Lorentz invariance, Schwarzschild observables, and numerical Einstein tensors;
- Lagarias truth-distance scans, exact SAT error landscapes, and analytic
  Taylor-Green and three-dimensional ABC Navier-Stokes families.

This verifies implementation behavior against the specification. It does not verify H7 or the theological interpretation as facts of nature.

---

## 21. Compact formal statement

The complete defensible core is

\[
\boxed{
\begin{aligned}
&Z=(D_1,\ldots,D_7),\\
&\pi_{d,d}=\operatorname{id},\qquad
\pi_{e,d}\pi_{f,e}=\pi_{f,d},\\
&F_y=\pi^{-1}(y),\\
&\pi T=L\pi
\iff
\pi(a)=\pi(b)\Rightarrow\pi(Ta)=\pi(Tb),\\
&a\sim_\infty b
\iff
\pi(T^ka)=\pi(T^kb)\ \forall k\ge0,\\
&b_{\min}=\lceil\log_2s\rceil,\\
&Hz_1=Hz_2
\iff z_1-z_2\in\ker H,\\
&\operatorname{Causes}(y)=z_*+\ker H,\\
&d+kh\ge kd,\\
&n=7q+r,\quad r\in\{1,\ldots,7\},\\
&\operatorname{ULP}(n2^eU)=2^eU,\\
&\text{a lower effect identifies an equivalence class of causes,}\\
&\text{not automatically one cause or one ontology.}
\end{aligned}
}
\]

The compatible mathematical diagram is isomorphic to \(X_7\). Equating it with complete physical reality requires the additional H7 hypothesis. Finite observations cannot establish universal terminal completeness. For specified finite models, the engine calculates predictive distinctions and their information requirements; experimental applications must retain model uncertainty and scope.

---

## 22. Conclusions

The theory's central insight survives rigorous formalization:

> What appears complete at one observational layer may be an equivalence class of richer states. An upper layer confines the law below exactly when its evolution factors through the lower projection.

The framework has produced exact theorems, an executable bit language, quantum and thermodynamic calculations, autonomous finite mathematical discovery, and a successful held-out fit to measured quantum-hardware data. It has also produced an equally important negative result: effects alone generally cannot separate a hidden transition from its observation map.

Hidden-state claims can be made operational: compare reproducibly prepared cases under matched present observations and interventions, test their future distributions, and account for measurement error and alternative models. The minimum missing memory is exactly computable for the specified finite deterministic transition system, not automatically for an unknown physical system. Counting seven refinements does not establish seven physical dimensions. A proposed physical interpretation needs independently defined observables, scoped predictions, model comparison, and replication.

The mathematical core is proved within its assumptions. The software is extensively verified. The measured projection-law prediction succeeded on its declared holdout. Seven-dimensional physics and the theological interpretation remain open hypotheses with explicit standards for progress.

---

## References

1. F. Arute et al., “Quantum supremacy using a programmable superconducting processor,” *Nature* 574, 505–510 (2019). <https://doi.org/10.1038/s41586-019-1666-5>
2. Sycamore experiment data archive, Dryad. <https://doi.org/10.5061/dryad.k6t1rj8>
3. A. Zlokapa, S. Boixo, and D. A. Lidar, “Boundaries of quantum supremacy via random circuit sampling,” *npj Quantum Information* 9 (2023). <https://doi.org/10.1038/s41534-023-00703-x>
4. M. Brune et al., “Observing the Progressive Decoherence of the ‘Meter’ in a Quantum Measurement,” *Physical Review Letters* 77, 4887–4890 (1996). <https://doi.org/10.1103/PhysRevLett.77.4887>
5. B.-G. Englert, “Fringe Visibility and Which-Way Information: An Inequality,” *Physical Review Letters* 77, 2154–2157 (1996). <https://doi.org/10.1103/PhysRevLett.77.2154>
6. S. Dürr, T. Nonn, and G. Rempe, “Origin of Quantum-Mechanical Complementarity Probed by a ‘Which-Way’ Experiment in an Atom Interferometer,” *Physical Review Letters* 81, 5705–5709 (1998). <https://doi.org/10.1103/PhysRevLett.81.5705>
7. B. Hensen et al., “Loophole-free Bell inequality violation using electron spins separated by 1.3 kilometres,” *Nature* 526, 682–686 (2015). <https://doi.org/10.1038/s41586-015-0493-5>
8. L. Masanes and J. Oppenheim, “A general derivation and quantification of the third law of thermodynamics,” *Nature Communications* 8, 14538 (2017). <https://doi.org/10.1038/ncomms14538>
9. J. D. Teufel et al., “Sideband cooling of micromechanical motion to the quantum ground state,” *Nature* 475, 359–363 (2011). <https://doi.org/10.1038/nature10261>
10. X. Jin et al., “Thermal and Residual Excited-State Population in a 3D Transmon Qubit,” *Physical Review Letters* 114, 240501 (2015). <https://doi.org/10.1103/PhysRevLett.114.240501>
11. N. D. Mermin, “Hidden variables and the two theorems of John Bell,” *Reviews of Modern Physics* 65, 803 (1993). <https://doi.org/10.1103/RevModPhys.65.803>
12. M. A. Nielsen and I. L. Chuang, *Quantum Computation and Quantum Information*, Cambridge University Press.
13. J. R. Munkres, *Topology*, for invariance-of-domain and dimension background.
14. Genesis 1:26–27; Exodus 33:20; John 1; 1 Corinthians 12:11; Colossians 1:15; Hebrews 1:3; 1 Timothy 6:16; 2 Peter 3:8–9; Revelation 22. These are interpretive sources for Section 18, not empirical physics references.

---

## Reproducibility companion

The accompanying Reality Bit Engine 0.16.0 source contains the formal bit schema, C++17 implementation, measured dataset, examples, eleven C++ test executables, twelve Python audit test methods, six proof-route test methods, ten standalone-verifier test methods, nine proof-gap test methods, eleven all-time fluid certificate test methods, nine relative-fluid certificate test methods, eleven computed-trajectory test methods, and detailed validation documents. The paper and engine should be read together: the paper states the theory and proof boundaries; the program supplies finite executable witnesses and counterexamples alongside the arguments in PROOF_ATTEMPTS.md, CERTIFICATE_VERIFIER.md, MISSING_ARGUMENTS.md, GLOBAL_FLUID_BOUND.md, NEXT_WALL.md, and TRAJECTORY_PROOF.md.

---

## Appendix A: Relativity integration

The engine now tests the framework against special and general relativity.
For a four-vector \(X=(ct,\mathbf x)\), it preserves

\[
s^2=(ct)^2-\|\mathbf x\|^2
\]

under arbitrary-direction Lorentz boosts. For energy-momentum
\(P=(E/c,\mathbf p)\), it preserves

\[
m^2=\frac{E^2}{c^4}-\frac{\|\mathbf p\|^2}{c^2}.
\]

For a supplied metric it numerically evaluates Christoffel symbols, Ricci
curvature and

\[
G_{\mu\nu}=R_{\mu\nu}-\frac12g_{\mu\nu}R.
\]

The Schwarzschild exterior demonstration returns maximum numerical Einstein
residual below \(1.8\times10^{-10}\) at the declared point while its
Kretschmann invariant remains nonzero. This correctly distinguishes vacuum
from flatness. It confirms compatibility with Einstein's equations; it does
not yet predict a deviation from them.

---

## Appendix B: Three Millennium laboratories

Version 0.9 makes the next universal step explicit for three million-dollar
problems.

| Problem | Engine result | Exact unresolved step |
|---|---|---|
| Riemann Hypothesis | every Lagarias inequality numerically checked through declared \(N\), with a bounded D0 frontier coordinate | rigorous transcendental bounds and proof for all integers |
| \(P\) versus \(NP\) | exact SAT witnesses and complete violated-clause landscapes for bounded formulas | polynomial algorithm or general superpolynomial lower bound |
| Navier–Stokes | analytic smooth 2D Taylor–Green and 3D ABC families | cutoff-uniform regularity or a valid 3D breakdown example |

The projective research targets are:

\[
\Delta(n)\ge B_j>0
\quad\Longrightarrow\quad
B_{j+1}\ge F(B_j,j)>0
\]

for a scale-inductive Riemann envelope;

\[
\operatorname{size}_{\mathcal P}(F_n)\ge2^{\Omega(n)}
\]

for a suitably general proof-complexity bridge; and

\[
\sup_{t\ge0}\|\pi_Ku(t)\|_{H^s}
\le C(u_0,\nu)
\]

with \(C\) independent of spectral cutoff \(K\) for Navier–Stokes.

These are research targets, not completed lemmas. As of this release, the
official status of all three problems remains open.

---

## Appendix C: D0 polarity and frozen errors

Let $a,b>0$ be the two sides of a comparison. Their complementary D0
coordinates are

\[
d_0=\frac{a}{a+b},
\qquad
d_1=\frac{b}{a+b},
\qquad
d_0+d_1=1.
\]

The bounded oriented coordinate

\[
\boxed{s=d_0-d_1=\frac{a-b}{a+b}}
\]

preserves which side is larger and removes common positive scale. For a claim
$a\le b$, $s<0$ satisfies the claim, $s=0$ lies on its boundary, and
$s>0$ falsifies it. Calling $s$ a coordinate does not make both signs true;
it makes the size and direction of failure explicit.

For Lagarias, $a=\sigma(n)$ and $b=H_n+e^{H_n}\log H_n$. The full configured
scan through five million integers found no positive coordinate. Its closest
normalized nontrivial frontier was

\[
s(12)=-0.00571425357027553127.
\]

For a CNF formula, the discrete coordinate

\[
E_F(a)=\#\{i:C_i(a)=0\}
\]

equals zero exactly on satisfying assignments. Exhaustive enumeration of the
four-pigeons/three-holes formula found minimum distance one with 60 assignments
at that distance and no zero-distance state.

For Navier-Stokes, residual coordinates are typed:

\[
r_{div}=\nabla\cdot u,
\qquad
r_{mom}=\partial_tu+(u\cdot\nabla)u+\nabla p-\nu\Delta u.
\]

They may be combined only after nondimensionalization. Version 0.9 freezes the
largest normalized residual with its $(x,y,z,t)$ address. For the implemented
three-dimensional ABC eigenflow, 179,712 sampled coordinates produced maximum
raw momentum residual $1.1774\times10^{-18}$ and worst dimensionless residual
$7.3537\times10^{-20}$. Analytic substitution proves that family, but not
global regularity for arbitrary three-dimensional data.

Thus the reusable error object is

\[
\boxed{
\mathcal E=(\text{input coordinate},\text{typed residuals},
\text{normalization},\text{distance},\text{evidence status},\text{hash}).
}
\]

An error value can be a correct coordinate on the model's map while the
candidate that generated it remains false. This distinction is the formal
content of “freezing the error.”

---

## Appendix D: Independent falsification and the limits of D0

Version 0.10 tests whether frozen residuals identify actual faults, using an
independent companion and declared synthetic negative controls. The experiment
does not assume every failure to reach zero is an arithmetic mistake.

First, an inequality and an equality have different zero-defect functions:

$$
V_{a\le b}(a,b)=\max(0,a-b),\qquad V_{F=0}(x)=|F(x)|.
$$

The negative signed barycentric slack of a satisfied inequality is not an
error that must be driven to equality. A nonzero defect may reflect
inconsistent constraints, a wrong model, a wrong parameter, rounding, or
discretization. Its cause does not follow from its magnitude alone.

### D.1 Exact finite RH certificate

The Python companion computes outward-rounded dyadic intervals in arbitrary
integers at scale 2^128. It encloses harmonic sums and uses nonnegative
exponential/logarithm series with proved geometric tail bounds. This removes
reliance on floating exp/log values when determining the sign for the listed
finite cases.

It certifies every integer from 1 to 4,096 plus the five integers
5,040; 55,440; 720,720; 4,324,320; and 5,000,000. All 4,101 inequalities are
satisfied. Deliberately replacing sigma(12)=28 by 29 is correctly classified
as a violation. Insufficient series resolution returns unresolved precision.

This does not upgrade the entire five-million-case floating scan to an exact
certificate and does not prove the all-integer Lagarias criterion.

### D.2 Independent SAT check

A separate implementation enumerates all 4,096 assignments and agrees with
the full C++ violation histogram. All assignments violate at least one clause.
Exactly 60 minimize the cost at one, but none satisfies the formula.
A candidate at distance one remains false; the constraints are inconsistent,
not incorrectly added or multiplied.

### D.3 Fault recovery with a frozen bias

The C++ program exports only the ABC field values u,v,w,p. An independent
Python implementation estimates derivatives using centered differences.
At 24 training points it fits pressure and viscosity coefficients; it
evaluates them on 40 untouched points at each of three declared step sizes.
The audit variables are nondimensional; the common spatial/temporal numeric
step is specified in these scaled coordinates.

Injected faults multiply pressure by 1.125 and substitute an assumed viscosity
of 0.008 for the generating value 0.01. At step 0.005, the held-out momentum
RMS falls from approximately 0.285102 to 1.55e-13 after fitting.

However, the recovered viscosity is 0.01000008340041921, not exactly 0.01.
This bias shrinks approximately fourfold when the differentiation step is
halved. For this eigenflow the finite-difference operators imply

$$
\nu_h=\frac{h\sinh(\nu k^2h)}{4\sin^2(kh/2)}=\nu+O(h^2).
$$

Therefore a near-zero numerical residual can conceal parameter error.
The fitted parameter has partly absorbed discretization error.

An extra synthetic velocity mode leaves a held-out RMS around 0.113499 even
after fitting. The two-coefficient correction model is insufficient for that
candidate. This failure is retained, not redefined as success at another layer.
Corrupting held-out data changes evaluation while leaving the training fit
unchanged, verifying the declared separation.

### D.4 Residual is not distance to solution

The exact counterexample

$$
F(x)=10^{-12}x,\quad x=10^6
$$

has residual 10^-6 but lies 10^6 from its unique zero. A useful inference
from residual to state error requires a stability bound with controlled
constant, such as dist(x,S)<=C||F(x)||.

The revised frozen object therefore includes the constraint type, full input
and residual vector, precision or step size, normalization, evidence status,
source identity, and—where applicable—model-identifiability and refinement
checks. Its purpose is to preserve counterevidence as well as successes.

The complete companion derivation and exact/numerical records are provided in
INDEPENDENT_AUDIT.md and results/residual-audit-v010.json inside the release.
No new physical measurements, seven-dimensional physics proof, or
Millennium solution has been obtained.

## Appendix E. Restricted proofs and rejected universal estimates

Version 0.11.0 adds written arguments that cover restricted infinite families,
supported by exact arithmetic in `tools/proof_routes.py`. This goes beyond
extending a finite numerical scan, but it does not resolve any of the three
Millennium targets. Full proofs are in `PROOF_ATTEMPTS.md` in the engine archive.
No novelty or proof-assistant verification is claimed.

### E.1 A restricted Lagarias theorem

For every positive integer with at most two distinct prime factors,

\[
\sigma(n)\le H_n+e^{H_n}\log H_n.
\]

The argument splits at 1,024. Below the cutoff, all 1,023 integers are checked
with outward-rounded integer intervals. Above it, the prime-factor product
gives \(\sigma(n)<3n\). Convexity of \(1/x\) gives
\(H_n>\log n+1/2\), and therefore \(e^{H_n}>3n/2\).
The exact rational series bounds

\[
\log2>\frac{842}{1215}>\frac{69}{100},
\qquad
e^2<\frac{134800213}{18243225}<\frac{37}{5}
\]

imply \(H_n>e^2\) for \(n\ge1024\). Thus \(\log H_n>2\), making the right-hand
side strictly larger than \(3n\). This proves the restricted theorem.

It does not cover integers with arbitrarily many distinct prime factors.
The unrestricted Lagarias statement is equivalent to RH; the restriction is
therefore the unresolved quantifier, not a numerical rounding issue.

### E.2 All-size pigeonhole error formulas

For the engine's pairwise pigeonhole CNF, with \(p\) pigeons and \(h\) holes,
the exact minimum number of violated clauses is

\[
\min_x E(x)=\max(0,p-h).
\]

For \(p\ge h\), the number of minimizing assignments is

\[
N_{\min}(p,h)=\sum_{d=0}^{\min(h,p-h)}
\binom hd\frac{p!}{(p-h-d)!2^d}.
\]

For \(p<h\), it is \(h!/(h-p)!\). An exact decomposition of clause cost into
\(p-h\) plus nonnegative incidence-count terms proves the lower bound. The
equality conditions give the count: all holes are occupied, no pigeon is
assigned twice, and each hole contains one or two pigeons. This explains the
previous minimum of one with 60 minimizing assignments at \(p=4,h=3\).

The new companion checked the formulas against direct clause enumeration and
an independent graph decomposition on 112,938 assignments across 21 sizes.
The written proof covers all positive sizes of this encoding. It supplies no
algorithm or lower bound for unrestricted SAT and does not resolve P versus NP.

### E.3 Exact fluid counterexamples to proposed estimates

On the periodic three-torus of side \(2\pi\), let energy and enstrophy use
normalized spatial means. The globally smooth exact family

\[
u_N=e^{-\nu N^2t}(\sin Nz,\cos Nz,0)
\]

has \(E(0)=1/2\) and \(Z(0)=N^2/2\). Thus initial energy alone cannot give a
finite, initial-data-independent upper bound on enstrophy. Bounds depending
on further initial norms or positive time remain possible.

A second proposed invariant is \(L_\lambda=E+\lambda Z\) with fixed
\(\lambda>0\). For smooth divergence-free initial data

\[
u_0=A(\sin y,0,\cos x-\cos(x+y)),
\]

the exact initial rates are

\[
E'(0)=-2\nu A^2,\qquad
Z'(0)=\frac{A^3}{4}-3\nu A^2,
\]

and hence

\[
L_\lambda'(0)=\frac{\lambda A^3}{4}-(2+3\lambda)\nu A^2.
\]

For any fixed positive \(\lambda,\nu\), choosing
\(A>4\nu(2+3\lambda)/\lambda\) makes this positive. Therefore no such fixed
positive weighted sum is universally nonincreasing. At
\(A=1,\nu=1/100,\lambda=1\), the exact rates are \(-1/50\), \(11/50\), and
\(1/5\) respectively. Sparse Fourier algebra independently obtains them using
rational real and imaginary coefficients, without a numerical grid.

This rejects a particular sufficient-estimate route. Instantaneous enstrophy
growth is not finite-time blowup; no Navier–Stokes breakdown example or
general global-smoothness proof is supplied.

### E.4 The meaning of a frozen failure

An injective coordinate change \(f\) with \(f(0)=0\) satisfies
\(f(r)=0\iff r=0\). It cannot turn a false equality into a true one. A positive
minimum SAT violation can represent inconsistent constraints, not a wrongly
computed factor. The engine therefore freezes counterexamples to proposed
lemmas with their exact assumptions and retains them as rejected proof routes.

The six new test methods and the complete exact-arithmetic companion run
passed. The frozen records, exact RH base-case endpoints, SAT histograms,
Fourier identities, and source/document SHA-256 values are in
`results/proof-routes-v1.json`. The retained v0.10 numerical suite remains
historical; it is not relabeled as a fresh v0.11 run. All three Millennium
problems remain unresolved by this work.

## Appendix F. Independent certificates and exact counting projections

Version 0.12.0 adds a standalone Python verifier with no imports from the
engine or the proof-result generator. It accepts a narrowly scoped portable
certificate and recomputes its evidence using different methods. It does not
accept a producer's pass flag or an unsupported broader theorem claim.

### F.1 Rational lower witnesses

For each of the 1,023 RH base cases, the verifier independently computes the
exact harmonic sum and divisor sum. It checks a portable lower coordinate
\(h=a/2^{24}\) satisfying \(h\le H_n<h+2^{-24}\). With
\(z=(h-1)/(h+1)\), positive-term series imply

\[
H_n+e^{H_n}\log H_n\ge
h+\left(\sum_{j=0}^{40}\frac{h^j}{j!}\right)
\left(2\sum_{j=0}^{31}\frac{z^{2j+1}}{2j+1}\right).
\]

The right side is rational. It strictly exceeds the exact divisor sum for
every tested \(n>1\), with equality at 1. This calculation shares neither the
producer's rounded transcendental interval code nor its divisor-pair summation.
The analytic and number-theoretic steps extending the result to integers with
at most two distinct prime factors remain written proofs, not formalized
proof terms.

### F.2 A counting law that survives projection

For the specified pairwise pigeonhole CNF, let \(c\) record the occupancies
of the holes after some rows have been chosen, and let \(e\) record total
clause cost. A new row chooses a subset \(M\) of holes and adds

\[
\Delta(c,M)=\mathbf1_{M=\varnothing}+\binom{|M|}{2}
+\sum_{j\in M}c_j.
\]

Project through \(\alpha(c,e)=(\operatorname{sort}(c),e)\). For every two
occupancy vectors related by a hole permutation, that permutation gives a
bijection between row subsets. Corresponding subsets give the same cost
increment and sorted successor. Their multisets of projected transitions
therefore agree.

Consequently the exact row-counting recurrence is

\[
F_{r+1}\left(\operatorname{sort}(c+\mathbf1_M),e+\Delta(c,M)\right)
\mathrel{+}=F_r(c,e),
\]

starting from the unique empty assignment and summing over all subsets.
Induction on rows proves that its final histogram counts every labeled
Boolean assignment once. For counting transfer operators,
\(\alpha_*T=\bar T\alpha_*\). This is an exact projection rule under the
stated symmetry; it is not a descent result for arbitrary individually fixed
labeled updates or asymmetric rules.

The cost coordinate is necessary. The two assignments with rows
\(((1,0),(0,1))\) and \(((1,1),(0,0))\) have the same column occupancies but
costs 0 and 2. Dropping the cost would identify states that the requested
histogram must distinguish.

The recurrence recovered all 21 earlier histograms, representing 112,938
assignments, using 4,632 row transitions and at most 215 intermediate states.
This is a count of different computational operations, not a measured speed
ratio or a general polynomial-time SAT claim.

### F.3 Independent fluid algebra

The checker expands \(\cos(x+y)\) in real sine/cosine variables and evaluates
polynomial moments with exact rational coefficients. It computes enstrophy
production from vortex stretching rather than the producer's differentiated
transport expression. It reproduces the exact rates \(E'=-1/50\) and
\(Z'=11/50\), along with the documented weighted and high-frequency examples.
These are checks of the specified smooth periodic fields, not certificates
for arbitrary fluid evolutions.

### F.4 Rejection tests and remaining trust

The valid portable certificate passed. All 26 deliberate corruptions were
rejected, including altered arithmetic, incomplete coverage, forged broader
claims, changed physical assumptions, and malformed numeric representations.
Ten new test methods passed. The checker was also run in isolation with only
its source and the certificate present, and rejection remained active under
Python's `-O` option.

These checks increase independence from the producer, but the verifier's own
mathematics and implementation remain trusted. No Lean or other established
proof kernel checked these arguments. A valid formal proof would also need a
faithful formal statement of the intended claim. The complete theory is not
made irrefutable by finite tests or by certificate acceptance.

The detailed derivations, trust boundary, and Windows/Python reproduction
instructions are in `CERTIFICATE_VERIFIER.md`. The frozen data are
`results/portable-certificate-v1.json`, `results/verifier-v012.json`, and
`results/certificate-challenges-v012.json`. Earlier proof-route source and
document fingerprints were preserved unchanged. This release makes no new
claim of physical validation or resolution of a Millennium problem.

## Appendix G. Arguments at the remaining proof gaps

Version 0.13.0 supplies further written reductions and obstructions in
`MISSING_ARGUMENTS.md`, supported by `tools/proof_gap_lab.py`. No mathematical
novelty or formal-kernel verification is claimed. The full Millennium problems
remain unresolved by this work.

### G.1 Removing the prime-count restriction from a bounded RH verification

Put \(G(h)=h+e^h\log h\), \(B_n=G(H_n)/n\), and \(R_n=\sigma(n)/n\).
The new discrete argument proves \(B_{n+1}>B_n\) for every \(n\). For
\(n\ge64\), integral harmonic bounds and \(n>2(1+\log n)^2\) imply

\[
nG'(H_n)-(n+1)G(H_n)>0.
\]

Convexity of \(G\) then gives the desired adjacent inequality. The 63
remaining adjacent cases and the logarithm constant used in the analytic
argument are checked with exact outward-rounded intervals.

Any integer can be replaced by a no-larger product of consecutive primes
with nonincreasing positive exponents, while increasing or preserving its
divisor ratio. Missing smaller primes can replace larger ones. Increasing
exponent pairs can be exchanged because \(S_b(r)/S_a(r)\) is increasing for
\(b>a\), where \(S_a(r)=1+r+\cdots+r^a\). Strict integer descent terminates
at a canonical profile.

Consequently it suffices, below any fixed cutoff, to check successive record
divisor ratios among all canonical profiles. For each \(n\) there is a retained
record \(r\le n\) with \(R_r\ge R_n\), so

\[
R_n\le R_r\le B_r\le B_n.
\]

The associated reduction to superabundant numbers is already in the
literature, including [MacArevey's February 2026 preprint](https://arxiv.org/html/2602.15905v2).
It is incorporated here with a discrete proof and executable finite coverage,
not claimed as a discovery of this framework.

The run generated 172,513 canonical profiles through \(10^{24}\) and certified
all 148 record cases. Under these written reductions and correct execution
of the exact-arithmetic code, this verifies the Lagarias inequality for every
integer \(1\le n\le10^{24}\). It is a finite computer-assisted verification,
not RH and not a claimed verification record.

Large harmonic indices use the rigorous integral bracket

\[
H_M+\log\frac{n+1}{M+1}\le H_n\le H_M+\log\frac nM,
\qquad M=4096.
\]

The computation uses 192-bit integer ticks, 256-term interval bounds, and
rational logarithm range reduction. Its record extraction agrees with a
separate full divisor sieve through 100,000. The all-integer finite coverage
depends on the written enumeration and reduction proofs, not on that smaller
comparison alone. The previous standalone verifier was not extended to
formally check this new argument.

### G.2 A lower-bound extension that cannot work

Consider the proposed sufficient bound

\[
L_c(n)=\log n+c+n e^c\log(\log n+c).
\]

For every fixed \(c<\gamma\), it fails to dominate \(\sigma(n)\) infinitely
often. Indeed,

\[
\frac{L_c(n)}{n\log\log n}\longrightarrow e^c,
\qquad
\limsup_{n\to\infty}\frac{\sigma(n)}{n\log\log n}=e^\gamma>e^c,
\]

where the second statement is Gronwall's theorem, stated in
[Lagarias's paper](https://arxiv.org/html/math/0008177v2). The code also certifies
a concrete failure at \(c=1/2,n=12\), while the actual Lagarias inequality
there remains satisfied. Increasing precision in a fixed constant below
\(\gamma\) cannot supply the missing global argument.

### G.3 An exact continuation-summary bound

Suppose an evaluator must recover the equality relation on \(n\)-bit strings
from a deterministic summary of its first argument:

\[
g(\alpha(x),y)=\mathbf1_{x=y}\quad\text{for every }x,y.
\]

Then \(\alpha\) must be injective. If \(x\ne x'\) share a summary, choosing
\(y=x\) requires the same evaluator input to produce two different answers.
Thus at least \(2^n\) summary states, or \(n\) fixed-width bits, are necessary.
For \(d\) coordinates with \(b\) bits each, the condition is \(db\ge n\).

This refutes a particular fixed-capacity lossless summary strategy. Equality
itself is easy to compute, and this bound is not a general SAT time lower
bound. Twelve explicit collision witnesses accompany the all-size proof.
P versus NP still requires an argument about general algorithms, not only
this architecture.

### G.4 Why the quadratic fluid strategy fails more generally

For a differentiable homogeneous quadratic functional, amplitude scaling of
Navier–Stokes initial data gives

\[
\left.\frac{d}{dt}Q(Av)\right|_{t=0}
=A^3C_Q(v)-\nu A^2D_Q(v).
\]

If the cubic coefficient is nonzero for some \(v\), choose the amplitude's
sign and size so the cubic term is positive and dominates. Universal
nonincrease at every amplitude therefore requires the inviscid cubic
production to vanish identically. This is a necessary condition for that
strategy, not for every useful regularity estimate.

For \(u_0=A(\sin y,0,\cos x-\cos(x+y))\), a Fourier multiplier with weights
\(m_1,m_2\) at squared frequencies 1 and 2 has the exact initial derivative

\[
Q'(0)=\frac{A^3}{4}(m_2-m_1)-\nu A^2(m_1+m_2).
\]

For homogeneous Sobolev quadratic energy of any real order \(s>0\), use
\(m_1=1,m_2=2^s\). Choosing
\(A>4\nu(1+2^s)/(2^s-1)\) makes the derivative positive. Eight integer-order
Fourier probes and sign controls check the implementation; the displayed
algebra proves the statement for every positive real order. Initial norm
growth is not a singularity, and a continuation-strength bound remains
missing.

### G.5 Evidence boundary

All nine new test methods passed. The full finite run, exact record margins,
monotonicity base cases, failed lower bound, summary collisions, and weighted
fluid probes are frozen in `results/proof-gaps-v013.json`. The supplementary
proofs identify which proposed arguments are valid, which are false, and
which unbounded steps remain unproved. Neither finite coverage nor a new
coordinate interpretation supplies those remaining steps automatically.

## Appendix H. A computable sufficient condition for all-time fluid smoothness

Version 0.14.0 adds a positive conditional result for the full unforced
three-dimensional Navier–Stokes equation on the fixed torus \([0,2\pi]^3\).
The mathematical method is established Fourier small-data theory. It is
related to the periodic global existence and analyticity result of
[Ambrose, Lopes Filho, and Nussenzveig Lopes](https://arxiv.org/abs/2205.12383).
The new contribution to this project is an explicit written sufficient
estimate and an exact initial-data checker; mathematical novelty is not
claimed.

### H.1 The conditional all-time statement

Write \(u(x,t)=\sum_k u_k(t)e^{ik\cdot x}\), with real, smooth,
mean-zero, divergence-free initial data. For viscosity \(\nu>0\), define

\[
S_0=\sum_{k\ne0}\frac{\|u_k(0)\|_2}{|k|}.
\]

If an exact upper bound satisfies \(S_0\le B<\nu\), set
\(\delta=(\nu-B)/2\). Then the solution remains smooth for all \(t\ge0\)
and satisfies

\[
\boxed{
\sum_{k\ne0}e^{\delta t|k|}\frac{\|u_k(t)\|_2}{|k|}
\le S_0e^{-\delta t}\le Be^{-\delta t}.
}
\]

At positive times this controls every spatial derivative and gives spatial
analyticity with radius at least \(\delta t\). The statement concerns every
future time of the full equation, including modes generated by nonlinear
interactions. It does not assume a finite evolving spectrum.

### H.2 The estimate and its continuation argument

Incompressibility and the norm-one Leray projection give

\[
\|N_k\|_2\le |k|\sum_{p+q=k}\|u_p\|_2\|u_q\|_2
\]

for the projected nonlinear term. To avoid assuming analyticity in advance,
use capped weights \(w_K(k,t)=e^{\delta t\min(|k|,K)}\) and define
\(A_{j,K}=\sum_{k\ne0}|k|^jw_K\|u_k\|_2\). Submultiplicativity of the
weights and Cauchy–Schwarz imply

\[
D^+A_{-1,K}\le-\nu A_{1,K}+A_{0,K}^2+\delta A_{0,K}
\le-(\nu-\delta-A_{-1,K})A_{1,K}.
\]

Here \(A_{0,K}^2\le A_{-1,K}A_{1,K}\), and the torus's nonzero
frequencies satisfy \(|k|\ge1\). A barrier argument preserves
\(A_{-1,K}\le B\), leaving a dissipation margin of at least \(\delta\).
Thus \(A_{-1,K}(t)\le S_0e^{-\delta t}\). Monotone convergence as
\(K\to\infty\) supplies the uncapped exponential estimate.

For a multiindex of order \(m\), it follows that

\[
\|\partial^\alpha u(t)\|_\infty
\le\left(\frac{m+1}{e\delta t}\right)^{m+1}Be^{-\delta t}
\qquad(t>0).
\]

If a maximal smooth time were finite, these bounds would control all
spatial derivatives on an interval ending at that time and bounded away
from zero. Standard smooth local existence and continuation then extend
the solution, giving a contradiction. Those standard analytic theorems
are explicit dependencies of the proof, not Python operations. The full
argument and normalization are in `GLOBAL_FLUID_BOUND.md`.

### H.3 What the computer verifies

The input is an exactly specified finite rational Fourier polynomial.
Unlisted initial coefficients must be exactly zero; a measured truncation
with an unknown tail does not qualify. The checker verifies all reality,
mean, divergence, viscosity, and schema assumptions using exact arithmetic.

Each term of \(S_0\) is the square root of a nonnegative rational number.
Integer square roots produce lower and upper enclosures at chosen
increment \(2^{-p}\), with rational roots evaluated exactly. Summed bounds
\(L\le S_0\le B\) determine three possible valid-input outcomes:

- \(B<\nu\): the sufficient theorem applies;
- \(L\ge\nu\): this sufficient condition fails, without a blowup conclusion;
- \(L<\nu\le B\): precision is insufficient to decide the threshold.

All eleven new test methods passed. Six exact fields produce seven expected
outcomes when one borderline field is checked at both 8 and 96 bits. The
frozen report is `results/global-fluid-v014.json`. It explicitly disclaims
formal-kernel verification and any Millennium solution.

### H.4 The boundary that remains

For the interacting field
\(u_0=10^{-3}(\sin y,0,\cos x-\cos(x+y))\), the initial norm is
\(10^{-3}(2+1/\sqrt2)<1/100\), so the theorem certifies global smoothness
at viscosity \(1/100\). Increasing the amplitude to one violates this
sufficient hypothesis; the checker then gives no regularity conclusion.

Failure of strict smallness is not a singularity certificate. The exact
shear \(u=(\nu e^{-\nu t}\sin y,0,0)\) is globally smooth even though its
initial norm equals \(\nu\), exactly at the rejected threshold.

The general large-data gap is the absence of a bound controlling the full
evolution when the dissipative coefficient in the displayed estimate loses
its sign. Additional arithmetic precision cannot change that mathematical
obstruction for an exact input above threshold. This restricted fluid
result also does not close the all-integer RH argument, the general
algorithmic P-versus-NP argument, or the empirical gap between a seven-layer
representation and physical reality.

## Appendix I. Relative error control beyond the total-smallness condition

Version 0.15.0 implements a sufficient stability theorem around an exact
decaying shear. It certifies a concrete initial field outside Appendix H's
total-smallness condition. This is an expansion of the engine's implemented
coverage, with no claim of mathematical novelty. Error estimates around
approximate solutions are an established approach; see
[Morosi and Pizzocchero](https://arxiv.org/abs/1104.3832).

### I.1 A reference trajectory and its full error

For the same unforced periodic equation and Fourier normalization, write
\(u=v+w\), where \(v\) is a smooth, real, mean-zero, divergence-free reference.
Define \(\|f\|_{\chi^j}=\sum_{k\ne0}|k|^j\|\widehat f(k)\|_2\), and let

\[
E=\|w\|_{\chi^{-1}},\quad E_1=\|w\|_{\chi^1},\quad
V=\|v\|_{\chi^0},\quad
R=\|\partial_tv-\nu\Delta v+P[(v\cdot\nabla)v]\|_{\chi^{-1}}.
\]

The Fourier bilinear estimate, Cauchy–Schwarz, and Young's inequality give

\[
D^+E+\frac\nu4 E_1\le\frac{4V^2}{\nu}E+R,
\qquad\text{provided }E\le\nu/2.
\]

All nonlinear interactions of the error are retained in deriving this
estimate. A rigorously bounded scalar comparison function can preserve the
error threshold and control the integrated gradient of the full solution.
Standard high-order Sobolev estimates and local continuation then justify
existence throughout the certified interval. The complete derivation is in
`NEXT_WALL.md`.

### I.2 An all-time specialization the engine computes

For \(v=(Ae^{-\nu t}\sin y,0,0)\), the reference is an exact solution,
so \(R=0\). If

\[
\boxed{E(0)e^{2(A/\nu)^2}\le M<\nu/2,}
\]

then a barrier and integration argument give

\[
\|u(t)-v(t)\|_{\chi^{-1}}\le Me^{-\nu t/4},\qquad
\int_0^\infty\|u-v\|_{\chi^1}\,dt
\le\frac4\nu\left(E(0)+2M(A/\nu)^2\right).
\]

The finite integrated gradient bound prevents finite-time smooth breakdown.
The implemented checker supports \(|A|/\nu\le8\); the theorem itself permits
larger amplitudes with smaller perturbations. The engine uses exact
rational square-root bounds and outward dyadic exponential enclosures.

The included example, with \(\nu=1\) and \(\varepsilon=10^{-5}\), is

\[
u(x,0)=(2\sin y+\varepsilon\sin z,\ \varepsilon\sin(2x),\
         \varepsilon\sin y).
\]

It has exact zero divergence, three independent spatial Fourier directions,
and nonzero projected nonlinear interactions. Its total initial norm exceeds
two, violating the previous sufficient threshold of one. Relative to the
known shear, however, \(E(0)=1/40000\) and
\(E(0)e^8\approx0.074524<1/2\), so the new theorem applies.

At \(T=4\), the certified full norm is below \(0.065\). The smooth solution
then satisfies the previous small-data theorem and its analyticity bound,
with the time origin shifted to \(T\). This invokes the written theorem on
the full evolving field; it does not presume a finite spectrum at \(T\).

### I.3 The next engineering prerequisite

Nine new test methods and six expected example outcomes passed. Independent
Fourier convolution caught an omitted quadratic perturbation coefficient in
an initial hand calculation; the corrected expression and rejected value
are preserved in `results/relative-fluid-v015.json`. The general error
estimate already included that interaction. No formal proof kernel has
checked the analysis.

The next implementation target is a general reference trajectory with
certified continuous-time residual, all omitted Fourier output modes, and
a rigorously enclosed scalar comparison solution. If it controls the error
up to a time at which the full norm becomes small, it can connect initially
larger data to Appendix H. Finite-time samples alone cannot establish these
bounds. This proposed general validator is not yet implemented in v0.15.

The useful measure of progress is a stronger valid certificate on a frozen
previously unresolved input, followed by a clearly stated extension to an
input class. Neither this restricted stability theorem nor its coordinate
representation supplies arbitrary large-data regularity, an all-integer RH
proof, or a general complexity separation.

## Appendix J. Computed references with continuous-time and full-mode certificates

Version 0.16.0 implements the three engineering components proposed in
Appendix I: bounds on the full residual throughout time intervals, a
certified error-envelope constructor, and independent checking with
counterexample searches. The supported representation is finite Fourier
support and piecewise polynomial time dependence. The written theorem and
exact certificate format are in `TRAJECTORY_PROOF.md`.

### J.1 Continuous-time polynomial bounds

On each interval, use normalized time \(\theta\in[0,1]\). For a vector
polynomial with Bernstein expansion

\[
p(\theta)=\sum_{i=0}^d b_i\binom di\theta^i(1-\theta)^{d-i},
\]

nonnegativity and partition of unity imply

\[
\|p(\theta)\|_2\le\max_i\|b_i\|_2
\quad\text{for every }\theta\in[0,1].
\]

Rational squared-norm witnesses certify the control-vector bounds.
The verifier reconstructs the controls by a different algorithm from the
producer. These finite witnesses cover the entire interval by the stated
convexity argument; they are not observations at a finite list of times.

### J.2 Every omitted output mode of the reference residual

For reference support \(K\), the full projected residual

\[
r=\partial_tv-\nu\Delta v+P[(v\cdot\nabla)v]
\]

has support in \(K\cup(K+K)\). Both implementations reconstruct every
nonzero residual mode there, including those outside the reference support.
One uses the ordered convection expression, while the other uses divergence
of the velocity tensor. Exact incompressibility makes these expressions
equivalent. The independent checker rejects missing residual mode bounds.

The residual vanishes outside this finite union because the reference is
finite. The true solution may have modes arbitrarily far beyond it; those
are covered by the full relative-error estimate, not set to zero.

### J.3 An affine supersolution checked by algebra

Given interval bounds \(V\ge\|v\|_{\chi^0}\) and
\(R\ge\|r\|_{\chi^{-1}}\), set \(b=4V^2/\nu-\nu/4\). On an error-envelope
segment of physical duration \(\Delta\), the proposed affine function joins
\(q_L\) and \(q_R\). The checker verifies

\[
\frac{q_R-q_L}{\Delta}\ge bq_L+R,
\qquad
\frac{q_R-q_L}{\Delta}\ge bq_R+R,
\qquad 0\le q_L,q_R<\nu/2.
\]

Since the right-hand side is affine in time, these inequalities cover every
interior time. Joined with the certified initial error and continuous
reference/envelope joins, comparison gives
\(\|u-v\|_{\chi^{-1}}\le q<\nu/2\). Integrating the stronger error inequality
controls the velocity gradient in time; standard local existence and Sobolev
continuation exclude a smooth breakdown during the certified interval.

The final checked inequality

\[
\|v(T)\|_{\chi^{-1}}^{\rm upper}+q(T)<\nu
\]

invokes Appendix H's all-time theorem. The evolved exact field need not have
finite support. The reference may have derivative jumps between polynomial
intervals; its checked continuity avoids impulses, and the differential
estimates integrate across the finitely many joins.

### J.4 Executed evidence and its limits

The main run uses viscosity one and the same form of interacting initial
field as Appendix I, now with perturbation amplitude \(10^{-7}\). Its total
initial norm exceeds two. The computed reference uses six signed Fourier
wavevectors, 32 polynomial intervals of degree six, and 158 affine envelope
segments, reaching time four. For this support the retained Galerkin
convection vanishes, while the full residual is nonzero and includes omitted
nonlinear output modes.

The independent checker accepted the entire certificate. It covers 384
omitted-mode occurrences across intervals, bounds the error below
0.000045 throughout, and gives a full norm bound below 0.036664 at time
four. This initial field was already inside the prior shear-perturbation
theorem; the new result is a different implemented validation mechanism,
not a claim of a previously unknown regularity class.

All eleven new test methods passed. All seventeen deliberate corruptions
were rejected, including omitted residual modes and a perturbation hidden
from endpoint values and derivatives. A search over small polynomials found
an exact counterexample to the endpoint-only maximum shortcut. A second
counterexample has zero retained-mode residual but nonzero full residual.

The bounded refinement search retained a failed degree-one/eight-interval
case, then accepted and independently verified a degree-four/sixteen-interval
case. It stopped at that success; it makes no global optimality claim.
The checker also passed an isolated run with only its source and certificate
present and rejected a corruption under optimized Python.

These methods belong to established approximate-solution validation; see
[Morosi and Pizzocchero](https://arxiv.org/abs/1104.3832). The analysis has not
been formalized in a proof assistant. Arbitrary large-data regularity, the
unbounded RH step, general P versus NP, and empirical seven-dimensional
physics remain unresolved by this engine.

## Appendix K. A sharper error estimate that certifies a previously unresolved input

**Engine addition:** 0.17.0. This appendix describes a concrete expansion of
the engine's certificate coverage. The complete written argument, exact
records, and PC commands are in `SHARP_TRAJECTORY_BOUND.md` in the archive.
It is not a claim of mathematical novelty or of a solved Millennium problem.

### K.1 The same input before and after the changes

On the fixed periodic torus, with viscosity one, take

\[
u_0=(2\sin y+\varepsilon\sin z,\varepsilon\sin 2x,
\varepsilon\sin y),\qquad\varepsilon=1/10.
\]

The exact data are real, mean-zero and divergence-free. Their initial
\(\chi^{-1}\) norm is \(\sqrt{4+\varepsilon^2}+3\varepsilon/2>2\),
outside the total-smallness condition. The older exact-shear sufficient
certificate also fails on this input.

The frozen comparison fixes degree-six polynomial references, 32 intervals
of width \(1/8\), time four, and 64-bit outward norm witnesses:

| Reference | Prior estimate | Sharper estimate |
|---|---|---|
| Six initial wavevectors | Envelope unresolved | Envelope unresolved |
| Eighteen wavevectors retaining nonlinear interactions | Envelope unresolved | Independently accepted |

This establishes that both changes are needed among these four tested
configurations. It says nothing about all other possible reference choices,
norms, certificates, or known mathematical approaches.

### K.2 Exact constrained quadratic optimization

For the complete error \(w=u-v\), use
\(E=\|w\|_{\chi^{-1}}\), \(E_0=\|w\|_{\chi^0}\), and
\(E_1=\|w\|_{\chi^1}\). The full bilinear Fourier estimate gives

\[
D^+E+\nu E_1\le2VE_0+E_0^2+R,
\qquad E_0^2\le EE_1,\qquad E_1\ge E.
\]

Here \(V=\|v\|_{\chi^0}\) and \(R=\|r\|_{\chi^{-1}}\) include
the full reference residual. Choose an error cap \(c>0\), reserved
dissipation \(\mu>0\), and \(D=\nu-c-\mu>0\). While \(E\le c\),

\[
D^+E+\mu E_1\le-DE_1+2V\sqrt{EE_1}+R
\le\Lambda(V,D)E+R,
\]

where completing the square proves

\[
\Lambda(V,D)=\max_{x\ge1}(-Dx^2+2Vx)
=\begin{cases}
V^2/D,&V\ge D,\\
2V-D,&0\le V<D.
\end{cases}
\]

For positive error, \(x=\sqrt{E_1/E}\); zero error is handled directly
without division. The piecewise bound is monotone in \(V\), so an
interval-wide reference cap \(V_j\) can replace \(V\). Consequently

\[
\boxed{D^+E\le b_jE+R_j,\qquad
b_j=\Lambda(V_j,D)-\mu,\quad E\le c.}
\]

The release sets \(c=\mu=\nu/8\). The affine supersolution checks from
Appendix J use this coefficient and the strict cap \(q<c\). Positive
reserved dissipation bounds \(\int E_1dt\), which supplies the same
Sobolev continuation argument. A final full-norm bound below \(\nu\)
then invokes Appendix H's all-time theorem.

The scalar maximum is exact. The upstream norm estimates and the overall
PDE criterion are not claimed to be optimal. Setting the reserved
dissipation to zero would invalidate this particular continuation argument
and is rejected by the checker.

### K.3 Nonlinear references with accounted-for rounding

The new reference retains the twelve projected interaction wavevectors
generated from the initial six. Their initial coefficients are zero but
their subsequent time-polynomial coefficients are nonzero. This retains
actual nonlinear interactions that the previous approximation omitted.

Intermediate Taylor coefficients are rounded to control rational arithmetic
growth, then projected and reflected to preserve exact divergence freedom
and conjugate symmetry. A highest-degree correction gives an exactly
shared rational endpoint between adjacent polynomial intervals. The main
references use 48-bit intermediate rounding, separately from the 64-bit
outward norm/envelope witnesses.

The independent checker trusts none of the reference's generation history.
It reconstructs the entire residual polynomial, including rounding,
endpoint corrections, and every nonzero omitted output mode. Bernstein
witnesses cover all times in each interval. The exact solution's evolving
spectrum is never assumed finite.

### K.4 Checked outcome and the next failure

For \(\varepsilon=0.1\), the accepted certificate has 32 polynomial
intervals, 76 affine error segments, and 1,792 omitted-output-mode
occurrences across intervals. Its exact rational witnesses establish

\[
\max_{0\le t\le4}\|u(t)-v(t)\|_{\chi^{-1}}<0.04424,
\qquad \|u(4)\|_{\chi^{-1}}<0.05144<1.
\]

Together with the written sufficient argument and its standard analytic
dependencies, these imply global smoothness for this specified input.
The independent program check is not formal verification in a proof
assistant, and it does not by itself prove the analytic dependencies.

At \(\varepsilon=0.01\), the sharper six-mode reference has maximum
certified error below 0.026679; retaining nonlinear interactions reduces
that bound to below 0.002382. These compare certified upper bounds,
not exact observed errors. All twelve new targeted test methods pass,
and all 23 deliberate corruptions of the nonlinear 0.01 certificate
are rejected. An exact scalar counterexample also rejects use of the
small-\(V\) branch outside its valid domain.

At \(\varepsilon=0.5\), the same configuration fails its sufficient
envelope on the second interval. The failed estimate is retained as the
next fixed research target. It is not a singularity or evidence that the
exact solution loses smoothness.

These developments extend established approximate-solution validation;
see [Morosi and Pizzocchero's smooth-solution analysis](https://arxiv.org/abs/1405.3421).
They establish an implementation improvement on a fixed input, without
claiming a previously unknown mathematical regularity class. General
large-data Navier–Stokes regularity, RH, P versus NP, and empirical
seven-dimensional physics remain unresolved by this engine.

## Appendix L. Directing the engine toward valid extension arguments

**Engine addition:** 0.18.0. The complete implementation record and mathematical
arguments are in `PROOF_SEARCH_DIRECTION.md` in the archive.

### L.1 The shared objective

The three prize targets share a proof-engineering difficulty: a finite
calculation must be connected to the entire domain of the claim by a valid
extension argument. For RH this concerns all integers, for P versus NP a
uniform complexity statement, and for Navier–Stokes the required set of
initial data and times. This is a useful organizational principle, not a
reduction proving the three problems equivalent.

The engine is now directed toward sound abstraction, inductive closure,
residual-guided refinement, reuse of expensive certificate work, and
independent verification of the precise claimed scope. Encoding an answer
as a bit or a distance to zero does not make that bit easy to compute.
Representation size, search cost, and proof coverage remain separate.

### L.2 A finite certificate with unbounded path-length coverage

For a transition relation \(T\), let \(\operatorname{Post}(S)\) be every
possible successor of a state in \(S\). With initial set \(S_0\) and bad
set \(B\), a candidate invariant \(I\) must satisfy

\[
S_0\subseteq I,\qquad\operatorname{Post}(I)\subseteq I,
\qquad I\cap B=\varnothing.
\]

Induction then proves that every reachable state avoids \(B\), at every
finite path length and hence at every prefix of an infinite trajectory.
This does not prove termination or a liveness property.

The new finite-system engine searches a quotient graph of a supplied state
partition. An abstract path to a bad block is checked for a concrete
realization using backward suffix-predecessor sets. A real path becomes a
counterexample certificate. A spurious path identifies a proper block split,
and the search repeats. Every split adds a block, so at most \(N-1\)
splits are possible from one block on \(N\) states. A smaller resource
budget can leave the search unresolved.

The standalone checker uses a different calculation: bit-mask inclusions
and direct closure under every concrete edge. It checks initial coverage,
complete partition coverage, safety, and transition closure without trusting
the refinement history. It can bind the certificate to a separately
provided requested system.

On the five-state safe example, three spurious paths are removed and the
invariant \(\{0,1\}\) is certified. Changing one initial outgoing edge
produces the real counterexample \(0,2,3,4\). Another 160 seeded finite
systems agree with independent concrete reachability. These results concern
fixed explicitly supplied systems, not all programs or input sizes. Small
abstract state counts do not imply cheap construction or a compact membership
map.

### L.3 Refine the distinctions responsible for the fluid residual

The previously frozen fluid failure has viscosity one and

\[
u_0=(2\sin y+\varepsilon\sin z,\varepsilon\sin2x,
\varepsilon\sin y),\qquad \varepsilon=1/2.
\]

The engine ranks an omitted conjugate Fourier pair by the exact rational
sum of interval width times its residual norm caps. Seven pairs account
for approximately 91.214% of the old integrated omitted-residual bound.
Adding them yields a thirty-two-mode reference. Its first interval's full
residual cap falls from about 0.115940 to 0.020422.

This is a heuristic for constructing a candidate reference: the approximate
dynamics change when modes are added. The new full residual therefore must
be recomputed and independently checked. Intermediate rounding and endpoint
corrections remain included. A selected mode can round to zero at lower
precision; the release preserves the counterexample to the false expectation
that every selected mode must remain active.

### L.4 Local error budgets

The sharper inequality from Appendix K can be applied on each affine
envelope segment with a different cap \(c_\ell\). A fixed reserve
\(\mu>0\) remains positive throughout. The checker requires

\[
0\le q_L,q_R<c_\ell,\qquad c_\ell+\mu<\nu,
\]

\[
b_\ell=\Lambda(V_j,\nu-c_\ell-\mu)-\mu,
\]

\[
\frac{q_R-q_L}{\Delta}\ge b_\ell q_L+R_j,
\qquad
\frac{q_R-q_L}{\Delta}\ge b_\ell q_R+R_j.
\]

These inequalities hold throughout the affine segment. Exact joins and
the checked initial error propagate the bootstrap across the whole finite
partition. All remaining damping values are positive; their finite minimum
is positive. Integrating the stronger error inequality with the fixed
positive reserve supplies the existing Sobolev continuation argument.
The final full norm below viscosity again invokes the all-time small-data
theorem, without assuming finite evolving support for the exact solution.

The local policy uses reserve \(\nu/1024\), caps on a \(\nu/1024\)
grid, and a declared segment-resolution ladder. These are computational
parameters, not fundamental physical units. The complete policy changes
the caps, reserve, and resolution; the comparison does not isolate one
parameter as the sole cause of success.

### L.5 The new accepted input and the common audit

For the same \(\varepsilon=0.5\) input, all five fixed-cap configurations
fail on both the eighteen-mode and thirty-two-mode references. All three
local-control resolutions fail on the old reference. On the refined
reference, sixteen segments per interval fail and sixty-four pass the
independent checker. The next listed fallback is not visited.

The accepted certificate covers 32 degree-six reference intervals,
2,048 affine segments, and 4,032 omitted-output-mode occurrences. Exact
rational witnesses give the outward comparisons

\[
\max_{0\le t\le4}\|u(t)-v(t)\|_{\chi^{-1}}<0.73674,
\qquad \|u(4)\|_{\chi^{-1}}<0.70240<1.
\]

The maximum local cap is 0.7373046875, the minimum remaining damping is
0.26171875, and the positive reserve is \(1/1024\). Subject to the
written analytic dependencies, this supplies global smoothness for the
specified initial field. It expands the engine's coverage; it is not a
claim of a new regularity class or a general all-data theorem.

Reference bounds are cached across candidate searches. Their content hashes
check identity, not mathematical truth: a deliberately falsified and
rehashed cache still fails independent verification. The final checker
reconstructs the entire residual and checks all continuous-time witnesses.
No timing result is presented as a globally optimal or portable performance
guarantee.

The proof director also independently rechecks 1,023 portable RH base cases,
21 bounded SAT instances, and the finite inductive certificate. It compares
property, domain, problem identity, and bounded versus unbounded coverage
before allowing evidence to support a claim. A finite prefix does not
cover all integers; one fluid datum does not cover all data; a formula-family
calculation does not bound every algorithm.

Twenty-three targeted test methods pass, including the 160 finite-system
comparisons. All 24 deliberate corruptions of a small local-cap certificate
are rejected. Earlier proof sources and their frozen documents remain
unchanged.

The new work uses independently implemented programs and written mathematical
arguments, without formal proof-assistant verification or a novelty claim.
General RH, P versus NP, and all-data Navier–Stokes remain open in the
engine's audit. The next substantive work is the missing extension argument
for each target, broader verified initial-data coverage, and formalization
of selected analytic dependencies. The shared workflow supports that work;
it does not substitute for it.

---

## Appendix M. Hypothetical universes and checked proof transfer

Version 0.19 makes the proposal to connect possible answers precise. Let
R denote RH, P denote P=NP, and N denote all-data global smoothness for the
unforced, viscosity-one, mean-zero periodic three-dimensional fluid problem.
Classical logic has eight truth assignments to these three propositions.
The engine enumerates them, without asserting that each assignment is
consistent with arithmetic or is an actual physical universe. No new
unconditional implication between the targets has been established;
all eight assignments remain open in this audit.

A case split can be a valid proof method. If a common base theory proves
P implies R and not-P implies R, then R follows by excluded middle. The
new checker verifies the conditional propositional inference and retains
both premises as unproved. In contrast, even assuming the cycle R implies
P implies N implies R leaves both the all-false and all-true assignments.
The engine explicitly records this remaining ambiguity. A screen of 24
signed implications tests propositional tautologies only; it cannot decide
which additional implications arithmetic or analysis might prove.

### M.1. A contracting comparison can close an error loop

Let d be a finite nonnegative real vector, A a nonnegative matrix, and b a
nonnegative forcing vector. Suppose

\[
d\le Ad+b,\qquad B\ge0,\qquad B(I-A)=I.
\]

Multiplying the rearranged inequality by B gives the exact conditional bound

\[
\boxed{d\le Bb.}
\]

When b=0, nonnegativity forces d=0. This is a proved comparison lemma;
the standalone checker verifies the matrix identities over exact rationals.
For the three-cycle with each gain 1/2, B has rows (8,4,2)/7,
(2,8,4)/7, and (4,2,8)/7. The positive weight vector B*1=(2,2,2) has
weighted contraction factor 1/2. Forcing (1/100,0,0) produces the exact
bound (2/175,1/350,1/175). Nonzero forcing is retained explicitly.

More generally, d1<=a*d2, d2<=b*d3, d3<=c*d1 with nonnegative gains
forces the three finite errors to vanish if a*b*c<1. At unit gains,
d=(1,1,1) is a counterexample to that conclusion. The finite-vector and
strict-contraction conditions are essential.

The engine has not constructed actual defects for RH, P versus NP, and
Navier–Stokes satisfying these inequalities. Such a construction must prove
what zero means, finiteness, full-domain coverage, and the coupling bounds.
Naming an arbitrary coordinate after a prize does not supply those proofs.
This elementary comparison theorem is not claimed as a new mathematical
discovery or a solution of the three targets.

### M.2. A finite dynamics-to-SAT bridge with complete coverage

For a finite transition system S with N states, any reachable bad state
has a path of at most N-1 edges: a shortest such path cannot repeat a state,
because the repeated segment could be removed. Add self-loops for padding
and compile a one-hot CNF Phi_K requiring an initial state at time zero,
legal padded transitions, and a bad state at time K. Then

\[
\operatorname{SAT}(\Phi_K(S))
\iff \text{a bad state is reachable within K steps},
\]

so

\[
\boxed{\operatorname{UNSAT}(\Phi_{N-1}(S))
\iff \text{S is safe at every path length}.}
\]

The checker independently reconstructs the formula and checks either all
bits of a satisfying assignment or a finite DPLL refutation tree with
clause-justified unit inferences and both children of every split. Shorter
horizons retain bounded conclusions. Explicit graph size is tracked; a
polynomial-size encoding in N does not give polynomial cost in a succinct
description with only log(N) state bits. Ordinary graph traversal already
decides this finite reachability problem, so the bridge gives no general
polynomial SAT algorithm.

### M.3. Transfer through a verified state map

A source-to-target map must preserve initial states, send bad source states
to bad target states, and send every source edge to a target edge or a
stutter. A hypothetical bad source path would then map to a bad target
path after stutters are removed. Therefore checked target safety implies
source safety. The converse is not supplied by these hypotheses.

The frozen demonstration reduces five source states to three target states,
and the corresponding SAT instance from 25 variables and 77 clauses to
9 variables and 20 clauses. Every source edge is checked. A separate unsafe
example yields [0,2,3,4]. This is actual finite proof transfer between two
representations, with no claim that either graph exactly encodes the full
fluid PDE or the universal arithmetic and complexity targets.

The new modules pass 16 targeted test methods, including 640 graph/horizon
comparisons, 80 small CNF comparisons, and 64 exact matrix examples.
Six positive certificate controls pass and 32 deliberate corruptions are
rejected. Earlier analytic certificates remain frozen and are not enlarged
by this release. The detailed proofs, schemas, executed records, and Windows
commands are in `UNIVERSES_AND_PROOF_BRIDGES.md` and the v0.19 engine.

The official problem statements remain the reference for the universal
claims: [RH](https://www.claymath.org/wp-content/uploads/2022/05/riemann.pdf),
[P versus NP](https://www.claymath.org/wp-content/uploads/2022/06/pvsnp.pdf),
and [Navier–Stokes](https://www.claymath.org/wp-content/uploads/2022/06/navierstokes.pdf).
These program checks and written arguments are not formal proof-assistant
verification. No universal bridge, new seven-dimensional physical law, or
Millennium solution follows from this release.

---

## Appendix N. Null-axis probes, observability, and a fluid projection obstruction

Version 0.20 tests the proposal that observing both sides of a controlled
push can expose hidden coordinates. It gives exact conditions for success,
complete ambiguity descriptions in a nonlinear model, and a written limit
on one proposed finite projection of fluid dynamics. No universal prize
problem or new physical dimension is established.

### N.1. Linear hidden directions and allowed transformations

For a linear readout y=Cx, two states are indistinguishable precisely when
their difference lies in ker(C). Known additive pushes do not remove this
ambiguity because C(x+u)-C(x'+u)=C(x-x'). Hidden distinctions occur at any
observed coordinate, not only at a chosen zero origin. A zero observation
does not imply a perfect shape or an absence of upper-state information.

If known allowed transformations A_i can mix hidden and visible coordinates,
form W from all rows C*A_last*...*A_first over finite experiment words.
Then

\[
x\sim x'\iff x-x'\in W^\perp.
\]

The algorithm starts with rows of C and closes their span under right
multiplication by every allowed map. At most n independent rows survive
for an n-dimensional state. Each has an explicit word witness. Inclusion
of C and closure prove coverage of every finite word by induction; the
witnesses prove that no unattainable measurement was added. A standalone
checker verifies both directions and a full basis of the annihilator.

The three-coordinate example has one hidden direction under identity
dynamics and none when an x-z quarter rotation is allowed. A fourth untouched
coordinate remains hidden. A seven-coordinate shift exposes the initially
seventh coordinate only after six transitions. The number seven is an
example of the general finite-dimensional result, not a physical deduction.
Combining multiple experiment words requires reproducible initial conditions
or a justified measurement protocol.

### N.2. Signed recovery from paired squared-distance readings

With h(x)=||x||^2, paired displacements give

\[
h(x+u)-h(x-u)=4u\cdot x.
\]

Choosing u=delta*e_i recovers x_i exactly by dividing the contrast by
4*delta, for nonzero delta. The implemented example recovers
(3/5,-2/7,4/3) with delta=1/8. Under reading errors individually bounded
by eta, the coordinate error is at most eta/(2|delta|). Smaller probes can
therefore amplify reading error.

The complete data also contain the common squared norm through paired sums.
Let U contain the probe vectors, d their contrasts divided by four, and s
the squared norm inferred from each sum. If x0 is the minimum-norm solution
of Ux=d, the exact fiber is

\[
\{x_0+v:v\in\ker U,\ \|v\|^2=s-\|x_0\|^2\}.
\]

Orthogonality of x0 and ker(U) proves the formula in both directions.
The checker distinguishes an empty fiber, a unique point, two points, or
a sphere of compatible states. A zero remaining radius can give uniqueness
even with a rank-deficient probe matrix. These results assume the stated
sensor law and repeated access to the same initial state.

### N.3. No autonomous law for any fixed energy-derivative jet of all shears

For a periodic smooth shear with positive finite energy weights w_k,

\[
u(y,t)=\left(2\sum_k\sqrt{w_k}e^{-\nu k^2t}\sin(ky),0,0\right),
\]

incompressibility and zero advection reduce the full unforced Navier–Stokes
equation to the heat equation. With normalized energy E=average(|u|^2)/2,

\[
E^{(j)}(0)=(-2\nu)^j\sum_k w_k k^{2j}.
\]

For every finite m, take m+2 squared wave numbers lambda_i=i^2 and
v_i=1/product_(l!=i)(lambda_i-lambda_l). Comparing highest coefficients
in the Lagrange interpolation formula gives

\[
\sum_i v_i\lambda_i^j=0\ (0\le j\le m),\qquad
\sum_i v_i\lambda_i^{m+1}=1.
\]

The positive and negative parts of v therefore realize two global smooth
shears with equal energy derivatives through order m and unequal next
derivatives. Their common projected state z_m=(E,...,E^(m)) has different
projected time derivatives. Consequently no single-valued autonomous rule
z_m'=F(z_m) describes all these shears. This is an explicit failure of the
earlier law-descent criterion for that projection.

At viscosity one and wave numbers 1,2,3,4, integer-scaled weights
(0,14,0,2) and (7,0,9,0) give equal E=16, E'=-176, E''=2944, but
E'''=-72704 and -52544 respectively. An integer-amplitude realization is
checked independently with the frozen full Fourier algebra, including zero
divergence, zero nonlinear term, and all four energy derivatives.

When exactly four distinct frequencies are known and no others occur,
four energy derivatives recover the weights through a Vandermonde system.
The new observability engine executes that recovery. This determines energy
evolution in the specified shear model; it does not identify every velocity
phase or prove a finite cutoff for the general PDE. Both alias examples
remain globally smooth. The theorem concerns finite energy-derivative jets,
not every possible observation or intervention.

### N.4. Two further shortcut failures and validation

The entire polynomial F(s)=w^4+(15/8)w^2+289/256, w=s-1/2, has reflection
and conjugation symmetry but roots s=1/2 +/- 1/4 +/- i. Exact Gaussian
arithmetic checks all four. Thus those symmetries alone do not force roots
onto the central line; this polynomial is not the Riemann xi function and
does not refute RH.

For n>=3, the satisfiable CNF consisting of the all-positive clause and
every implication x_i implies x_j (i!=j) has violation count
H(k)=1_(k=0)+k(n-k), where k is the number of true variables. The all-zero
state has cost one, each one-bit neighbor has cost n-1, and the all-one
state has cost zero. Local one-bit descent can therefore stop above the
satisfying set. This is not a lower bound for all algorithms.

The release passes 18 targeted test methods and 31 corruption rejections.
Twenty-three standalone certificates cover the main examples and shear
orders zero through twelve. The all-m theorem has the interpolation proof
above and in `NULL_AXIS_OBSERVABILITY.md`; finite runs are not its proof.
There is no mathematical novelty or formal proof-assistant verification claim.

The three prize targets retain their original quantified definitions:
[RH](https://www.claymath.org/wp-content/uploads/2022/05/riemann.pdf),
[P versus NP](https://www.claymath.org/wp-content/uploads/2022/06/pvsnp.pdf),
and [Navier–Stokes](https://www.claymath.org/wp-content/uploads/2022/06/navierstokes.pdf).
An assumed finite grid would require a proved equivalence to those domains
before its computation could settle them. Consistency with a proposed model
does not itself establish decidability, efficient solvability, or truth of
every conjecture expressed in it.

## Appendix O. Active experiments, model rejection, and correction of the hierarchy

Revision 1.14 is accompanied by Rope Tomography Lab 0.2.0 and the focused report Reality_Revision_1.14.md. It makes a substantive correction to Section 17: complete deterministic closure for a fixed transition family is idempotent. Seven strict repeated complete closures therefore cannot satisfy the former theorem's premises. A seven-bit shift register demonstrates a valid seven-stage finite-horizon hierarchy, with no implication for the number of physical dimensions.

### O.1. Intervention-relative cause classes

For allowed experiments \(\mathcal E\) and predicted responses \(F_e(\theta)\), define

\[
\theta\sim_{\mathcal E}\theta'
\iff F_e(\theta)=F_e(\theta')\quad\forall e\in\mathcal E.
\]

Use equality of full response distributions for a general stochastic model. The recoverable object is an equivalence class in \(\Theta/{\sim_{\mathcal E}}\). Increasing the allowed experiments can refine that quotient, but repeating observationally equivalent measurements need not refine it.

In the lab's conditionally independent common-noise model, adaptive choice cannot distinguish causes equivalent under every allowed experiment. Equal past-data distributions induce equal distributions for the next experiment chosen from that history; its response distributions also agree. Induction establishes equality of the entire transcript distributions.

This formalizes the distinction between improving numerical estimation and obtaining a new identifying observation. Optimal experimental design is an existing research area; a relevant primary research preprint studies control inputs for parameter identifiability and model discrimination. The lab implements a finite catalogue method rather than that paper's continuous optimal-control construction. [Liu, Maini and Baker, 2025](https://arxiv.org/html/2506.11311v1)

### O.2. Choose a discriminating experiment before seeing its outcome

For fixed candidate means \(\mu_{ei}\), weights \(w_i\), known common Gaussian covariance \(\Sigma_e\), and positive cost \(c_e\), the implemented criterion is

\[
S(e)=\frac1{c_e}\sum_{i<j}w_iw_j
(\mu_{ei}-\mu_{ej})^\top\Sigma_e^{-1}(\mu_{ei}-\mu_{ej}).
\]

The algorithm maximizes this pairwise-KL criterion over a finite affordable catalogue. It returns no informative selection when all affordable scores are numerically zero. It does not optimize over every physically realizable apparatus or generate every plausible hidden mechanism. Its current pairwise evaluation costs \(O(|\mathcal E|M^2r)\), excluding the cost of predicting means.

It updates relative candidate probabilities, then separately checks absolute residual compatibility against the known-noise distribution. If every candidate fails, it records model inadequacy rather than reporting its highest-posterior candidate as established truth. The statistical cutoff is per observation and is not a sequentially adjusted error guarantee.

### O.3. A concrete intervention that breaks an exact delay ambiguity

When \(\tau\) is transit delay and \(\delta\) local pure delay, passive measurements can identify only their sum. If a calibrated intervention multiplies the path delay by \(\lambda\ne1\) while preserving speed and the local attachment,

\[
t_1=\tau+\delta,\qquad t_\lambda=\lambda\tau+\delta.
\]

Solving gives

\[
\tau=\frac{t_\lambda-t_1}{\lambda-1},\qquad
\delta=\frac{\lambda t_1-t_\lambda}{\lambda-1}.
\]

These equations use unwrapped delays. Wrapped phases still require cycle resolution; unknown intervention side effects invalidate the assumed two-equation model. The new simulation discriminates fixed candidate pairs and supplies no physical implementation of the intervention.

### O.4. Evidence and limits of this release

The recorded protocol fixes six scenarios, each with 2,000 synthetic observations, plus a 2,000-observation missing-model control. Integer-frequency phase aliases, path/local-delay sums, and uncalibrated gain/depth products remain ambiguous. Respectively, an off-harmonic probe, a known path scaling, and independent coupling calibration distinguish the selected pairs.

In the three informative scenarios, forced nearest-mean errors were 12, 16, and 37 out of 2,000. These counts include observations where the absolute check rejected both candidates; they are not error rates conditional on acceptance. All three passive controls retained tied candidate probabilities in every trial.

In all 2,000 missing-model trials, one incorrect candidate had over 99% relative probability, yet the absolute check rejected both. This is a constructed counterexample to using posterior concentration alone as model validation.

An independent audit recalculated all 14,000 observation records and catalogue scores without importing the selector or simulator. The combined unit suite passed 15 groups. The exact closure audit passed 7,662 partition/transition cases and checked the seven-bit shift example. Written proofs establish the stated general identities; numerical experiments are scoped implementation evidence. No new physical measurements, extra-dimensional discovery, P=NP proof, or formal proof-assistant certification is supplied.

### O.5. A revisable theory

The machine-readable claim ledger stores assumptions, status, evidence, and failure conditions. It explicitly retracts the former seven-stage closure argument. Earlier evidence remains tied to its named release and source, rather than being silently promoted to support later claims.

The continuing research task is to identify a precise missing distinction, declare competing models, design a feasible discriminating experiment, and test a prediction excluded from construction. If the candidate set is inadequate, revise the models and reserve fresh validation data. If the permitted observations cannot distinguish the candidates, preserve the equivalence class. Metaphysical \(G\) is not a numerical repair term for model failure.

## Appendix P. Capture, generator discovery, and the limits of an observed cycle

Revision 1.15 incorporates an audit of RRLS Studio v0.27 and the uploaded dimensional-future report. The repaired source is release 0.27.1, documented in RRLS_v27_SOURCE_AUDIT.md. Original source snapshots and failures are retained. New results are distinct from the original benchmark record.

### P.1. Keep the event model's semantics explicit

The modular event solver finds integer ticks satisfying simultaneous rational linear congruences. It does not enforce continuous non-crossing. For example, \(x_A(t)=t\) and \(x_B(t)=1-t\) modulo 10 never coincide at an integer tick, because \(2t\equiv1\pmod{10}\) is impossible for integer \(t\). They nevertheless intersect at \(t=1/2\).

Persistent point capture is an additional rule. Each nontrivial merge reduces the cluster count, so an initially \(N\)-cluster merge-only model has at most \(N-1\) merge events. That bound does not count the internal deformation dynamics of extended strands. Bond creation changes a contact graph; non-crossing motion alone does not establish a change of knot or link type.

The uploaded mechanics text also mixed torque and angular impulse. For an ideal instantaneous contact at offset \(\mathbf r_c\),

\[
\Delta\mathbf L=\mathbf r_c\times\mathbf J_t,
\]

where \(\mathbf J_t\) is tangential impulse. It already integrates force over time and must not receive another time factor. General offset rigid-body impacts require rotational inertia and relative contact-point velocities. The repaired source's impulse helper remains explicitly limited to central/point-mass contact.

### P.2. Finite observations do not select a universal generator language

For any positive integer \(N\), the compact periodic laws

\[
a(t)=0,\qquad b(t)=\mathbf1[t\bmod(N+1)=N]
\]

agree on the first \(N\) integer samples but differ on the next. A deterministic learner given that common prefix has identical input in both cases and therefore cannot guarantee the correct next value for both. A previously known exact period would be an additional assumption; a finite observed repetition does not establish it.

This obstruction does not rule out useful generator learning within a restricted language.

### P.3. Positive theorem: continuation in a bounded recurrence class

Fix a prime field and a bound \(K\). Suppose the complete noiseless sequence satisfies a homogeneous constant-coefficient linear recurrence of order at most \(K\), valid from its initial state.

**Theorem.** Its first \(2K\) consecutive terms uniquely determine its continuation within that class.

**Proof.** For two candidate sequences, let \(P\) and \(Q\) be monic annihilating polynomials in the forward shift, each of degree at most \(K\). Their product annihilates the difference and has degree at most \(2K\). Agreement of the first \(2K\) terms gives enough zero initial values for this monic recurrence to force the difference to remain zero. \(\square\)

This determines the sequence, not necessarily a unique coefficient representation. It neither discovers nor proves the physical applicability of the recurrence class. Linear feedback synthesis is established mathematics; the new implementation uses bounded-order modular elimination rather than claiming a new general method. [Massey, 1969](https://ieeexplore.ieee.org/document/1054260/)

The implementation requires exact consecutive field-valued samples and declares \(p\le65521\), \(K\le64\). It rejects insufficient sample coverage or a prefix incompatible with every tested order. Once a recurrence is fitted, companion-matrix exponentiation computes a future term in \(O(r^3\log t)\) field operations, with arithmetic bit costs still present.

### P.4. New computation and repaired failures

The source audit reproduced weak-damping cancellation that returned zero for positive winding and zero time for a positive wrapping target. Stable expm1/log1p evaluation repairs those cases. The audit also repaired an earliest-event error for negative starting ticks, a verifier that accepted unsupported impossibility claims, and an impulse helper that did not distinguish approach from separation under the stated normal convention.

The new ten-group suite passed 1,200 rational pair worlds checked over full relative periods, 120 capture regressions, 21 high-precision damping configurations, 500 learned recurrence sequences, and 14,000 values excluded from fitting. Six exact finite-prefix examples retain the unrestricted-discovery obstruction. The five original compact self-tests also passed; the complete inherited benchmark corpus was not rerun.

From \(0,1,1,2,3,5\) over GF(257), with order bound three, the new learner recovers coefficients \((1,1)\) and computes

\[
a_{10^{96}}=160\pmod{257}.
\]

Independent Fibonacci fast doubling agrees. This validates a scoped learned-generator jump, not a universal predictor or a physical dimensional claim. The earlier P versus NP proof requirements remain unchanged.

## Appendix Q. Repeated model checks with a bounded false-rejection budget

Revision 1.16, 6 September 2026, adds Rope Tomography Lab 0.3.0. The source paper was its latest saved v1.15; the rope archive was 0.2.0; the requested v0.27 file identity now resolves to RRLS_Studio_v27_Audited_Source.zip, including the subsequent reconciled audit. The v0.28 generator, v0.29 semantic-canonicalization, and v0.30 dual-phase reports were also consulted to avoid treating their existing results as new. No RRLS source was changed or its inherited corpus rerun here.

### Q.1. A single-observation cutoff is not an investigation-wide guarantee

Appendix O correctly limited its 1% gate to each observation. If a correct candidate is tested on independent, continuously distributed, correctly calibrated residuals at that same level for T looks, the probability of at least one false rejection is

\[
1-(1-0.01)^T.
\]

At T=200 this is about 86.602%. Two observationally identical correct candidates are rejected together, so this is also an explicit counterexample to interpreting repeated all-candidate rejection as having a 1% investigation-wide error rate. This does not retract the earlier single-observation results; it rules out an unsupported extension of them.

### Q.2. Written guarantee and its exact assumptions

Fix a finite candidate catalogue before validation. Experiment e_t may be selected using previous observations, but before seeing the next one. Assume at least one fixed candidate i* supplies the correct conditional Gaussian mean and known positive diagonal covariance for every selected experiment. Fresh standardized residual coordinates must have the specified conditional standard-normal law; marginal Gaussianity alone is insufficient after adaptive selection. No parameter fitting to the scored observation is permitted.

Let p_(i,t) be the upper chi-square tail probability of the squared standardized residual. Allocate

\[
\alpha_t=\frac{\alpha}{t(t+1)},\qquad
\sum_{t=1}^{T}\alpha_t=\alpha\frac{T}{T+1}<\alpha.
\]

Permanently remove candidate i if p_(i,t)<alpha_t. Conditional calibration implies P(p_(i*,t)<alpha_t | past)<=alpha_t. Taking expectations and applying the countable union bound gives

\[
\Pr\{\text{ever remove }i_*\}\le\sum_{t\ge1}\alpha_t=\alpha.
\]

If i* is present initially, ever removing every candidate implies removing i*, so it has the same bound. No factor for candidate count is required for this specific claim. Simultaneously preserving every true candidate is a different, stronger claim and is not asserted. Stopping or choosing experiments based on past data does not break this argument. Selecting the largest of two current residuals does: even if each arm alone has a valid p-value, its selected p-value rejects with probability 1-(1-alpha_t)^2>alpha_t for independent arms.

This is a written probability proof, not a proof-assistant certificate. It is standard error-budget reasoning, not a newly discovered statistical theorem. The numerical implementation uses chi-square survival probabilities directly, avoiding late-look cancellation in 1-alpha_t; it explicitly supports alpha>=10^-12 and at most 10^9 looks. Floating-point tail evaluation is not interval-certified.

### Q.3. Fresh synthetic evidence and an independent check

The protocol froze source hashes, seed 606202603, two calibrated experiment settings, and three scenarios before generating validation data. Each scenario has 2,000 independent histories of 200 observations. Experiment choice depends on the preceding standardized first residual. Within each experiment the two candidates make identical predictions: retaining them must preserve their ambiguity.

| Scenario | Repeated 1% gate: histories ever rejected | Budgeted gate: histories ever rejected |
|---|---:|---:|
| Correct conditional noise | 1,717 / 2,000 | 14 / 2,000 |
| Persistent first-coordinate shift of 0.35 noise standard deviations | 1,864 / 2,000 | 26 / 2,000 |
| Actual noise standard deviation twice the declared value | 2,000 / 2,000 | 2,000 / 2,000 |

Only the first row satisfies the full null model. The second measures limited detection power; the third shows why a correct mean with incorrect noise calibration is not covered by the guarantee. Non-rejection establishes neither truth nor good sensitivity. For the correct-noise scenario, independent product formulas give finite-horizon false-rejection probabilities 86.6020% and 0.991530%; the sample rates need not equal them.

An audit that imports neither the engine, simulator, nor scipy.stats reconstructs the adaptive choices and uses the analytic two-coordinate tail exp(-q/2). It agrees on 2,400,000 rejection bits from 1,200,000 stored synthetic observations. It shares the declared assumptions, observations, and NumPy; this is independent implementation checking, not external replication. The stateful API was additionally replayed on ten histories per scenario, not counted as extra data. All 25 unit tests pass, including the 15 inherited tests.

### Q.4. Research context, model revision, and next test

Time-uniform inference is established research; Howard et al. (Annals of Statistics, April 2021) develop more sophisticated confidence sequences: https://arxiv.org/abs/1810.08240v9. Koning and van Meer (preprint revised 21 December 2025) give sequential tests matching a specified fixed-horizon test at its terminal horizon: https://arxiv.org/abs/2501.03982v5. A newly located application by Grob, Daniele, and Ziegel (preprint submitted 24 August 2026) evaluates forecast calibration sequentially with e-values: https://arxiv.org/abs/2608.23064v1. These are relevant statistical methods and applications, not physical evidence for this framework. This release implements conservative alpha spending, not those papers' constructions.

The next bounded test is a frozen cumulative Gaussian likelihood-ratio/e-process against small persistent shifts, compared at the same 1% ever-false-rejection budget on new seeds. It is not implemented or validated here. In a continuing investigation, refitting candidates requires fresh validation and an explicitly allocated remaining or cross-revision error budget. Resetting a test to 1% after every revision does not retain a global 1% guarantee. Neither this gate nor a concentrated posterior resolves observational equivalence, identifies God as a residual, establishes seven physical dimensions, or decides P versus NP. All v1.14 and v1.15 corrections, including the initial-state bounded recurrence-class assumption, remain in force.

## Appendix R. Accumulating weak residual evidence without optional-stopping inflation

Revision 1.17, 7 September 2026, adds Rope Tomography Lab 0.4.0. It starts from the saved paper v1.16 and rope archive v0.3.0. The requested RRLS v0.27 identity still resolves to the audited v0.27 source and was read for its current claim ledger; RRLS code and evidence are unchanged in this revision.

### R.1. Bounded target and assumptions

Revision 1.16 found that allocating a shrinking significance level to each new residual controls false rejection but detected only 26 of 2,000 histories containing a persistent standardized mean shift of 0.35. The bounded target here is that same failure mode, not arbitrary misspecification.

Fix a candidate i, a finite catalogue of experiments, a significance level alpha, a positive target shift delta, and one unit direction u_e in whitened measurement coordinates for each experiment. The experiment and direction at time t may depend on the past but must be selected before observing Y_t. If i is correct, assume

\[
Z_{i,t}=u_{e_t}^{\mathsf T}\Sigma_{e_t}^{-1/2}(Y_t-\mu_{e_t,i})
\mid\mathcal F_{t-1}\sim N(0,1).
\]

The conditional statement is essential. Marginal normality, reused data, current-response-dependent direction choice, fitted predictions scored on the same response, or misspecified covariance do not satisfy it.

Put S_(i,t)=sum_(s=1)^t Z_(i,s). For fixed alternatives with standardized shifts +delta and -delta, define

\[
L_{i,t}^{+}=\exp\!\left(\delta S_{i,t}-\frac{t\delta^2}{2}\right),\qquad
L_{i,t}^{-}=\exp\!\left(-\delta S_{i,t}-\frac{t\delta^2}{2}\right),
\]

and their equal mixture

\[
E_{i,t}=\frac{L_{i,t}^{+}+L_{i,t}^{-}}{2}
=\exp\!\left(-\frac{t\delta^2}{2}\right)\cosh(\delta S_{i,t}).
\]

### R.2. Written anytime-valid guarantee

Under the declared null,

\[
\mathbb E[\exp(\pm\delta Z_{i,t}-\delta^2/2)\mid\mathcal F_{t-1}]=1.
\]

Therefore L+ and L- are nonnegative martingales starting at one, and so is their mixture E. Ville's inequality gives

\[
\boxed{\Pr_i\!\left(\sup_{t\ge0}E_{i,t}\ge1/\alpha\right)\le\alpha.}
\]

This is a written probability proof, not a proof-assistant artifact or a new theorem. If one correct candidate is present, ever rejecting the entire candidate set by this rule implies rejecting that candidate and inherits the bound. It does not simultaneously protect every correct candidate without an additional multiple-testing argument. Nor may this 1% process and the separate 1% point-residual gate be joined with an OR rule and called a 1% procedure; a valid combined rule must allocate or merge their error budgets.

The engine stores log e-values using a stable log-cosh identity. It copies the candidate predictions, noise scales, and normalized directions at construction. It cannot detect dishonest data reuse, post-response experiment selection, or a caller resetting the object. Floating-point evaluation is tested but not interval-certified.

### R.3. Frozen synthetic validation

Before outcome generation, the protocol fixed source hashes, seed 1709202604, 2,000 histories, 200 looks, alpha=0.01, delta=0.35, two experiment settings, and a past-dependent experiment policy. Both modeled candidates make identical predictions, so their evidence paths must remain tied. The e-process and alpha-spending gate are compared separately at 1%; this table is not a combined decision rule.

| Synthetic scenario | Cumulative e-process ever rejected | Alpha-spending gate ever rejected | Median e-process first rejection among rejected |
|---|---:|---:|---:|
| Correct conditional Gaussian model | 20 / 2,000 | 23 / 2,000 | 60 |
| Constant +0.35 shift in monitored standardized coordinate | 1,896 / 2,000 | 28 / 2,000 | 75 |
| Constant -0.35 shift in monitored standardized coordinate | 1,906 / 2,000 | 27 / 2,000 | 74 |
| Alternating +0.35/-0.35 shift | 18 / 2,000 | 31 / 2,000 | 72 |
| Constant +0.35 shift in the orthogonal coordinate | 15 / 2,000 | 22 / 2,000 | 64 |
| Actual noise scale twice the declared scale | 865 / 2,000 | 2,000 / 2,000 | 50 |

Only the first row satisfies the complete null. Its observed 1% e-process rejection fraction is a finite sample, not the proof of the guarantee. The two constant-shift rows answer the prespecified power question: the cumulative process detects about 95% of those histories, whereas the shrinking point gate detects about 1.4%. The alternating and orthogonal rows are deliberate counterexamples to an omnibus interpretation. The variance row shows complementary behavior: the point-residual gate detects all tested histories, while the mean-shift e-process does not.

An independent audit imports neither the cumulative engine, simulator, nor SciPy. It reconstructs the predictable experiment choices, uses a separate scalar stable-log-cosh calculation, and independently evaluates the exact two-degree-of-freedom survival formula for the comparison gate. It agrees on 4,800,000 decision states from 2,400,000 stored synthetic observations. The stateful public API is replayed on 60 of those histories, not counted as new evidence. All 33 unit tests pass, including all 25 inherited tests. The audit shares the raw records, assumptions, standard-library mathematics, and NumPy loading/median; it is implementation independence, not external replication.

### R.4. Relation to current primary research

The use of nonnegative martingales and Ville thresholds belongs to established safe anytime-valid inference, not to the physical claims of this framework. Kaufmann and Koolen construct mixture martingales for time-uniform inference under adaptive sampling in exponential-family bandits: https://jmlr.org/papers/volume22/18-798/18-798.pdf. Arnold, Gavrilopoulos, Schulz, and Ziegel's *Sequential model confidence sets*, published 13 May 2026, uses e-processes, confidence sequences, and multiple-testing corrections to retain time-uniform nonasymptotic model-set coverage: https://academic.oup.com/jrsssb/advance-article/doi/10.1093/jrsssb/qkag066/8676743. Wang and Ramdas's Gaussian-mean work, updated 24 August 2026, develops e-processes when variance is unknown: https://arxiv.org/html/2310.03722v4. The present lab implements only the elementary known-variance two-point mixture above; it does not reproduce those papers' broader results.

### R.5. What changed and what remains open

The supported change is methodological: weak, directionally consistent residuals can be accumulated with an anytime false-rejection guarantee rather than tested in isolation. This does not validate the rope physics, identify a unique hidden cause, or turn non-rejection into truth. Delta=0.35 and the monitored direction were prespecified from revision 1.16's named failure; performance for other magnitudes and directions is not inferred.

The next bounded test is a variance-robust or self-normalized e-process, frozen before new outcomes, because the current known-covariance assumption is the most consequential fragility. A separate design should then combine complementary point and cumulative detectors with one total error budget. All earlier corrections remain: complete deterministic closure for one fixed transition family is idempotent; integer-tick point capture is not continuous non-crossing; angular impulse has no extra time factor; finite prefixes do not identify arbitrary generators; and the 2K continuation theorem assumes an exact noiseless homogeneous constant-coefficient recurrence of order at most K valid from the initial state. No unconstrained residual is identified with God, and neither seven physical dimensions nor P=NP is established.

## Appendix S. Unknown fixed scale without pretending to know the variance

Revision 1.18, 8 September 2026, adds Rope Tomography Lab 0.5.0. It starts from the saved paper v1.17, rope archive 0.4.0, and the current claim ledgers. The requested RRLS identity still resolves to the audited v0.27 package. Its code and evidence are unchanged, and no inherited RRLS benchmark is relabeled as fresh validation.

### S.1. Bounded question and exact model class

Appendix R's cumulative test assumes the variance used to standardize residuals is correct. Its doubled-noise control rejected 865 of 2,000 histories, even though that experiment changed scale rather than mean. The bounded question here is whether a sequential mean check can remain valid when one common positive Gaussian residual scale is unknown.

Fix one candidate and, before each response, a scalar projected residual $X_t$. Under the null assume

\[
X_1,X_2,\ldots\overset{\mathrm{iid}}{\sim}N(0,\kappa^2)
\]

for one fixed unknown \(\kappa>0\). Nominal per-coordinate scales may define the projection, but after that projection the residual stream must have this common-scale law. Independence, Gaussianity, a zero mean, and a time-constant scale are all assumptions. Past-adaptive experiment selection is not covered merely because it is predictable; it is permitted only when the resulting projected stream still has the stated iid law.

Let $S_n=\sum_{t=1}^nX_t$, $V_n=\sum_{t=1}^nX_t^2$, and predeclare $c>0$. The implemented statistic is the scale-invariant Gaussian likelihood mixture of Wang and Ramdas:

\[
G_n^{(c)}=
\sqrt{\frac{c^2}{n+c^2}}
\left(
\frac{(n+c^2)V_n}{(n+c^2)V_n-S_n^2}
\right)^{n/2}.
\]

For nonzero $V_n$, Cauchy--Schwarz and $c>0$ make the denominator positive. Multiplying the complete prefix by any nonzero constant leaves $G_n^{(c)}$ unchanged. The implementation uses $c=1$, stable logarithms, and reports exact all-zero prefixes as insufficient scale information instead of treating the formula's degenerate case as evidence.

### S.2. Established theorem and a crucial filtration boundary

Wang and Ramdas prove that $G_n^{(c)}$ is a nonnegative martingale with initial value one for the scale-invariant Gaussian null on their reduced, scale-invariant filtration. Ville's inequality therefore gives

\[
\Pr\!\left\{\sup_{n\ge1}G_n^{(c)}\ge1/\alpha\right\}\le\alpha
\]

when rejection and stopping are measurable using that scale-invariant evidence. This is an application of their published mathematical result, not a new theorem proved by this project. It is also narrower than a canonical-filtration claim: stopping because a raw residual magnitude or a running variance estimate crossed a threshold may reveal scale information and is not licensed by this argument. Data reuse, refitting on scored observations, resetting the process, post-response direction choice, non-Gaussian errors, dependence, or changing variance can likewise invalidate the guarantee.

Source: Hongjian Wang and Aaditya Ramdas, *Anytime-valid t-tests and confidence sequences for Gaussian means with unknown variance*, arXiv:2310.03722v4, updated 24 August 2026: https://arxiv.org/html/2310.03722v4.

### S.3. Frozen finite computation and synthetic tests

Before generating outcomes, the protocol fixed source hashes, seed 1809202605, $c=1$, alpha=0.01, 2,000 histories, 200 looks, and six scenarios. Two candidates had identical zero predictions, so the engine also had to preserve their observational ambiguity. The scale-two Gaussian records were exact positive rescalings of their scale-one partners; they are a pathwise invariance control, not an independent sample.

| Synthetic scenario | Scale-invariant process ever rejected | v1.17 known-scale process ever rejected | Median first scale-invariant rejection among rejected |
|---|---:|---:|---:|
| iid (N(0,1)) | 15 / 2,000 | 12 / 2,000 | 51 |
| exact 2x rescaling, iid (N(0,4)) | 15 / 2,000 | 848 / 2,000 | 51 |
| iid (N(0.35,1)) | 1,811 / 2,000 | 1,915 / 2,000 | 94 |
| exact 2x rescaling, iid (N(0.70,4)) | 1,811 / 2,000 | 2,000 / 2,000 | 94 |
| alternating means (+0.35,-0.35) with unit Gaussian noise | 9 / 2,000 | 16 / 2,000 | 25 |
| variance-one scaled Student (t_3) | 3 / 2,000 | 37 / 2,000 | 12 |

Only the first two rows satisfy the full mean-zero theorem. Their 15/2,000 count is finite synthetic evidence, not the proof of the 1% bound. Exact equality of both scale-invariant decision paths is a finite computational check of the algebraic scale invariance. The old process's 848/2,000 scale-two count is not a failure of its theorem: those data violate its known-unit-variance null.

The paired shift rows have the same standardized effect and exactly the same scale-invariant decisions. They show 1,811/2,000 targeted detections at this horizon, less than the v1.17 known-scale process's scale-one count. The alternating row is a retained counterexample: a constant-mean detector is not an omnibus adequacy test. The Student row violates Gaussianity; its low rejection count neither proves validity nor adequacy under heavy tails.

Nine new unit tests compare the formula to high-precision arithmetic, check scale and sign invariance, preserve tied candidates, and exercise invalid and zero-energy inputs. Together with 33 inherited tests, all 42 pass. An audit importing neither production evidence code, the simulator, nor SciPy independently reconstructs all 4,800,000 decision states from 2,400,000 stored synthetic residuals and agrees exactly. It shares the data and assumptions and is not external replication. An initially truncated result archive is recorded as a packaging failure; the unchanged frozen sources and seed reproduced the same summary before the intact archive was audited.

### S.4. Result, limits, and next test

The supported correction is methodological: known variance is unnecessary for this particular scale-invariant Gaussian mean test. It does not make the overall theory variance-free, distribution-free, or physically validated. In particular, it deliberately ignores pure scale changes; the doubled-scale null is acceptable within its model class. Detecting variance misspecification requires a separate detector and a shared error budget.

The next bounded test is a joint point-plus-cumulative procedure with one total 1% anytime false-rejection budget, evaluated against both mean and scale errors on fresh data. Candidate or detector redesign must reserve a new seed and validation set. All earlier corrections remain in force: complete deterministic closure for one fixed transition family is idempotent; integer-tick point capture is not continuous non-crossing; angular impulse has no extra time factor; finite prefixes do not identify arbitrary generators; and the 2K continuation theorem assumes an exact noiseless homogeneous constant-coefficient recurrence of order at most K valid from the initial state. No unconstrained residual is identified with God, and neither seven physical dimensions nor P=NP is established.

## Appendix T. One shared error budget for complementary model checks

Revision 1.19, 9 September 2026, adds Rope Tomography Lab 0.6.0. It starts from the saved paper v1.18, rope archive 0.5.0, and their current ledgers and revision notes. The audited RRLS v0.27 package was checked for the retained capture, impulse, generator-class, and P-versus-NP boundaries. RRLS code and evidence are unchanged.

### T.1. The composition problem

Appendix R warned that joining two 1% detectors with an OR rule does not create a 1% procedure. Appendix S then supplied a cumulative mean process whose martingale theorem holds on a reduced scale-invariant filtration. The point-residual gate, by design, reads raw standardized magnitudes. Therefore these two particular processes are not simply averaged and declared to be one e-process; doing so would require a common validity argument that this release does not possess.

Research on merging sequential e-values shows that legitimate merging is a substantive construction, not a license to combine arbitrary evidence streams. See Vladimir Vovk and Ruodu Wang, *Merging sequential e-values via martingales*, Electronic Journal of Statistics 18 (2024), 1185--1205: https://projecteuclid.org/journals/electronic-journal-of-statistics/volume-18/issue-1/Merging-sequential-e-values-via-martingales/10.1214/24-EJS2228.full. Wang and Ramdas also emphasize that filtration choice matters for unknown-variance sequential tests and describe union-bound constructions for simultaneous mean and variance confidence sets: https://arxiv.org/html/2310.03722v4.

### T.2. Written joint guarantee

Fix a candidate before validation and a total level $\alpha$. Allocate $\alpha_m=w\alpha$ to the scale-invariant cumulative mean process and $\alpha_p=(1-w)\alpha$ to the alpha-spending point-residual gate, with fixed $0<w<1$. This implementation uses $w=1/2$.

Let $A_m$ be the event that the mean process ever removes the specified correct candidate and $A_p$ the analogous event for the point gate. Under all assumptions of both component theorems,

\[
\Pr(A_m)\le\alpha_m,\qquad \Pr(A_p)\le\alpha_p.
\]

No independence between these two events is required. The union bound gives

\[
\boxed{\Pr(A_m\cup A_p)\le\alpha_m+\alpha_p=\alpha.}
\]

The joint engine removes a candidate if either component removes it. If at least one fixed correct candidate was present, ever eliminating the entire candidate set implies eliminating that candidate and inherits the same bound. This does not guarantee simultaneous preservation of every correct candidate.

The joint assumptions are stronger than Appendix S alone: fresh standardized residual vectors must have the candidate's declared conditional Gaussian mean and covariance for the point gate, while the monitored projection must also satisfy the iid fixed-scale Gaussian and scale-invariant-stopping contract of the mean process. No reset, scored-data refit, current-response selection, or reuse is allowed. This is a written probability argument, not a proof-assistant certificate or new statistical theorem.

### T.3. Frozen synthetic validation

Before outcomes, the protocol fixed source hashes, seed 1909202606, 2,000 independent histories per scenario, 200 looks, total alpha=0.01, equal 0.005 component budgets, and $c=1$. Two modeled candidates had identical scalar predictions, so the implementation had to preserve their ambiguity. The invalid comparator gives both components a full 1% and ORs them; it is recorded as a counterexample, not labeled a valid 1% rule.

| Synthetic scenario | Shared-budget joint rejection | Mean component | Point component | Invalid full-1%-each OR |
|---|---:|---:|---:|---:|
| iid $N(0,1)$ | 25 / 2,000 | 8 | 17 | 45 |
| iid $N(0.35,1)$ | 1,714 / 2,000 | 1,712 | 22 | 1,814 |
| iid $N(0,4)$ | 1,927 / 2,000 | 3 | 1,927 | 1,969 |
| iid $N(0.70,4)$ | 1,997 / 2,000 | 1,724 | 1,982 | 1,998 |
| alternating means $+0.35,-0.35$ with unit Gaussian noise | 31 / 2,000 | 5 | 26 | 54 |
| iid $N(0,1.25^2)$ | 141 / 2,000 | 8 | 133 | 208 |
| variance-one scaled Student $t_3$ | 1,049 / 2,000 | 4 | 1,046 | 1,113 |

Only the first row satisfies the full joint null. Its observed 25/2,000 rate is 1.25%, not a proof or disproof of the mathematical bound. A post-outcome diagnostic gives probability 0.155651 of at least 25 rejections under 2,000 independent Bernoulli trials with probability 0.01, and a two-sided 95% exact binomial interval of 0.8105% to 1.8397%; this diagnostic was not part of the frozen protocol. The theorem, not agreement of a finite count with 1%, supplies the guarantee.

The shared rule retained complementary power: 1,714/2,000 for the specified persistent mean shift, 1,927/2,000 for doubled scale, and 1,997/2,000 when both were present. Tightening the mean threshold from 1% in v1.18 to its 0.5% share reduced its mean-shift detections from 1,811/2,000 in a different seeded dataset to 1,712/2,000 here; cross-seed counts are descriptive, not a paired causal estimate of the budget cost.

The failures are equally important. Alternating shifts remained mostly invisible. A moderate scale increase to 1.25 was detected in only 141/2,000 histories because the point gate waits for isolated extremes rather than accumulating weak scale evidence. Heavy-tailed $t_3$ noise has the same variance as the null after scaling but violates Gaussianity; the joint rule rejected 1,049/2,000 histories. This is a direct counterexample to calling the procedure distribution-free or equating variance calibration with tail calibration.

Eight new unit tests and all 42 inherited tests pass. A separate scalar audit imports neither production engine, simulator, nor SciPy; it recomputes the scale-invariant statistic and uses the analytic one-dimensional Gaussian tail via `erfc`. It agrees on 11,200,000 component and joint decision states from 2,800,000 stored synthetic residuals. It shares data and assumptions and is not external replication.

### T.4. Supported change and next test

The supported change is a valid shared-budget decision rule with measured complementary synthetic power. It is not a universal model validator, does not prove any physical rope mechanism, and turns neither rejection nor non-rejection into ontology. The severe heavy-tail result says the current Gaussian point gate can confuse tail misspecification with candidate failure; whether that is useful detection or false rejection depends on the explicitly declared model.

The next bounded test is a Gaussian variance likelihood-ratio e-process that accumulates moderate scale evidence, compared against the current alpha-spending point gate at the same reserved budget and on fresh data. A later robust-tail design would require a new null class rather than relabeling Gaussian results. All earlier corrections remain in force: complete deterministic closure for one fixed transition family is idempotent; integer-tick point capture is not continuous non-crossing; angular impulse has no extra time factor; finite prefixes do not identify arbitrary generators; and the 2K continuation theorem assumes an exact noiseless homogeneous constant-coefficient recurrence of order at most K valid from the initial state. No unconstrained residual is identified with God, and neither seven physical dimensions nor P=NP is established.

## Appendix U. Accumulating moderate Gaussian scale evidence

Revision 1.20, 10 September 2026, adds Rope Tomography Lab 0.7.0. It starts from the saved paper v1.19, rope archive 0.6.0, and their current ledgers and revision notes. The requested RRLS identity still resolves to the audited v0.27 package. RRLS code and evidence are unchanged.

### U.1. Bounded question and fixed alternatives

Appendix T found that the alpha-spending point gate detected only 141 of 2,000 synthetic histories whose Gaussian residual scale was 1.25 times the declared scale. The bounded question is whether squared-residual evidence can be accumulated without sacrificing an anytime false-rejection guarantee. It is not a search over arbitrary variance paths.

For a fixed candidate, let the fresh observation at look \(t\) have declared mean \(\mu_t\) and positive diagonal covariance \(\Sigma_t\), with experiment selection completed before seeing that observation. Under the null assume

\[
Z_t=\Sigma_t^{-1/2}(Y_t-\mu_t)\mid\mathcal F_{t-1}\sim N(0,I_{d_t}).
\]

The validation predeclares two covariance multipliers,

\[
\lambda_H=1.25^2=1.5625,\qquad \lambda_L=0.8^2=0.64,
\]

with equal weights. They represent complete-covariance scale corrections, not physical mechanisms. Put \(Q_n=\sum_{t=1}^n\lVert Z_t\rVert^2\) and \(D_n=\sum_{t=1}^n d_t\). For any fixed positive \(\lambda\), the alternative-to-null likelihood ratio is

\[
L_n(\lambda)=\lambda^{-D_n/2}
\exp\!\left[\frac12\left(1-\frac1\lambda\right)Q_n\right],
\]

and the implemented evidence process is

\[
E_n=\tfrac12L_n(\lambda_H)+\tfrac12L_n(\lambda_L).
\]

### U.2. Written anytime guarantee and interpretation boundary

For a correct candidate, each one-step factor is the conditional density ratio of \(N(0,\lambda I_{d_t})\) to \(N(0,I_{d_t})\), evaluated at \(Z_t\). Its conditional null expectation is one. Therefore each \(L_n(\lambda)\) is a nonnegative martingale beginning at one, and their fixed mixture \(E_n\) is also one. Ville's inequality gives

\[
\boxed{\Pr\!\left\{\sup_{n\ge0}E_n\ge1/\alpha_v\right\}\le\alpha_v.}
\]

This is a written probability proof, not a proof-assistant artifact or a new statistical theorem. The engine uses stable log-sum-exp arithmetic and permanently removes a candidate after the running maximum reaches the threshold. Floating-point calculations are tested but not interval-certified.

The assumptions are material: the candidate and mixture must be fixed before scoring; residuals must obey the declared conditional multivariate Gaussian law; the covariance correction must scale all scored coordinates uniformly; and there may be no reset, reuse, current-response selection, or refitting on scored observations. Predictable experiment choice is allowed only while preserving that conditional law. Crossing says a prespecified predictive correction outperformed the source strongly enough for this test. It does not identify whether the cause was sensor calibration, latent dynamics, a mean error, non-Gaussian tails, or anything theological.

This interpretation follows the general construction in Seungjin Choi, *Anytime-Valid Evidence for Prespecified Predictive Corrections*, submitted 8 August 2026: https://arxiv.org/abs/2608.08174. That primary research establishes the likelihood-ratio method; it is statistical methodology, not measured evidence for this theory.

### U.3. Frozen synthetic test and independent replay

Before outcomes, the protocol fixed the four scientific source hashes, seed 201009202607, 2,000 independent histories per scenario, 200 looks, the two scale alternatives, equal mixture weights, and the same reserved \(\alpha_v=0.005\) used by Appendix T's point component. The two detectors were compared separately; they were not OR-combined. Two modeled candidates had identical predictions, so their evidence paths had to remain tied.

| Synthetic scenario | Variance e-process ever rejected | Point gate ever rejected | Median variance-process first rejection among rejected |
|---|---:|---:|---:|
| iid \(N(0,1)\) | 4 / 2,000 | 4 / 2,000 | 80.5 |
| iid \(N(0,1.25^2)\), prespecified high scale | 1,774 / 2,000 | 125 / 2,000 | 87 |
| iid \(N(0,0.8^2)\), prespecified low scale | 1,669 / 2,000 | 0 / 2,000 | 113 |
| iid \(N(0,2^2)\), off-target large scale | 2,000 / 2,000 | 1,941 / 2,000 | 13 |
| iid \(N(0,1.10^2)\), smaller untuned scale | 251 / 2,000 | 34 / 2,000 | 119 |
| iid \(N(0.35,1)\), mean error with correct variance | 58 / 2,000 | 18 / 2,000 | 107 |
| variance-one scaled Student \(t_3\) | 1,047 / 2,000 | 996 / 2,000 | 81 |

Only the first row satisfies the full null. Its four rejections are a finite synthetic outcome; the martingale proof, not this count, supplies the 0.5% anytime bound. The targeted result is a large paired-method improvement for the 1.25-scale failure at the same budget. The matched low-scale result exposes an asymmetry of the upper-tail point gate, while the 1.10-scale row preserves a power limitation rather than extrapolating from the targeted alternative.

The mean-shift and heavy-tail rows are counterexamples to identifying a crossing with a unique variance cause. In particular, the scaled Student \(t_3\) data have unit variance but violate Gaussianity, and 1,047 of 2,000 histories crossed. The process is therefore neither distribution-free nor a variance-mechanism detector.

Nine new unit tests and all 50 inherited tests pass. A separate scalar audit imports neither the production variance engine, validation runner, nor SciPy. It recomputes the likelihood mixture and Gaussian point tail and agrees on all 5,600,000 stored decision states. It shares the synthetic records, null assumptions, floating-point convention, and threshold and is not external replication or physical evidence.

### U.4. Supported change, limitations, and next test

The supported change is a standalone cumulative Gaussian scale diagnostic for fixed candidate catalogues. It improves the named moderate-scale failure under the tested model, but it does not yet replace Appendix T's point component: changing the joint decision rule after inspecting these results requires a fresh joint validation set. It also cannot turn rejection into ontology or non-rejection into truth.

The next bounded test is a frozen paired comparison of the existing mean-plus-point joint rule against a mean-plus-variance joint rule under the same total 1% budget, including persistent scale errors and isolated one-look shocks. A heavy-tail-robust guarantee would require a new null class, not reuse of the Gaussian proof. All earlier corrections remain in force: complete deterministic closure for one fixed transition family is idempotent; integer-tick point capture is not continuous non-crossing; angular impulse has no extra time factor; finite prefixes do not identify arbitrary generators; and the 2K continuation theorem assumes an exact noiseless homogeneous constant-coefficient recurrence of order at most K valid from the initial state. No unconstrained residual is identified with God, and neither seven physical dimensions nor P=NP is established.

## Appendix V. A common-filtration mean-plus-variance process

Revision 1.21, 11 September 2026, adds Rope Tomography Lab 0.8.0. It starts from the saved paper v1.20, rope archive 0.7.0, and their current ledgers and revision notes. The audited RRLS v0.27 package was checked for the retained collision, angular-impulse, recurrence-class, and complexity boundaries. RRLS code and evidence are unchanged.

### V.1. Why this is not a detector substitution

Appendix T combined an unknown-scale mean process and a point-residual gate by splitting one error budget. The mean theorem there applies on a reduced scale-invariant filtration, while the point gate reads raw magnitudes, so the release correctly refused to average them into one e-process. Appendix U then supplied a canonical-filtration variance process but left open whether it should replace the point component.

The bounded question is whether a mean and variance process can share one Gaussian observation filtration and one 1% threshold, and how that choice trades persistent-error sensitivity against isolated-shock sensitivity. This requires changing the mean component back to the known-covariance process of Appendix R. That is not a free strengthening: it abandons Appendix S's unknown-scale null. The variance component already requires calibrated covariance, so the new joint null makes that stricter assumption explicit.

### V.2. Written fixed-mixture guarantee

Fix a candidate, a positive mean-shift magnitude \(\delta\), one unit direction \(u_e\) in whitened coordinates for each experiment, positive covariance multipliers and their fixed weights, and an outer weight \(0<\rho<1\). Experiments may be chosen from the past before the current response. Under a correct candidate require

\[
Z_t=\Sigma_t^{-1/2}(Y_t-\mu_t)\mid\mathcal F_{t-1}
\sim N(0,I_{d_t}).
\]

Let \(E^{(m)}_n\) be Appendix R's equal mixture of likelihood ratios for projected shifts \(+\delta\) and \(-\delta\). Let \(E^{(v)}_n\) be Appendix U's fixed mixture of likelihood ratios for uniform covariance multipliers. Both are nonnegative martingales starting at one on the same canonical observation filtration under this null. Therefore

\[
E^{(j)}_n=\rho E^{(m)}_n+(1-\rho)E^{(v)}_n
\]

is also a nonnegative martingale starting at one, and

\[
\boxed{\Pr\!\left\{\sup_{n\ge0}E^{(j)}_n\ge1/\alpha\right\}\le\alpha.}
\]

The implementation predeclares \(\delta=0.35\), covariance factors \(1.5625\) and \(0.64\), equal inner weights, \(\rho=1/2\), and \(\alpha=0.01\). Alpha is applied once to the outer mixture. When only one component grows, its half weight imposes roughly the same evidence penalty as a 0.5% component allocation; when both grow, their current evidence can combine. The mixture uses current e-values, not separately maximized component values.

This is a written probability proof and an elementary common-filtration mixture, not a proof-assistant artifact or new statistical theorem. Vovk and Wang's work shows that merging sequential e-values is a substantive construction: https://projecteuclid.org/journals/electronic-journal-of-statistics/volume-18/issue-1/Merging-sequential-e-values-via-martingales/10.1214/24-EJS2228.full. Choi's prespecified-correction framework likewise interprets likelihood-ratio crossings relative to declared predictive corrections rather than as mechanism identification: https://arxiv.org/abs/2608.08174. The present implementation does not claim the full generality of either paper.

### V.3. Frozen paired comparison

Before outcomes, the protocol fixed four scientific source hashes, seed 211109202608, 2,000 independent histories per scenario, 200 looks, and all mixture parameters. Both rules received one total 1% anytime budget and were evaluated on identical fresh residual paths. Two candidates had identical predictions and remained tied.

| Synthetic scenario | Existing mean-plus-point rule | New mean-plus-variance mixture |
|---|---:|---:|
| iid \(N(0,1)\) | 12 / 2,000 | 10 / 2,000 |
| iid \(N(0.35,1)\) | 1,744 / 2,000 | 1,878 / 2,000 |
| iid \(N(0,1.25^2)\) | 140 / 2,000 | 1,807 / 2,000 |
| iid \(N(0,0.8^2)\) | 4 / 2,000 | 1,667 / 2,000 |
| iid \(N(0,2^2)\) | 1,941 / 2,000 | 2,000 / 2,000 |
| iid \(N(0.35,1.25^2)\) | 1,175 / 2,000 | 1,976 / 2,000 |
| alternating means \(+0.35,-0.35\) | 28 / 2,000 | 60 / 2,000 |
| one exactly signed 6-sigma residual at look 100 | 2,000 / 2,000 | 197 / 2,000 |
| one exactly signed 10-sigma residual at look 100 | 2,000 / 2,000 | 2,000 / 2,000 |
| variance-one scaled Student \(t_3\) | 1,053 / 2,000 | 1,031 / 2,000 |

Only the first row satisfies both complete nulls. Its finite counts neither prove nor compare the general probability bounds; the written arguments do that under their respective assumptions. The persistent scale and combined-error rows establish the intended synthetic improvement. The 6-sigma row is a constructed counterexample to unconditional dominance: the point gate reacts to the isolated event, whereas the cumulative variance alternatives dilute it among roughly null-sized earlier residuals. At 10 sigma both tested designs reject every history.

The known-covariance mean component also improves the targeted mean row in this paired design, but that result cannot be exported to Appendix S's unknown-scale class. Alternating shifts remain weakly detected by either procedure. Both rules reject about half the unit-variance Student-\(t_3\) histories, directly refuting distribution-free interpretations.

Nine new unit tests and all 59 inherited tests pass. An independent scalar audit imports neither production joint engine, evidence kernels, validation runner, nor SciPy. It reconstructs and matches all 24,000,000 stored component and joint decision states. The audit shares the synthetic observations, assumptions, and floating-point threshold convention; it is implementation independence, not external replication or measured evidence.

### V.4. Supported change and unresolved design choice

The supported change is a separately selectable \(JointGaussianMeanVarianceEProcess\) for investigations that predeclare persistent mean and uniform covariance-scale alternatives under a known conditional Gaussian model. The existing \(JointMeanPointAdequacy\) remains available for isolated-residual sensitivity and for its distinct unknown-scale mean component. Selecting between them is an experimental-design decision, not an ontological conclusion.

A three-component rule cannot give the point, mean, and variance detectors their existing full allocations and still inherit a 1% bound. The next bounded test is to predeclare a three-way mixture or budget allocation and optimize its mean, persistent-scale, and isolated-shock tradeoff on training simulations, then evaluate exactly one frozen choice on new validation data. Heavy-tail robustness requires a different null class.

All earlier corrections remain in force: complete deterministic closure for one fixed transition family is idempotent; integer-tick point capture is not continuous non-crossing; angular impulse has no extra time factor; finite prefixes do not identify arbitrary generators; and the 2K continuation theorem assumes an exact noiseless homogeneous constant-coefficient recurrence of order at most K valid from the initial state. No unconstrained residual is identified with God, and neither seven physical dimensions nor P=NP is established.


## Appendix W. A trained, frozen mean/variance/point allocation

Revision 1.22, 12 September 2026, adds Rope Tomography Lab 0.9.0. It starts from the saved paper v1.21, rope archive 0.8.0, and their current claim ledgers and revision notes. The current audited RRLS v0.27 package was read to confirm that its collision, angular-impulse, recurrence-class, and complexity boundaries remain present. RRLS code and evidence are unchanged.

### W.1. Bounded question and predeclared separation

Appendix V left one concrete design problem: can one procedure keep most of the persistent mean/scale sensitivity of the common-filtration mixture while regaining the point gate's response to an isolated shock, without spending more than one total 1% anytime error budget?

The lab used two distinct data roles. A deterministic training grid used seed 221209202609, 1,000 histories per target, 200 looks, point-budget fractions in \(\{0.05,0.10,0.15,0.20,0.25,0.30\}\), and mean weights in \(\{0.25,0.375,0.50,0.625,0.75\}\). The four training targets were a constant 0.35 mean, scales 1.25 and 0.8, and one noisy signed 6-sigma shock at look 100. The predeclared lexicographic objective maximized the minimum target detection count, then the total, then preferred a smaller point fraction, then a mean weight nearer one half.

Training selected a 0.30 point-budget fraction and, inside the remaining mean/variance mixture, weights 0.25 for mean and 0.75 for variance. Its training counts were 915, 909, 852, and 837 of 1,000. These counts selected the configuration; they are not validation evidence. The runner, selected configuration, validation seed 221209202610, scenario definitions, source hashes, and audit were then frozen before the one validation run. The runner is designed to refuse silent reuse of its reserved output directory. The adjacent configuration with mean weight 0.375 had a worst-target count of 836 rather than 837, so the selected grid optimum is not evidence of a uniquely optimal continuous allocation and may be training-seed sensitive.

This separation follows the general principle that data used to choose a procedure must not be counted again as held-out evidence. See Angelopoulos et al., *Learn then Test* (2021), https://arxiv.org/abs/2110.01052. That work is methodological context; the finite grid and guarantee below are the claims implemented here.

### W.2. Written anytime bound

Let \(\alpha=0.01\), let \(q=0.30\), and set \(\alpha_p=q\alpha=0.003\) and \(\alpha_{mv}=(1-q)\alpha=0.007\). Under a specified correct candidate, require fresh whitened residuals to satisfy

\[
Z_t\mid\mathcal F_{t-1}\sim N(0,I),
\]

with each experiment and monitored direction selected from the past before the current response, fixed alternatives, and no scored-data reuse, reset, or refit. Let \(E^{(m)}_n\) and \(E^{(v)}_n\) be the known-covariance mean and covariance-scale processes of Appendices R and U on this same canonical Gaussian filtration. With the training-selected fixed weight \(w=0.25\),

\[
M_n=wE^{(m)}_n+(1-w)E^{(v)}_n
\]

is a nonnegative martingale starting at one. Therefore Ville's inequality gives

\[
\Pr\!\left\{\sup_n M_n\ge 1/\alpha_{mv}\right\}\le \alpha_{mv}.
\]

The point gate is not asserted to be part of that e-process. At look \(t\), it uses the two-sided Gaussian tail allocation \(\alpha_{p,t}=\alpha_p/[t(t+1)]\). Since \(\sum_{t\ge1}1/[t(t+1)]=1\), a union bound gives probability at most \(\alpha_p\) that it ever fires for the specified correct candidate. A second event-level union bound then gives

\[
\boxed{\Pr\{\text{hybrid ever rejects the specified correct candidate}\}
\le \alpha_{mv}+\alpha_p=0.01.}
\]

Choosing \(q\) and \(w\) from data independent of all scored observations does not spend this bound. Conditional on the training data, they are fixed and the preceding inequality holds; averaging over the independent training data preserves it. Selection on the scored sequence, repeated re-selection, or reusing validation data would require a different proof.

This is a written probability argument, not a proof-assistant artifact and not a new statistical theorem. The within-filtration mixture follows the martingale logic discussed by Vovk and Wang, *Merging sequential e-values via martingales* (2024), https://projecteuclid.org/journals/electronic-journal-of-statistics/volume-18/issue-1/Merging-sequential-e-values-via-martingales/10.1214/24-EJS2228.full. Choi's *Anytime-Valid Evidence for Prespecified Predictive Corrections* (submitted 8 August 2026), https://arxiv.org/abs/2608.08174, supports interpreting a crossing against declared predictive alternatives, not as identification of a unique mechanism. The point gate remains separate precisely because the earlier scale-invariant mean construction and raw point statistic do not share the filtration needed for the claimed merge.

### W.3. One held-out synthetic validation

The frozen comparison used 2,000 independent 200-look histories per scenario. All three rules were evaluated on the same new residual paths: the revision 1.19 mean-plus-point rule, the revision 1.21 50/50 mean-plus-variance mixture, and the selected hybrid.

| Synthetic scenario | v1.19 mean + point | v1.21 mean + variance | selected hybrid |
|---|---:|---:|---:|
| iid \(N(0,1)\) | 17 / 2,000 | 22 / 2,000 | 21 / 2,000 |
| iid \(N(0.35,1)\) | 1,736 / 2,000 | 1,866 / 2,000 | 1,811 / 2,000 |
| iid \(N(0,1.25^2)\) | 122 / 2,000 | 1,780 / 2,000 | 1,781 / 2,000 |
| iid \(N(0,0.8^2)\) | 6 / 2,000 | 1,683 / 2,000 | 1,689 / 2,000 |
| iid \(N(0.35,1.25^2)\) | 1,187 / 2,000 | 1,986 / 2,000 | 1,985 / 2,000 |
| alternating means \(+0.35,-0.35\) | 28 / 2,000 | 75 / 2,000 | 82 / 2,000 |
| one noisy signed 6-sigma shock at look 100 | 1,673 / 2,000 | 379 / 2,000 | 1,621 / 2,000 |
| one exactly signed 6-sigma residual at look 100 | 2,000 / 2,000 | 222 / 2,000 | 2,000 / 2,000 |
| variance-one scaled Student \(t_3\) | 1,028 / 2,000 | 1,061 / 2,000 | 1,508 / 2,000 |

Only the first row satisfies the complete Gaussian null. Its 21 selected-hybrid rejections are a finite synthetic observation, not the proof of the 1% bound. The selected design bridges the targeted tradeoff on these paths: compared with v1.21 it gives up 55 detections for the constant mean target, is essentially unchanged on the persistent scale and combined targets, and gains 1,242 detections for the noisy-shock target. It exactly matches the point-based rule on the constructed exact-shock row but remains 52 detections below it on the noisy-shock row. These differences are finite paired results for one horizon, parameter set, grid, training seed, and validation seed; they do not establish universal ordering.

The heavy-tail control is a decisive counterexample to robustness. Although the scaled Student \(t_3\) paths have unit variance, Gaussianity is false and the selected hybrid rejects 1,508 of 2,000—more than either comparator. A crossing therefore indicates incompatibility with the specified Gaussian prediction and alternatives, not a unique variance, physical, sensor, metaphysical, or theological cause. Alternating shifts also remain weakly detected.

Eight new unit tests and all 68 inherited tests passed before validation, for 76 total. A separate scalar audit imports neither production engines, evidence kernels, tuner, validation runner, SciPy, nor their decision functions. It reconstructed the training-grid argmax and matched all 18,000,000 stored validation decision states. It shares the saved synthetic paths, assumptions, floating-point convention, and summary operations; this is finite implementation independence, not external replication or measured physical evidence.

### W.4. Supported change, limits, and next test

The supported change is a separately selectable \(ThreeWayGaussianAdequacy\) engine with a frozen 0.7% mean/variance mixture and 0.3% point gate. It is useful when persistent mean errors, uniform covariance-scale errors, and isolated residuals are all predeclared concerns under the conditional Gaussian known-covariance null. The two older rules remain available because they encode different assumptions and priorities.

No measured physical data were collected. The mathematical result is the stated conditional probability bound; the computation is one finite grid search; the validation and counterexamples are synthetic; the claim that this allocation will transfer to an instrument is a hypothesis; theological interpretation remains outside the statistic. Non-rejection does not establish truth, and rejection does not identify a generator or ontology.

Because the winning training configuration beat a nearby grid point by only one worst-target history, the next bounded test is training-only allocation-stability analysis across preregistered independent seeds. It should estimate how often each grid cell wins and how sensitive held-out target power is to that choice, without reopening or reusing this validation set. A heavy-tail-safe detector would require a separately declared null class and proof rather than relabeling this Gaussian engine.

All earlier corrections remain in force: complete deterministic closure for one fixed transition family is idempotent; integer-tick point capture is not continuous non-crossing; angular impulse has no extra time factor; finite prefixes do not identify arbitrary generators; and the 2K continuation theorem assumes an exact noiseless homogeneous constant-coefficient recurrence of order at most K valid from the initial state. No unconstrained residual is identified with God, and neither seven physical dimensions nor P=NP is established.

## Appendix X. Allocation-selection stability and a boundary warning

Revision 1.23, 13 September 2026, adds Rope Tomography Lab 0.10.0. It starts from the saved paper v1.22, rope archive 0.9.0, and their current claim ledgers and revision notes. The current audited RRLS v0.27 package was read to confirm the retained collision, angular-impulse, recurrence-class, and complexity boundaries. RRLS code and evidence are unchanged.

### X.1. Bounded question and sealed-data rule

Revision 1.22 selected a point-budget fraction of 0.30 and an inner mean weight of 0.25 from one 30-cell synthetic training grid. Its worst-target score beat a nearby cell by one history. The bounded question is whether that selected cell is repeatable under independent draws from the same four declared training scenarios.

This is a training-selection stability study, not a second validation of the adequacy engine. The frozen v1.22 validation observations and summary were not read by the new runner or audit. Before outcomes, the protocol fixed 64 seeds, 1,000 histories per target and seed, 200 looks, the original six point fractions and five mean weights, the original lexicographic selection objective, and all reported metrics. It also fixed a separate reference seed with 20,000 histories for each of the four targets. That 80,000-history reference sample estimates performance within this synthetic selection study; it is not measured evidence or permission to retune against the sealed validation set.

This separation addresses a standard model-selection problem: optimizing a noisy empirical criterion can overfit the selection sample even when the underlying candidates are unchanged. See Cawley and Talbot, *On Over-fitting in Model Selection and Subsequent Selection Bias in Performance Evaluation* (2010), https://www.jmlr.org/papers/v11/cawley10a.html. Bouthillier et al., *Accounting for Variance in Machine Learning Benchmarks* (2021), https://proceedings.mlr.press/v139/bouthillier21a.html, provides related primary evidence that repeated sources of variation matter when comparing procedures. These papers supply methodological context, not evidence for the present physical theory.

### X.2. Finite stability results

The 64 independent training runs selected four cells:

| Point-budget fraction | Mean weight inside mixture | Wins | Frequency |
|---:|---:|---:|---:|
| 0.30 | 0.25 | 49 / 64 | 76.6% |
| 0.30 | 0.375 | 10 / 64 | 15.6% |
| 0.30 | 0.50 | 3 / 64 | 4.7% |
| 0.25 | 0.25 | 2 / 64 | 3.1% |

For the v1.22 cell's 49/64 frequency, the descriptive 95% Wilson interval is 64.9% to 85.3%. It quantifies binomial sampling uncertainty under this exact simulation protocol; it is not a theorem about other target sets, horizons, grids, or physical instruments.

The primary worst-target count was tied between the winner and runner-up in 31 of 64 runs. Its median advantage was one history and its maximum was nine. The secondary total-count and fixed tie breakers therefore materially affected many selections. This confirms that “the optimizer returned one cell” is stronger language than the data support.

At the same time, the selections occupied a narrow region. A point fraction of 0.30 won 62 of 64 times; 0.25 won twice; no smaller point fraction won. Mean weight 0.25 won 51 times, 0.375 won ten times, and 0.50 won three times; larger weights never won. Thus the original cell is usually repeatable within the tested grid, although exact selection is not deterministic.

### X.3. Separate synthetic reference and counterexample to optimality

On the separately seeded 20,000-history-per-target reference sample, the original v1.22 cell also maximized the declared finite-grid objective:

| Target | Detections | Rate |
|---|---:|---:|
| constant mean 0.35 | 18,029 / 20,000 | 90.15% |
| scale 1.25 | 17,938 / 20,000 | 89.69% |
| scale 0.8 | 16,788 / 20,000 | 83.94% |
| noisy signed 6-sigma shock | 16,273 / 20,000 | 81.37% |

The worst-target reference rate was 81.365%. When each seed-selected cell was scored on this same reference sample, its minimax-rate regret relative to the reference grid winner had mean 0.0498 percentage points, median zero, 90th percentile 0.0600 points, and maximum 0.9200 points. These are finite Monte Carlo summaries on a shared reference set, not confidence bounds on real-world power.

The key counterexample is geometric rather than numerical: the winner sits on two grid boundaries. The largest tested point fraction won 62 of 64 seeds, while the smallest tested mean weight won 51. Therefore this experiment cannot distinguish “an optimum near 0.30 and 0.25” from “the objective would keep improving beyond one or both tested boundaries.” Repeatability inside a truncated search space does not establish a local or global optimum outside it.

Five new tests and all 76 inherited tests passed, for 81 total. An independent implementation imports neither the production evidence kernels, v1.22 tuner, stability runner, nor SciPy. It regenerated every synthetic history and formula from the frozen seeds and matched all 10,080,000 stored cell-target-history decisions. It shares the protocol, NumPy random generator, floating-point convention, and synthetic model, so it is implementation independence rather than external replication.

### X.4. Supported correction, evidence classes, and next test

The supported correction is narrower than adopting new weights. Revision 1.22's allocation is now described as usually repeatable and low-regret *within its declared grid and synthetic target distribution*, not as uniquely optimal. Its anytime false-rejection proof is unaffected because that proof applies to the already frozen allocation under its conditional known-covariance Gaussian assumptions. No core adequacy-engine code or v1.22 validation record changed.

The written mathematics remains the v1.22 martingale-plus-union-bound argument. This revision adds finite computation and synthetic stability evidence only. Transfer to a physical instrument remains a hypothesis; no measured observations were collected. Neither an optimizer boundary nor a residual is an ontological or theological identification.

The next bounded test is a training-only boundary-expansion study: preregister point fractions above 0.30 and mean weights below 0.25, repeat the same multi-seed stability analysis, and keep the v1.22 validation set sealed. If a materially different allocation is proposed afterward, it must receive a new frozen validation set before replacing the released design. Heavy-tail validity remains a separate problem requiring a new null class and proof.

All earlier corrections remain in force: complete deterministic closure for one fixed transition family is idempotent; integer-tick point capture is not continuous non-crossing; angular impulse has no extra time factor; finite prefixes do not identify arbitrary generators; and the 2K continuation theorem assumes an exact noiseless homogeneous constant-coefficient recurrence of order at most K valid from the initial state. No unconstrained residual is identified with God, and neither seven physical dimensions nor P=NP is established.

## Appendix Y. Expanded allocation boundaries refute the released grid optimum

Revision 1.24, 14 September 2026, adds Rope Tomography Lab 0.11.0. It starts from the saved paper v1.23, rope archive 0.10.0, and their current claim ledgers and revision notes. The audited RRLS v0.27 source was reread only to confirm its retained boundaries; RRLS code and evidence are unchanged.

### Y.1. Assumptions and frozen training-only protocol

Appendix X left one concrete ambiguity: its winner occupied the largest tested point-budget fraction, 0.30, and the smallest tested inner mean weight, 0.25. The bounded question is whether the released cell remains competitive after expanding both boundaries under the same four-target synthetic training objective.

Before generating outcomes, `boundary_expansion_protocol.json` fixed 64 new seeds disjoint from Appendix X, 1,000 histories per target and seed, 200 looks, and a separate 20,000-history-per-target reference seed. The complete old grid was retained. Point fractions were expanded to

\[
0.05,0.10,\ldots,0.60,
\]

and inner mean weights to

\[
0.05,0.10,0.15,0.20,0.25,0.375,0.50,0.625,0.75,
\]

for 108 cells. The targets, Gaussian generators, fixed alternatives, 1% total budget, lexicographic minimax objective, metrics, source hashes, and rule against automatic adoption were also frozen. The runner and audit do not read the v1.22 held-out validation observations or results.

This is a finite hyperparameter-selection study. Bergstra and Bengio, *Random Search for Hyper-Parameter Optimization* (2012), https://www.jmlr.org/papers/v13/bergstra12a.html, shows why a small axis-aligned search can miss important regions. Cawley and Talbot (2010), https://www.jmlr.org/papers/v11/cawley10a.html, and Bouthillier et al. (2021), https://proceedings.mlr.press/v139/bouthillier21a.html, motivate separating tuning from performance estimation and reporting selection variability. These primary methodological sources do not supply physical evidence for this framework.

### Y.2. Finite results and counterexamples

The released v1.22 cell won **0 / 64** new training runs; its descriptive Wilson 95% interval for the win frequency is 0% to 5.66%. Sixteen cells won at least once. The two modal cells, point fraction 0.45 with mean weight 0.10 and point fraction 0.50 with mean weight 0.10, each won only 12 of 64 runs. Aggregated over mean weights, point fractions 0.45 and 0.50 won 20 and 19 runs. Aggregated over point fractions, mean weight 0.10 won 39 runs and 0.15 won 19. Twenty-two runs tied the runner-up on the primary worst-target count; the median primary margin was one history and the maximum was six.

Thus the broader grid supplies a direct finite counterexample to treating the released cell as optimal for the declared training objective. It also supplies a second counterexample: opening the boundaries does not reveal one stable exact replacement at 1,000 histories per target. Selection instability must be retained rather than hidden by reporting only an argmax.

On the separately seeded 20,000-history-per-target synthetic reference sample, the expanded-grid winner was point fraction 0.45 and inner mean weight 0.10:

| Target | released 0.30/0.25 | reference candidate 0.45/0.10 |
|---|---:|---:|
| constant mean 0.35 | 18,072 / 20,000 | 17,096 / 20,000 |
| scale 1.25 | 17,956 / 20,000 | 17,916 / 20,000 |
| scale 0.8 | 16,886 / 20,000 | 16,798 / 20,000 |
| noisy signed 6-sigma shock | 16,360 / 20,000 | 16,715 / 20,000 |
| worst-target rate | 81.80% | 83.575% |

The declared minimax objective improves by 1.775 percentage points, but the candidate is not universally better: it loses 976 mean-shift detections, 40 high-scale detections, and 88 low-scale detections while gaining 355 shock detections. Changing the target set or utility weights can change the preferred cell. Among the 64 seed-selected cells, reference minimax regret had mean 0.637, median 0.430, 90th percentile 1.215, and maximum 3.260 percentage points.

Four new protocol tests and all 81 inherited tests passed before outcomes, for 85 total. A separate scalar implementation regenerated the synthetic histories and formulas and matched all 36,288,000 cell-target-history decisions. It imports neither the production evidence kernels, v1.22 tuner, stability runner, nor SciPy. It shares the frozen seeds, NumPy generator, synthetic model, and floating-point convention, so it is implementation independence rather than external replication.

### Y.3. Supported correction, limits, and next test

The supported correction is that revision 1.22's allocation is not optimal even within the newly declared 108-cell training grid. The 0.45/0.10 reference winner is a candidate, not a released replacement: exact training selection was dispersed across 16 cells, and the reference sample is tuning evidence rather than fresh validation. No core detector or sealed validation record changed.

The written 1% anytime bound is unchanged. It applies to any point fraction and inner mixture weight fixed independently of scored observations under the conditional known-covariance Gaussian assumptions of Appendix W. It does not prove that a tuned allocation has useful power, transfers to an instrument, or remains valid under heavy-tailed noise.

No measured physical data were collected. The probability bound is written mathematics; the grid search, replay, and counterexamples are finite computation on synthetic data; transfer is a hypothesis; and theological interpretation remains outside the statistic. A rejection cannot identify a unique physical, metaphysical, or theological cause.

The next bounded test is one preregistered, paired, fresh frozen validation comparing the released 0.30/0.25 design with the single 0.45/0.10 candidate. It should retain the declared mean, scale, and shock targets, include Gaussian-null and heavy-tail controls, fix one untouched seed and all metrics before outcomes, and adopt the candidate only if the preregistered decision rule is met. The v1.22 validation set must remain sealed.

All earlier corrections remain in force: complete deterministic closure for one fixed transition family is idempotent; integer-tick point capture is not continuous non-crossing; angular impulse has no extra time factor; finite prefixes do not identify arbitrary generators; and the 2K continuation theorem assumes an exact noiseless homogeneous constant-coefficient recurrence of order at most K valid from the initial state. No unconstrained residual is identified with God, and neither seven physical dimensions nor P=NP is established.

## Appendix Z. Fresh paired validation rejects a universal allocation replacement

Revision 1.25, 15 September 2026, adds Rope Tomography Lab 0.12.0. It starts from the saved paper v1.24, rope archive 0.11.0, and their current claim ledgers and revision notes. The audited RRLS v0.27 source was reread to confirm the retained collision, angular-impulse, recurrence-class, and complexity boundaries. RRLS code and evidence are unchanged.

### Z.1. Frozen question, assumptions, and decision rule

Appendix Y selected point fraction 0.45 and inner mean weight 0.10 on training-only synthetic evidence but prohibited adoption without a fresh validation. `candidate_validation_protocol.json` therefore froze one untouched seed, 20,000 paired 200-look histories per scenario, both allocation coordinates, scenario generators, metrics, source hashes, and decision rule before outcomes. The runner never reads the v1.22 validation observations or the v1.23-v1.24 training seeds and reference paths.

The released design uses point fraction 0.30 and mean weight 0.25. The candidate uses 0.45 and 0.10. Both retain the same total 1% budget, fixed mean alternative \(\delta=0.35\), fixed variance factors \(1.5625\) and \(0.64\), and equal variance weights. Under Appendix W's complete conditional Gaussian null, the written e-process-plus-union-bound argument applies to either allocation because the weights are fixed independently of the scored histories.

The preregistered engineering rule required all four conditions:

1. increase the minimum detection rate over the four declared targets by at least one percentage point;
2. give a positive descriptive paired 95% interval on the released design's bottleneck target;
3. keep every target's paired interval lower bound at or above minus two percentage points; and
4. increase the finite Gaussian-null rejection rate by no more than 0.5 percentage points.

The normal intervals summarize finite paired Bernoulli differences; they are not universal coverage guarantees. The one- and two-point thresholds express a declared utility, not facts derived from physics. Multi-objective optimization generally seeks a Pareto set rather than pretending that one coordinate dominates every objective; Inatsu et al., *Bounding Box-based Multi-objective Bayesian Optimization of Risk Measures under Input Uncertainty* (2024), https://proceedings.mlr.press/v238/inatsu24a.html, supplies relevant primary methodological context. Cawley and Talbot's selection-bias analysis, https://www.jmlr.org/papers/v11/cawley10a.html, supports keeping the candidate-selection samples separate from this evaluation. Neither paper supplies evidence for the physical framework.

### Z.2. Fresh synthetic results and counterexample

| Scenario | released 0.30/0.25 | candidate 0.45/0.10 | candidate minus released |
|---|---:|---:|---:|
| iid \(N(0,1)\) | 170 / 20,000 | 174 / 20,000 | +0.020 points |
| iid \(N(0.35,1)\) | 18,015 / 20,000 | 16,982 / 20,000 | **-5.165 points** |
| iid \(N(0,1.25^2)\) | 18,043 / 20,000 | 18,005 / 20,000 | -0.190 points |
| iid \(N(0,0.8^2)\) | 16,986 / 20,000 | 16,886 / 20,000 | -0.500 points |
| noisy signed 6-sigma shock | 16,287 / 20,000 | 16,674 / 20,000 | **+1.935 points** |
| iid \(N(0.35,1.25^2)\) | 19,822 / 20,000 | 19,767 / 20,000 | -0.275 points |
| alternating \(+0.35,-0.35\) means | 713 / 20,000 | 724 / 20,000 | +0.055 points |
| exact signed 6-sigma shock | 20,000 / 20,000 | 20,000 / 20,000 | 0 points |
| variance-one Student \(t_3\) | 15,253 / 20,000 | 15,422 / 20,000 | +0.845 points |

For the noisy-shock bottleneck, the paired difference was 1.935 points with descriptive normal 95% interval 1.740 to 2.130 points. The candidate therefore passed the minimum-improvement and positive-bottleneck checks. On persistent mean error, however, the paired interval was -5.473 to -4.857 points. This violates the frozen two-point loss guardrail, so the adoption decision is **retain the released design**.

The result is also a concrete counterexample to universal dominance. The candidate improves the chosen minimax bottleneck but materially weakens the persistent-mean target. Conversely, the released design is not uniformly better because it misses 387 additional noisy-shock histories. Which tradeoff is preferable cannot be inferred from predictive scores without an externally declared utility.

The Gaussian-null counts, 0.850% and 0.870%, are finite diagnostics rather than proofs of the 1% anytime bound. The Student \(t_3\) control again violates Gaussianity and produces rejection rates of 76.265% and 77.110%; neither design is distribution-free. Alternating shifts remain weakly detected. These controls preserve the previously known model failures.

Four new protocol tests and all 85 inherited tests passed before outcomes, for 89 total. An independent implementation regenerated every scenario and separately implemented the mean, variance, mixture, Gaussian-tail, interval, and adoption calculations. It matched all 72,000,000 saved look-by-design decision states and the final recommendation. It shares the seed, NumPy generator, synthetic model, and floating-point convention, so it is implementation independence rather than external replication.

### Z.3. Supported correction, evidence classes, and next test

The supported correction is to reject the 0.45/0.10 candidate as a balanced replacement under the frozen utility while retaining it as a documented shock-sensitive alternative. `selected_three_way_design.json` and the core runtime remain unchanged. The result does not retroactively alter the sealed v1.22 validation or Appendix W's mathematical guarantee.

The probability bound is written mathematics. The comparison, paired intervals, counterexamples, and audit are finite computation on synthetic data. No measured physical data were collected. Transfer to a real instrument remains a hypothesis, and neither predictive rejection nor a residual identifies a unique physical, metaphysical, or theological cause.

The next bounded test should address the shared failure rather than retune these Gaussian weights again: predeclare a heavy-tail null class and construct an anytime-valid robust process for that class. The first checkpoint must be a written supermartingale or other time-uniform validity argument; only afterward should implementation tests and fresh synthetic calibration be run. Until such a proof exists, preserve the Gaussian and heavy-tail models as distinguishable assumptions rather than relabeling either as reality.

All earlier corrections remain in force: complete deterministic closure for one fixed transition family is idempotent; integer-tick point capture is not continuous non-crossing; angular impulse has no extra time factor; finite prefixes do not identify arbitrary generators; and the 2K continuation theorem assumes an exact noiseless homogeneous constant-coefficient recurrence of order at most K valid from the initial state. No unconstrained residual is identified with God, and neither seven physical dimensions nor P=NP is established.

## Appendix AA. An anytime-valid sign process for a declared heavy-tail class

Revision 1.26, 15 September 2026, adds Rope Tomography Lab 0.13.0. It starts from the saved paper v1.25 and rope archive 0.12.0. The current claim ledger, revision notes, and unchanged audited RRLS v0.27 source were reread. RRLS's collision, angular-impulse, recurrence-class, and complexity boundaries remain unchanged.

### AA.1. The bounded question and null class

Appendix Z's next question was whether one useful adequacy check could remain anytime-valid without Gaussian tails. “Heavy-tail robustness” alone is not a mathematical model. This appendix narrows it to a sign-only claim for one specified candidate and one past-measurable projected residual. Let

\[
S_t=\operatorname{sign}(Z_t)\in\{-1,0,+1\},
\qquad
\mathbb E[S_t\mid\mathcal F_{t-1}]=0. \tag{AA.1}
\]

Here \(Z_t\) is the selected projected residual, and exact zeros are neutral. The candidate, direction, experiment choice, \(\rho\in(0,1)\), and threshold must be fixed or chosen from past information before \(S_t\) is observed. Equation (AA.1) is **conditional sign balance**. It permits Gaussian, Student-\(t\), Cauchy, heteroscedastic, and other magnitude laws if their current sign remains balanced conditional on the past. It is neither a mean-zero condition nor a consequence of marginal 50/50 signs.

This construction follows the nonnegative-martingale foundation and symmetry-testing perspective in Ramdas, Ruf, Larsson, and Koolen, *Admissible anytime-valid sequential inference must rely on nonnegative martingales* (revised 5 November 2022), https://arxiv.org/abs/2009.03167, and the broader time-uniform methodology of Howard et al., *Time-uniform, nonparametric, nonasymptotic confidence sequences* (2021), https://projecteuclid.org/journals/annals-of-statistics/volume-49/issue-2/Time-uniform-nonparametric-nonasymptotic-confidence-sequences/10.1214/20-AOS1991.full. The elementary sign mixture below is an application, not a new theorem or evidence for the physical framework.

### AA.2. Written probability proof

Fix \(\rho\in(0,1)\) before scoring and define

\[
M_n^+=\prod_{t=1}^n(1+\rho S_t),\qquad
M_n^-=\prod_{t=1}^n(1-\rho S_t),\qquad
E_n=\tfrac12(M_n^++M_n^-). \tag{AA.2}
\]

Every factor is nonnegative. Under (AA.1),

\[
\mathbb E[1\pm\rho S_t\mid\mathcal F_{t-1}]=1,
\]

so both \(M_n^+\) and \(M_n^-\) are nonnegative martingales starting at one. Their fixed convex mixture \(E_n\) is also a nonnegative martingale starting at one. Ville's inequality therefore gives

\[
\Pr\!\left(\sup_{n\ge0}E_n\ge\frac1\alpha\right)\le\alpha. \tag{AA.3}
\]

Writing \(A_n=\sum_{t\le n}S_t\) and \(N_n=\sum_{t\le n}\mathbf1\{S_t\ne0\}\), the implemented stable form is

\[
\log E_n=\frac{N_n}{2}\log(1-\rho^2)
+\log\cosh\!\left(\operatorname{atanh}(\rho)A_n\right). \tag{AA.4}
\]

This is a written mathematical guarantee for each specified correct candidate under (AA.1). It does not protect all candidates simultaneously, does not renew after resetting an object, and does not combine with the existing 1%-level Gaussian engines. Any OR-combination would require a shared error budget or another valid joint proof.

### AA.3. Frozen finite synthetic validation

`sign_validation_protocol.json` froze \(\alpha=0.01\), \(\rho=0.30\), a fresh seed, 5,000 histories, 200 looks, eleven scenarios, source hashes, and interpretation before outcomes. The known-scale Gaussian mean process was run separately at its own 1% threshold only as a diagnostic; the two procedures were not combined.

| Scenario | sign process | Gaussian comparator |
|---|---:|---:|
| iid \(N(0,1)\) | 53 / 5,000 | 36 / 5,000 |
| variance-one Student \(t_3\) null | 37 / 5,000 | 82 / 5,000 |
| exact doubled \(t_3\) null paths | 37 / 5,000 | 1,742 / 5,000 |
| standard Cauchy null | 33 / 5,000 | 4,994 / 5,000 |
| shifted \(t_3,+0.35\) | 4,989 / 5,000 | 4,793 / 5,000 |
| shifted \(t_3,-0.35\) | 4,992 / 5,000 | 4,800 / 5,000 |
| shifted Gaussian (+0.35) | 3,996 / 5,000 | 4,747 / 5,000 |
| mean-zero asymmetric (+1/-9) mixture | 5,000 / 5,000 | 4,177 / 5,000 |
| marginally balanced persistent sign chain | 3,772 / 5,000 | 3,835 / 5,000 |
| alternating shifted \(t_3\) | 12 / 5,000 | 72 / 5,000 |
| one signed six-unit \(t_3\) shock | 49 / 5,000 | 95 / 5,000 |

The Gaussian, unshifted \(t_3\), doubled \(t_3\), and Cauchy sign rows satisfy the sign theorem. Their finite crossing counts do not prove (AA.3); the martingale argument does. The doubled paths produced exactly the same sign decisions path by path, demonstrating scale invariance. The Gaussian comparator's guarantee is false on the non-Gaussian rows, and its Cauchy and doubled-\(t_3\) counts illustrate that failure rather than a calibrated comparison.

The shifted-\(t_3\) rows show strong finite targeted power at the frozen \(\rho\), while the Gaussian shift preserves the expected efficiency cost of discarding magnitudes. No universal power claim follows from one seed, horizon, or alternative.

### AA.4. Counterexamples and interpretation

The asymmetric mixture with probability 0.9 at \(+1\) and 0.1 at \(-9\) has exact expectation zero but violates sign balance. All 5,000 histories crossed. Thus “mean zero” cannot silently replace (AA.1). The stationary sign chain has marginal probability one half for each sign but retains its previous sign with probability 0.9, so its current conditional expectation is usually nonzero; 3,772/5,000 paths crossed. Thus marginal balance cannot silently replace conditional balance.

Positive rescaling never changes the statistic, so it cannot detect pure scale errors or tail thickness. The alternating-shift and isolated-shock rows show further power failures. A crossing supports incompatibility with conditional sign balance around the specified prediction; it does not identify a mean error, sensor failure, physical mechanism, metaphysical cause, or God.

Eight engine tests and four frozen-protocol tests were added; all 101 tests passed before outcomes. An independently written audit regenerated every scenario and separately recomputed the sign and Gaussian formulas, every crossing path, and every summary. It matched 22,000,000 saved decision states. The audit shares the seed, NumPy samplers, scenario definitions, and floating-point arithmetic, so it is implementation independence rather than external replication.

### AA.5. Supported change, evidence classes, and next test

`ConditionalSignEProcess` is added as a standalone selectable engine. It does not replace or silently broaden the Gaussian mean/variance/point engine. The conditional-sign anytime bound is written mathematics; unit tests, synthetic paths, counterexamples, and the independent replay are finite computation. No measured physical evidence was collected. Application to real residuals remains a hypothesis requiring instrument-specific checks of sign balance and dependence.

The next bounded test is the dependence boundary exposed above: construct a sign process valid under a declared dependence model or a defensible predictable conditional-sign model, then test it against stationary marginally balanced chains without reusing this validation set. Until that proof exists, correlated signs must remain outside the guarantee.

All earlier corrections remain in force: complete deterministic closure for one fixed transition family is idempotent; integer-tick point capture is not continuous non-crossing; angular impulse has no extra time factor; finite prefixes do not identify arbitrary generators; and the 2K continuation theorem assumes an exact noiseless homogeneous constant-coefficient recurrence of order at most K valid from the initial state. No unconstrained residual is identified with God, and neither seven physical dimensions nor P=NP is established.

## Appendix AB. Composite Born prediction, coded lift search, and projection collisions

Revision 1.27, 15 September 2026, adds Born Outcome Frontier Audit 0.2.0. It starts from the saved paper v1.26, the court-audit phase/Born formalization, and the current RRLS claim ledgers. Rope Tomography Lab 0.13.0 and RRLS Studio 0.27.1 are unchanged. No physical measurement data were collected.

### AB.1. One theory, three different uses of seven and eight

Three objects must remain separate:

1. \((D_1,\ldots,D_7)\) is the declared hierarchy of seven **typed descriptive layers**.
2. Q1 uses eight computational-basis states of three qubits and therefore eight raw basis phases.
3. Removing common phase leaves seven relative phase coordinates because

\[
\mathbb T^8/S^1\cong\mathbb T^7.
\]

The numerical equality between the number of typed layers and the number of relative Q1 phases is not yet a derived physical identification. A bridge between them must be a typed map with units, transformation law, dynamics, and a measurement consequence. Treating the matching numeral as proof would violate coordinate covariance A2.

### AB.2. Exact and composite Born walls

At trial \(t\), let \(\mathcal F_{t-1}\) contain all admissible previous information and let \(C_t\) contain the complete declared standard context. An outcome predictor \(q_t\) must be fixed using information available before \(Y_t\). For an exact Born conditional law \(p_t\),

\[
\mathbb E_{p_t}\!\left[
\log\frac{q_t(Y_t)}{p_t(Y_t)}
\middle|\mathcal F_{t-1},C_t
\right]
=-D_{\mathrm{KL}}(p_t\Vert q_t)\le0. \tag{AB.1}
\]

The likelihood-ratio product is therefore a nonnegative martingale. This is the Born Prediction Wall: more computation cannot extract information that is absent from its admissible inputs.

Real calibration does not usually specify one exact \(p_t\). Let \(\mathcal P_t\) be the predictable set of conditional distributions retained by the calibrated null and define

\[
u_t(y)=\sup_{p\in\mathcal P_t}p(y).
\]

For any normalized predictable \(q_t\), put

\[
e_t(Y_t)=\frac{q_t(Y_t)}{u_t(Y_t)}.
\]

If the actual \(p_t^*\in\mathcal P_t\), then

\[
\mathbb E_{p_t^*}[e_t(Y_t)\mid\mathcal F_{t-1},C_t]
=\sum_y p_t^*(y)\frac{q_t(y)}{u_t(y)}
\le\sum_yq_t(y)=1. \tag{AB.2}
\]

Thus \(\prod_{s\le t}e_s(Y_s)\) is a nonnegative supermartingale under every admissible sequence in the composite null, and Ville's inequality remains valid. The coordinatewise envelope is generally conservative; its power loss is preferable to converting state, POVM, detector, or drift uncertainty into fake new physics. Broader composite-likelihood methods belong to established universal inference [Wasserman, Ramdas, and Balakrishnan (2020)](https://www.pnas.org/doi/10.1073/pnas.1922664117).

Exact zeros require a declared support rule. If every null distribution assigns probability zero to an observed outcome, that outcome rejects the stated null family. If the zero was produced only by finite calibration or numerical truncation, the null was misspecified; an outcome-dependent floor is not a valid repair.

### AB.3. Prefix-coded dimensional lifts

Let \(L_k\) be a countable frozen grammar of pre-measurement lift compilers and let \(\ell_k\) be prefix-code lengths satisfying

\[
\sum_k2^{-\ell_k}\le1.
\]

If \(E_{t,k}\) is a valid e-process for lift \(k\), then

\[
E_t^K
=1-\sum_k2^{-\ell_k}
+\sum_k2^{-\ell_k}E_{t,k} \tag{AB.3}
\]

is a fixed convex mixture of valid e-processes and unspent capital. It is therefore valid. A simple sufficient condition for candidate \(k\)'s weighted term to cross the threshold is

\[
\log E_{t,k}\ge\log(1/\alpha)+\ell_k\log2. \tag{AB.4}
\]

Every description bit costs one bit of evidence. This supplies a lawful statistical interface between RRLS and outcome prediction: phase unwraps, winding registers, parity lifts, Walsh interactions, lags, and boundary invariants may be searched only through a grammar and code fixed before validation.

Description length controls search multiplicity, not computational complexity. The RRLS language must still count recognition, transformation, decoding, representation growth, arithmetic bit costs, and total work. A compact but exponentially expensive oracle is not a polynomial predictor.

For a balanced 51%-accurate binary candidate at one-sided five sigma and approximate 90% power, the implementation gives:

| Prefix-code length | Sufficient candidate \(\log E\) | Approximate trials |
|---:|---:|---:|
| 0 bits | 15.064 | 119,635 |
| 8 bits | 20.609 | 153,191 |
| 16 bits | 26.154 | 186,030 |
| 32 bits | 37.244 | 250,321 |

These are planning approximations derived from the Bernoulli log-likelihood mean and variance. The e-process, not the normal approximation, supplies the primary anytime decision rule.

### AB.4. Projection collisions are universal decoder failures

Let \(\Lambda\) be the full admissible pre-measurement record and let

\[
G=g(\Lambda,C)
\]

be an RRLS-produced canonical address. Conditional data processing gives the exact identity

\[
I(Y;\Lambda\mid C)
=I(Y;G\mid C)
+I(Y;\Lambda\mid G,C), \tag{AB.5}
\]

and hence

\[
I(Y;G\mid C)\le I(Y;\Lambda\mid C). \tag{AB.6}
\]

The compression is predictively lossless exactly when the discarded coordinate has zero remaining conditional information. A collision

\[
G(\lambda,C)=G(\lambda',C),
\qquad
P(Y\mid\lambda,C)\ne P(Y\mid\lambda',C) \tag{AB.7}
\]

proves that no decoder receiving only \(G\) can be universally outcome-sufficient. This is structurally identical to the retained RRLS collision in which a SAT and an UNSAT formula share the full implemented invariant profile: computation downstream cannot reconstruct a distinction the projection erased.

The repair target is boundary compatibility, not \(R\to0\). In the quantum experiment the shared boundary contains the prepared-state family, allowed settings, timing, device identity, and interventions. A canonicalizer must preserve the conditional relation from that boundary to future outcome distributions.

The audit's fixed finite table contains \(0.192744757022\) nats of outcome information before coarsening and zero afterward. Across 5,000 randomized finite tables, the greatest numerical data-processing excess was \(2.22\times10^{-16}\) in the production implementation and zero in the independent replay. This is consistent with floating-point roundoff and the theorem; it is not a measurement of nature.

### AB.5. Foundational assumption map

Outcome determinism does not imply accessible prediction. Bohmian mechanics is deterministic, but its quantum-equilibrium analysis recovers \(\rho=|\psi|^2\) statistics and an absolute-unpredictability boundary [Dürr, Goldstein, and Zanghì (1992)](https://link.springer.com/article/10.1007/BF01049004).

Colbeck and Renner's no-improved-prediction conclusion is conditional on their free-choice formulation and quantum correctness. Ghirardi and Romano exhibit predictively inequivalent ontological models under a different free-choice assumption [Ghirardi and Romano (2013)](https://link.aps.org/doi/10.1103/PhysRevLett.110.170404). This exposes an assumption boundary; it does not validate an arbitrary hidden variable.

Valentini's proposed nonequilibrium subquantum information supplies another logically possible route but also a sharp burden: deterministic hidden-variable nonequilibrium generally exposes instantaneous statistical signaling [Valentini (2002)](https://arxiv.org/abs/quant-ph/0112151). Any such Dimensional Reality route requires an operational no-signaling analysis rather than a verbal assurance.

Prediction-based likelihood ratios are established in Bell analysis. Zhang, Glancy, and Knill's PBR protocol remains valid against memory-dependent local-realistic models even when prepared states vary [Zhang, Glancy, and Knill (2011)](https://link.aps.org/doi/10.1103/PhysRevA.84.062118). This project does not claim to invent likelihood-ratio Bell testing. Its candidate synthesis is the combination of projection-collision search, physically specified winding addresses, a composite Born envelope, and a frozen prefix-coded lift language.

The attribution gates are strict:

\[
\text{predictive residual}
\not\Rightarrow
\text{causal variable}
\not\Rightarrow
\text{new dimension}
\not\Rightarrow
\text{complete physical theory}. \tag{AB.8}
\]

A candidate must successively survive immutable ordering, leakage controls, tomography and detector uncertainty, process-tensor memory tests, causal breaks, resets, sign and magnitude reversal, independent platforms, independent laboratories, Bell/contextuality/free-setting constraints, no-signaling, Lorentz behavior, and the classical limit.

### AB.6. Executed evidence, preserved failures, and next test

Born Outcome Frontier Audit 0.2.0 and its independent verifier pass. The independent file imports none of the production module and reconstructs:

- the fixed composite-null multiplier \(0.962307692308\);
- the fixed information loss \(0.192744757022\to0\) nats;
- 5,000 randomized composite-envelope and data-processing cases;
- a 120,000-trial Born-null path with maximum \(\log E=1.17364005363\);
- a synthetic accessible-51%-lift path with maximum \(\log E=28.3649956267\);
- all four prefix-code power rows.

The independent replay shares the formulas, Python standard library, pseudorandom generator, seeds, and floating-point convention. It is implementation independence, not external replication. The positive world was deliberately generated with a predictive variable and therefore cannot support a claim that a natural one exists.

The next bounded test is not another synthetic retuning. It is a protocol artifact frozen before fresh hardware data:

1. specify a Q1 state and measurement family;
2. specify the composite Born calibration set \(\mathcal P_t\);
3. specify a prefix-coded admissible lift grammar;
4. prohibit all setting, outcome, and downstream timing leakage;
5. commit setting-conditional predictions before delayed random settings;
6. spend one investigation-wide evidence budget;
7. advance any crossing through the full attribution gates without altering its predicted sign, magnitude, or exclusions.

No physical Born residual, winding coupling, extra dimension, soul-mediated outcome rule, P=NP proof, or universal generator has been found. The supported advance is a stricter research constitution that makes those claims harder to fake and easier to kill.

## Appendix AC. Information rate, adversarial tripwires, and the wall ledger

Revision 1.28, 15 September 2026, adds Wallbreaker Audit 0.1.0 and Wallbreaker Protocol P3 0.1.0. It starts from the saved paper v1.27 and preserves every earlier correction. Born Outcome Frontier Audit 0.2.0, Rope Tomography Lab 0.13.0, and RRLS Studio 0.27.1 are unchanged. No physical measurements were collected.

### AC.1. General Born-evidence decomposition

Let \(C\) be the complete declared standard context, \(\Lambda\) an admissible pre-measurement lift, and \(b(y\mid C)\) the declared Born baseline. Write

\[
\bar r(y\mid C)=P(Y=y\mid C),
\qquad
r(y\mid C,\Lambda)=P(Y=y\mid C,\Lambda)
\]

for the actual marginal and lift-conditional laws. For any normalized decoder \(q(y\mid C,\Lambda)\), insert and subtract \(\log r\) and \(\log\bar r\):

\[
\begin{aligned}
\mathbb E\log\frac{q(Y\mid C,\Lambda)}{b(Y\mid C)}
&=\mathbb E\log\frac{r(Y\mid C,\Lambda)}{\bar r(Y\mid C)}
+\mathbb E\log\frac{\bar r(Y\mid C)}{b(Y\mid C)}\\
&\quad-\mathbb E\log\frac{r(Y\mid C,\Lambda)}{q(Y\mid C,\Lambda)}\\
&=I(Y;\Lambda\mid C)
+\mathbb E_C D_{\mathrm{KL}}(\bar r\Vert b)
-\mathbb E_{C,\Lambda}D_{\mathrm{KL}}(r\Vert q).
\end{aligned}
\tag{AC.1}
\]

Equation (AC.1) separates three effects that must not be conflated:

1. \(I(Y;\Lambda\mid C)\) is genuine conditional predictive information in the lift;
2. \(D_{\mathrm{KL}}(\bar r\Vert b)\) is ordinary marginal disagreement with Born's rule;
3. \(D_{\mathrm{KL}}(r\Vert q)\) is information lost by an imperfect decoder.

For a Born-marginal-preserving extension, \(\bar r=b\), so

\[
\boxed{
\mathbb E\log\frac{q(Y\mid C,\Lambda)}{b(Y\mid C)}
=I(Y;\Lambda\mid C)
-\mathbb E D_{\mathrm{KL}}(r\Vert q)
\le I(Y;\Lambda\mid C).
}
\tag{AC.2}
\]

Equality holds exactly when \(q=r\) on the supported contexts. The missing resource is therefore conditional information, not computation. More computation can decrease decoder loss; it cannot produce information absent from the admissible boundary record.

Under stationary independent trials, an exact decoder, integrable log increments, and marginal Born preservation, the strong law gives

\[
\frac{1}{T}\log E_T\longrightarrow I(Y;\Lambda\mid C)
\quad\text{almost surely}. \tag{AC.3}
\]

For a prefix-coded candidate of length \(\ell\), the characteristic evidence scale is

\[
T_{\mathrm{scale}}
\approx
\frac{\log(1/\alpha)+\ell\log2}{I(Y;\Lambda\mid C)}. \tag{AC.4}
\]

This is an asymptotic planning scale, not an exact expected stopping time. Dependence, overshoot, composite-null conservatism, and misspecification change finite performance.

Positive information is not determinism. Exact outcome prediction requires

\[
H(Y\mid C,\Lambda)=0. \tag{AC.5}
\]

Fano's inequality relates remaining conditional entropy to the optimal classification error but does not turn a weak advantage into certainty.

### AC.2. No finite data determine an unrestricted continuation

Let \(S\) be a proper finite subset of an unrestricted context set \(\mathcal C\). Given any law \(p(y\mid c)\), choose \(c_*\notin S\) and construct \(p'\) equal to \(p\) on \(S\) but different at \(c_*\). Every observation confined to \(S\) has identical likelihood under both laws. Therefore no finite dataset restricted to \(S\) can decide their continuation at \(c_*\).

This **No-Free-Extension theorem** does not prevent generalization inside a restricted law class whose extension is fixed by a proved relation. It forbids upgrading finite agreement to an unrestricted universal claim without such a restriction.

The executable collision uses eight contexts. Two laws have total-variation distance zero on the first seven and distance \(0.8\) on the eighth. This is the predictive analogue of the RRLS retained-invariant collision: a continuation distinction not present in the observed boundary cannot be reconstructed downstream.

### AC.3. Three-channel tripwire

A claimed accessible lift is tested with three distinct processes:

\[
E_T^{\mathrm{outcome}},\qquad
E_T^{\mathrm{setting}},\qquad
E_T^{\mathrm{signal}}. \tag{AC.6}
\]

The outcome channel asks whether the lift predicts \(Y\) beyond the complete calibrated context. The setting channel asks whether it predicts a supposedly delayed free setting. The signal channel asks whether a local outcome depends on a remote setting. The target pattern is outcome evidence without setting or signaling evidence, plus structural timing isolation and affirmative equivalence controls. Failure to reject the latter nulls does not prove exact independence.

Six seeded 50,000-trial worlds give:

| Synthetic world | Complete-context outcome crossing | Setting crossing | Signaling crossing | Interpretation |
|---|---:|---:|---:|---|
| Born null | no | no | no | correct negative control |
| planted accessible lift | yes | no | no | target signature; synthetic only |
| future-setting leak | yes | yes | yes | chronology and measurement-independence failure |
| remote signaling | no | yes | yes | naive evidence vanishes after completing the context |
| apparatus memory | no | no | no | naive evidence vanishes after modeling memory |
| memory after reset | no | no | no | reset destroys the artifact |

The planted world is not a physical construction. It only verifies that the audit recognizes a deliberately inserted conditional-information channel.

The production and independent implementations also reproduce:

- a marginal-preserving optimal information rate \(0.368064207168\) nats;
- a nonpreserving example \(0.388475204429=0.368064207168+0.0204109972601\) nats;
- 5,000 randomized special and general decompositions;
- maximum algebraic disagreement \(1.33\times10^{-15}\) in production and \(1.11\times10^{-15}\) independently.

These are floating-point checks of the written identities, not measurements of nature.

### AC.4. Bell, contextuality, and measurement-independence routes

A hidden-variable proposal must identify its route before data. The relevant alternatives are not interchangeable:

| Route | Changed premise | Unavoidable burden |
|---|---|---|
| local and measurement-independent | none | cannot reproduce Bell-violating quantum correlations |
| measurement-dependent | free-setting condition | quantify lift-setting dependence and stop calling the setting independent |
| nonlocal equilibrium | local causality | explains determinism but ordinarily supplies no accessible BPW gain |
| nonlocal nonequilibrium | equilibrium and locality | derive preparation/access and test signaling |
| contextual | noncontextual value assignment | include the actual measurement context in the response law |
| retrocausal | forward-only causal order | define spacetime accessibility and precommit every setting-conditional prediction |
| superdeterministic | measurement independence | impose quantitative restrictions preventing unconstrained retrospective explanation |

Kochen and Specker exclude the relevant noncontextual value assignments, not every contextual ontology [Kochen and Specker (1967)](https://www.jstor.org/stable/24902153). Colbeck and Renner exclude improved prediction under their explicit QM and free-choice assumptions [Colbeck and Renner (2011)](https://www.nature.com/articles/ncomms1416). Hall constructs a deterministic no-signaling singlet model after relaxing measurement independence and quantifies that relaxation [Hall (2010)](https://link.aps.org/doi/10.1103/PhysRevLett.105.250404). Assumption sensitivity is real; it is not a license to omit the cost of the selected route.

Bell-certified randomness establishes a different direction: observed nonlocal correlations can certify randomness under stated independence and no-signaling assumptions [Pironio et al. (2010)](https://arxiv.org/abs/0911.3427). A Dimensional Reality outcome predictor must confront that evidence rather than assume that determinism automatically implies accessible foresight.

### AC.5. The thirteen walls and the next physical test

Wallbreaker Protocol P3 assigns a pass condition to thirteen walls:

1. internal typing;
2. Born information;
3. projection collision;
4. chronology;
5. measurement independence;
6. context completeness;
7. no-signaling;
8. contextuality;
9. Bell route;
10. Lorentz behavior, units, scale, and classical limit;
11. finite-context extension;
12. computational complexity;
13. external replication.

The first, second, and eleventh walls now have stronger mathematics. The audit provides machinery for the fourth through seventh. It does not physically break them. The next bounded experiment must instantiate P3 on fresh hardware data by freezing the apparatus, state/POVM family, composite calibration set, admissible timestamped fields, prefix grammar, prediction commitments, delayed settings, evidence budgets, equivalence margins, resets, reversals, and replication rules before outcomes.

No finite synthetic result can discharge that physical burden. No natural predictive lift, new spacetime coordinate, soul-mediated outcome law, Lorentz-covariant coupling, universal quantum speedup, or P=NP proof has been obtained. The advance is an exact wall map and an adversarial instrument capable of distinguishing several false breakthroughs from the target signature.

## Appendix AD. P3-R1: an executable single-device hardware frontier

### AD.1. Bounded physical question and claim ceiling

P3-R1 converts the next physical step into one bounded question: can one of three frozen eight-cycle coordinates predict a held-out three-qubit measurement result beyond a calibrated coordinatewise upper envelope for the ordinary Born/hardware context, when its complete prediction for every allowed setting is committed before the setting is randomly chosen?

The source state is the Q1 equal-magnitude phase state

\[
|\psi\rangle=\frac1{\sqrt8}\sum_{x=0}^{7}e^{i\phi_x}|x\rangle,
\]

with eight frozen phases. `Z_ALL` measures it directly; `X_ALL` applies three Hadamards and then measures. The target record is one three-bit outcome in Qiskit's frozen \(q_2q_1q_0\) order. Every validation trial is a separate circuit/PUB with one shot, so aggregated counts are not retroactively split into fictitious individual trials.

This is a single-device pilot. It is not spacelike separated and therefore cannot test physical no-signaling, contextuality, or Bell locality. Even a valid outcome crossing is capped at “the declared single-device composite null was rejected” until memory, drift, reset, reversal, causal, and external-replication gates pass.

### AD.2. An exactly marginal-preserving eight-cycle grammar

Let \(b_c(y)>0\) be the ideal Q1 Born reference for setting \(c\), let \(w=t\bmod8\), and let \(\sigma_j\) be one of three fixed permutations: forward, reverse, or binary-reflected Gray order. Define

\[
a_{j,w,c}=\sigma_j(w)\oplus m_c,
\qquad
d_{j,w,c}=a_{j,w,c}\oplus4,
\]

and

\[
\Delta_{j,w,c}=\varepsilon
\min\{b_c(a_{j,w,c}),b_c(d_{j,w,c})\},
\qquad \varepsilon=0.12.
\]

The candidate transfers \(\Delta) from \(d\) to \(a\):

\[
q_{j,w,c}=b_c+\Delta_{j,w,c}(e_a-e_d).
\tag{AD.1}
\]

Because \(0<\varepsilon<1\), (AD.1) remains strictly positive and normalized. Over a complete cycle, each target appears once; the winding whose target is \(a\oplus4\) executes the reverse transfer with the same minimum. Consequently,

\[
\boxed{\frac18\sum_{w=0}^{7}q_{j,w,c}=b_c}
\tag{AD.2}
\]

for every candidate and setting. This makes the probes conditional-information tests in their mathematical reference model rather than disguised marginal Born shifts. Their prefix lengths are 2, 3, and 3 bits. The resulting weights \(1/4,1/8,1/8\) use half the unit mass; the remaining half is cash.

Equation (AD.2) is a construction theorem, not a reason nature must implement the winding variable. The three candidates are deliberately small probes chosen before physical data, not a learned law.

### AD.3. Composite-null score and delayed-setting tripwire

Independent calibration supplies a frozen upper envelope \(u_c(y)\). The run builder uses a simultaneous fixed-sample Hoeffding radius plus a predeclared per-outcome drift allowance. That allowance is an empirical assumption requiring interleaved stress tests; calibration cannot logically guarantee future stationarity.

If the complete ordinary hardware law obeys \(p_t(y\mid C_t)\le u_t(y)\), then any normalized, predictable candidate satisfies

\[
\mathbb E\!\left[
\frac{q_{j,t}(Y_t)}{u_t(Y_t)}\middle|\mathcal F_{t-1}
\right]
=\sum_y p_t(y\mid C_t)\frac{q_{j,t}(y)}{u_t(y)}\le1.
\tag{AD.3}
\]

Thus each product is a nonnegative supermartingale, and the Kraft/cash mixture is one as well. The primary anytime threshold is \(1/(2.87\times10^{-7})\). A separate predictable setting process tests whether the same lift forecasts the delayed balanced setting at \(\alpha=10^{-3}\). A setting crossing disqualifies an outcome crossing; failure to cross is recorded only as `NO_CROSS_NOT_EQUIVALENCE`, never as proof of exact setting independence.

All setting-conditional outcome vectors and setting-tripwire probabilities enter a SHA-256 prediction chain before any validation setting is generated. Every setting points back to its prediction digest. The pre-run manifest binds the config, calibration, commitments, and settings; a second manifest binds provider job records, raw outcomes, and score. Local hashes detect alteration relative to the manifests but do not prove public time. A court-grade execution must externally timestamp the pre-run manifest before submission.

The current provider interface supplies job-submission and result-retrieval times, not authenticated device-level timestamps for individual shots. P3-R1 therefore proves that all settings existed before submission, not that the unavailable timestamp was measured.

### AD.4. Executed adversarial software evidence

Production and independent implementations were tested on three 20,000-trial seeded worlds:

| World | Maximum outcome log mixture | Outcome threshold crossed? | Maximum setting log mixture | Setting threshold crossed? | Interpretation |
|---|---:|---:|---:|---:|---|
| Exact Born reference | 0.279025 | No | 4.631600 | No | Correct negative control |
| Frozen forward lift | 25.026590 | Yes | 1.346171 | No | Intended synthetic target |
| Forward lift plus setting dependence | 18.835820 | Yes | 1094.733929 | Yes | Correctly disqualified |

The independent replay imports none of the production runner. It reconstructs the phase-state Born probabilities, all prediction vectors, the hash chain, chronology, marginal-preservation identity, and both e-processes. It agrees with the production scores. The run-specific maximum marginal error is \(5.55\times10^{-17}\). A further 5,000 randomized property audit keeps the eight-cycle marginal error below \(3.33\times10^{-16}\) and finds no violation of the composite-null one-step bound. Deliberately changing a committed trial makes the verifier fail at the file digest. These are software and synthetic tests only.

### AD.5. Activation boundary and next honest wall

The shipped configuration has three intentional physical blockers: no fresh run UUID, no explicitly selected accessible QPU, and no real calibration envelope/artifact. Qiskit and Qiskit Runtime are also not installed in the present audit environment. The runner refuses live freeze or submission until those conditions are met; credentials remain in the provider's saved-account mechanism and are never accepted in the config or command line.

Current IBM documentation describes authentication through `QiskitRuntimeService`, selection of an accessible backend, transpilation into that backend's instruction set, and execution through Runtime primitives [IBM Quantum hardware guide](https://quantum.cloud.ibm.com/docs/en/guides/hello-world). IBM Cloud access requires an account, API key, and instance identifier [IBM Quantum cloud setup](https://quantum.cloud.ibm.com/docs/en/guides/cloud-setup). Accessible QPUs depend on the account's instance and plan [IBM Quantum instances](https://quantum.cloud.ibm.com/docs/en/guides/instances).

The executable path is now calibration \(\to\) envelope \(\to\) prediction commitment \(\to\) delayed settings \(\to\) one-shot submissions \(\to\) independent replay. Nature has not yet supplied the final records. A negative pilot would reject only these three lifts at this scale. A positive pilot would trigger controls and replication, not a declaration that all randomness or every remaining wall had been broken.

## Appendix AE. Dimensional Chess C1: exact solution of a closed game universe

### AE.1. Finite does not mean computationally feasible

Once its rule state is complete and automatic termination is fixed, chess has a finite state graph and therefore a definite game-theoretic value at its initial position. That proves existence of a solution by finite backward induction. It supplies no useful bound on the work. Shannon's original estimate places the ordinary game tree near \(10^{120}\), illustrating why finiteness and feasible solvability must not be conflated [Shannon (1950)](https://vision.unipv.it/IA1/ProgrammingaComputerforPlayingChess.pdf).

C1 begins with a closed standard-board subuniverse: White king and rook against Black king; checkmate wins; stalemate, rook capture, or infinite play draws. Its complete state is

\[
s=(k_W,r_W,k_B,\tau).
\]

For full chess, visible piece placement is an invalid projection. Side to move, castling rights, en-passant state, halfmove clock, and repetition-relevant history can change the legal future while leaving the board picture unchanged. They must be retained as coordinates.

### AE.2. A collision-free symmetry lift

Let the dihedral group \(D_4\) act on the board and let

\[
\kappa(s)=\min_{g\in D_4}\operatorname{encode}(gs).
\]

The quotient is dynamically closed only if

\[
\{\kappa(s'):s'\in T(s)\}
=
\{\kappa(s'):s'\in T(gs)\}
\quad\text{for every }g\in D_4.
\tag{AE.1}
\]

C1 checks (AE.1), including check status, on every canonical state and all eight symmetries. All 400,120 comparisons pass. The 399,112 raw legal states collapse to 50,015 exact classes, a factor of \(7.979846046\ldots\). This is a lawful dimensional lift: it removes viewpoint redundancy without merging different future laws.

### AE.3. Time cycles become a least fixed point

Move repetition makes the game a graph rather than a tree. Starting from checkmates \(L_0\), C1 repeatedly applies

\[
W_{n+1}=W_n\cup\{s:\exists s'\in T(s),\ s'\in L_n\}
\]

and

\[
L_{n+1}=L_n\cup\{s:T(s)\neq\varnothing,\ T(s)\subseteq W_{n+1}\}.
\]

The complement of the least fixed point \(W\cup L\) is the draw region. Winning states minimize distance to mate; losing states maximize it. This treats a returned position as the same coordinate instead of creating an infinite sequence of duplicated nodes.

### AE.4. Exact result and proof certificate

The canonical graph contains 558,816 transitions including the terminal rook-capture draw:

| Value from side to move | Canonical states |
|---|---:|
| Win | 21,959 |
| Loss | 25,260 |
| Draw | 2,796 |

There are no White-to-move losses and no Black-to-move wins. The maximum forced mate is 32 plies, or 16 moves. One maximum state has White king a1, White rook b2, Black king c2, and Black to move. The stored optimal line contains 32 legal moves and ends with Black checkmated. The 16-move maximum agrees with independent published KRK verification [Stålmarck et al. (2013)](https://journals.sagepub.com/doi/pdf/10.3233/ICG-2013-36203?download=true).

An independent program imports none of the solver. It reconstructs all raw states, symmetry classes, transitions, terminal conditions, Bellman equations, distance equations, policy witnesses, and concrete principal-variation moves. It passes. After one WDL entry was changed and its file digest recomputed, the verifier still rejected the certificate because the local draw fixed-point equation failed. Hash integrity and mathematical correctness are therefore separately tested.

### AE.5. The wall exposed by chess

Retrograde tablebases and board symmetries are established ideas; C1 does not claim their invention. It demonstrates that the framework's core disciplines work on a complete finite universe:

- hidden coordinates restore transition closure;
- a projection is accepted only after collision testing;
- geometry supplies an exact symmetry quotient;
- cycles are solved as graph fixed points;
- the initial unknown value is reduced from \(\{W,D,L\}\) by constraints;
- every conclusion carries a local certificate;
- solving one restricted family never licenses an unrestricted extension.

Modern Syzygy infrastructure supplies exact tables for complete seven-piece endgames and partial eight-piece work [Lichess tablebase implementation](https://github.com/lichess-org/lila-tablebase). The C1 symmetry factor is powerful locally but leaves Shannon's scale essentially unchanged: \(10^{120}/8\approx1.25\times10^{119}\). Full chess therefore requires either immense enumeration or a new uniform value-preserving quotient whose recognition, transformation, solution, and verification costs are all bounded. Any proposed shortcut fails immediately if two merged positions have different minimax values or different quotient successor sets.

The experiment has solved its declared universe and identified the next missing variable precisely: not “more time,” but a drastically stronger collision-free canonicalizer.

## Appendix AF. Dimensional Chess C2: interval bounds, certified pruning, and the full-chess gate

### AF.1. Freeze the target before optimizing it

“Solve chess” now means determine the game-theoretic value of the normal initial
position under a frozen board-game ruleset and provide a strategy DAG that an
independent program can verify against every required adversarial reply. Playing
above human level, producing one principal variation, or solving additional
endgames does not meet this definition.

The frozen target uses legal moves, checkmate, stalemate, dead positions,
claimable threefold/50-move draws, and automatic fivefold/75-move draws from the
FIDE Laws effective 1 January 2023. Clocks, resignations, and negotiated draw
offers are excluded; a correct claim is an optional transition to a draw
terminal. Articles 9.2–9.6 define the relevant repetition and move-count rules
[FIDE Laws of Chess](https://handbook.fide.com/chapter/e012023). A 2026 preprint
derives a maximum game length of 17,697 plies under those Laws. That establishes
a finite depth, not a feasible tree size [Yim (2026)](https://arxiv.org/abs/2608.14762).

The Markov-complete state is

\[
S=(B,\tau,C,E,h,\mathcal R),
\tag{AF.1}
\]

where $B$ is current typed piece placement (including promoted material), $\tau$ is side to move, $C$ is
castling state, $E$ is en-passant state, $h$ is the reversible-move clock,
and $\mathcal R$ is the repetition-relevant path record. FIDE Article 9.2.3
explicitly makes possible moves, side to move, en passant, and castling rights
part of repetition identity. The same visible board with different hidden
coordinates can therefore have a different transition law or draw status.

### AF.2. D-/D+ becomes interval minimax

Use White-relative values $v(s)\in\{-1,0,+1\}$. An unresolved state carries

\[
I(s)=[D^-(s),D^+(s)],\qquad D^-(s)\le v(s)\le D^+(s).
\tag{AF.2}
\]

The computational zero axis begins as $[-1,+1]$, retaining all three possible
readouts without pretending they are equally probable. Exact terminals and
verified tablebases seed point intervals. Given sound bounds on every child,

\[
\begin{aligned}
D^-(s)&=\max_a D^-(T(s,a)),&D^+(s)&=\max_a D^+(T(s,a))&&\text{for White},\\
D^-(s)&=\min_a D^-(T(s,a)),&D^+(s)&=\min_a D^+(T(s,a))&&\text{for Black}.
\end{aligned}
\tag{AF.3}
\]

Only proved premises may tighten an interval. The ordinary W/D/L result is read
only when $D^-=D^+$. This preserves the framework's richer truth coordinate
while retaining exact minimax semantics.

### AF.3. Exact Disadvantage Pruning Theorem

Suppose move $a$ has sound interval $[L_a,U_a]$. At a White/max state, $a$
may be discarded when another move $b$ satisfies

\[
L_b\ge U_a.
\tag{AF.4}
\]

The proof is the chain $v_b\ge L_b\ge U_a\ge v_a$. At a Black/min state,
$a$ may be discarded when

\[
U_b\le L_a,
\tag{AF.5}
\]

because $v_b\le U_b\le L_a\le v_a$. Equality preserves the root value even
if it removes an equally optimal alternative. A merely heuristic score is not
a bound and cannot license either deletion. Sacrifices, quiet defensive moves,
and zugzwang make local-looking disadvantage unreliable.

The theorem formalizes the proposed shortcut: discard possibilities only after
their inability to affect the minimax result is certified. Alpha-beta bounds
are one special case.

### AF.4. Exact KQK extension

C2 independently implements the complete White-king-and-queen versus
Black-king universe under the same small-game terminal convention as C1:
checkmate wins; stalemate, queen capture, or infinite non-mating play draws.

| Quantity | KQK result |
|---|---:|
| Raw legal states | 368,452 |
| $D_4$-canonical states | 46,137 |
| Canonical transitions | 609,873 |
| Symmetry-transition comparisons | 369,096 |
| Side-to-move wins | 18,081 |
| Side-to-move losses | 25,160 |
| Draws | 2,896 |
| Maximum forced mate | 20 plies = 10 moves |

One maximum-distance state is White king a1, White queen b2, Black king e4,
Black to move. Its certified line contains 20 legal moves and ends in
checkmate. The verifier imports no production functions and reconstructs every
state, move, orbit, terminal, Bellman equation, distance, policy edge, and
concrete principal-variation move. It passes. This verifies a second movement
law but not interacting friendly pieces, cross-material captures, pawns, or
history.

### AF.5. Strong future-law identity fails as the large quotient

C2 also computes the coarsest strong-bisimulation refinement induced by side to
move, terminal kind, and successor blocks. Previous block identity is retained
at every round, making refinement monotone. Every final block is uniform in the
independently certified WDL/DTM values, and a separately ordered implementation
reproduces the counts.

| Universe | $D_4$ states | Bisimulation blocks | Compression | Rounds |
|---|---:|---:|---:|---:|
| KRK | 50,015 | 49,885 | 1.002606× | 13 |
| KQK | 46,137 | 45,249 | 1.019625× | 9 |

Thus identical complete future law is safe but much too strict, and its
construction already requires the complete graph. By contrast, after solving,
side/WDL/DTM yields only 34 KRK and 22 KQK semantic classes—1,471× and 2,097×
compression. That partition is circular because it reads the unknown answer.
The research target is a cheap pre-solution invariant that gives sound bounds
approaching those semantic classes.

### AF.6. Measured ceiling of disadvantage pruning

The solved values were used as an oracle to measure—not assume—the available
pruning:

| Universe | Original branching | Strictly worse WDL edges | WDL+DTM-optimal branching |
|---|---:|---:|---:|
| KRK | 11.1810 | 40,266 / 558,816 = 7.21% | 1.8880 |
| KQK | 13.2633 | 40,323 / 609,873 = 6.61% | 2.3062 |

WDL-only dominance removes relatively little because many moves share the same
eventual outcome. Adding fastest-mate/longest-defense preference leaves a much
narrower ideal policy kernel. The latter does not solve the recognition
problem: its labels were obtained from the completed tables. It is an empirical
ceiling for a future bound engine, not a free oracle.

### AF.7. Counterexample-guided dimensional refinement

The next candidate algorithm is:

1. map complete states to a polynomial-cost relational signature;
2. compute sound abstract $[D^-,D^+]$ bounds with may/must transitions;
3. prune only through (AF.4)–(AF.5), storing every witness;
4. if a block remains ambiguous, extract concrete states producing the conflict;
5. add the missing invariant and split only the necessary block;
6. repeat until the initial state has $D^-=D^+$.

Candidate invariants include material interaction, attack/defence dependency,
king-safety cuts, pawn reachability, promotion cones, zugzwang parity,
reversible-region components, and repetition phase. No coordinate is accepted
because it sounds higher-dimensional; it must remove a witnessed projection
collision while preserving sound bounds. Description, recognition,
transformation, refinement, and certificate costs all count, so an exponential
lookup table cannot masquerade as a short theory.

### AF.8. Chess, quantum algorithms, and physical dimensions are separate gates

Balanced NAND/AND-OR trees have quantum oracle algorithms with square-root
scaling. Farhi, Goldstone, and Gutmann give $\Theta(\sqrt N)$ Hamiltonian-oracle
time for their NAND-tree model [Farhi et al. (2008)](https://theoryofcomputing.org/articles/v004a008/v004a008.pdf).
Chess is not automatically that oracle: it has irregular branching,
transpositions, cycles, history, and a non-free legal-transition computation.

A quantum result must specify a reversible complete-state encoding, reversible
move and terminal predicates, oracle construction cost, logical qubits, gate
count and depth, error assumptions, success probability, and a matched
classical baseline. A simulator validates circuit semantics but not hardware
advantage. Solving chess would validate the computational lift; it would not by
itself prove an eight-dimensional spacetime ontology.

Generalized $n\times n$ chess is EXPTIME-complete
[Fraenkel and Lichtenstein (1981)](https://doi.org/10.1016/0097-3165(81)90016-9).
This does not prevent solving the fixed 8×8 initial state, but a claimed uniform
polynomial method would have to confront that result.

### AF.9. Claim gate and next rung

“Full chess solved” remains prohibited until the normal initial state's bounds
collapse and an independent verifier validates a strategy DAG against every
required reply under the frozen rules. “Quantum advantage” remains prohibited
until the full reversible/oracle cost is compared experimentally or
theoretically with the best classical method. “Physical dimensional theory”
remains prohibited until independent natural data distinguish it from standard
physics.

The next executable rung is KBBK/KBNK, then capture transitions between material
strata, followed by a table-hidden D-/D+ solver that must rediscover the KRK/KQK
root values using only sound bounds and counterexample refinement. Pawns,
promotion, castling, en passant, halfmove phase, and repetition phase follow.
The full search should target a shared strategy DAG from the initial state, not
materialize every state unless forced.

## Appendix AG. Dimensional Chess C3: the complete state reaches the initial position

### AG.1. The legal-transition wall is crossed, not the solution wall

C3 implements the frozen full-chess state

\[
S=(B,\tau,C,E,h,n,\mathcal R),
\tag{AG.1}
\]

where $n$ is included for standards-compatible position serialization even
though it does not alter legal choices. The engine generates all orthodox piece
moves, check and self-check legality, both castling sides, en passant, all four
promotion choices, and the irreversible changes to castling rights after king
moves, rook moves, and rook captures. It retains repetition-relevant history
and implements claimable threefold/50-move draws and automatic
fivefold/75-move draws with checkmate precedence.

The en-passant component of the repetition key is retained only when the side
to move has a legal en-passant capture. This is necessary because FIDE
Article 9.2.3 defines positional identity partly through possible moves, not
merely through a visually identical board
[FIDE Laws of Chess](https://handbook.fide.com/chapter/e012023).

The supplied dead-material predicate deliberately covers standard
material-insufficient cases only. Determining whether no legal sequence can
possibly produce checkmate in every composed position is a broader reachability
problem. Until that predicate or an equivalent exact terminal mechanism is
completed, C3 cannot certify the whole FCS-2023 graph.

### AG.2. Move-generation audit

An independent driver checks six reference positions from the official
[Stockfish perft suite](https://github.com/official-stockfish/Stockfish/blob/master/tests/perft.sh).

| Position | Depth | Reference nodes | C3 |
|---|---:|---:|---|
| Initial position | 4 | 197,281 | PASS |
| Kiwipete | 3 | 97,862 | PASS |
| Position 3 | 4 | 43,238 | PASS |
| Position 4 | 3 | 9,467 | PASS |
| Position 5 | 3 | 62,379 | PASS |
| Position 6 | 3 | 89,890 | PASS |
| **Total** |  | **500,117** | **PASS** |

Ten targeted groups also pass: castling execution and rights, rejection of
castling through check, en-passant execution, rejection and repetition-key
normalization of pinned en passant, promotion multiplicity, rook-capture rights
mutation, threefold path state, automatic 75-move draw, checkmate precedence,
and basic dead material. Passing these tests strongly constrains the
implementation errors they target; it is not a proof over all reachable
states.

### AG.3. First exact interval probe of the normal start

C3 expands the ordinary starting position to four plies. Every nonterminal
horizon state keeps the zero-information interval $[-1,+1]$. Only exact
terminals and rule-authorized draw actions may introduce point intervals. No
piece values, engine evaluation, opening data, or learned predictor enters the
calculation.

| Quantity | Exact finite result |
|---|---:|
| Root legal moves | 20 |
| Nodes including root | 206,604 |
| Legal edges | 206,603 |
| Horizon leaves | 197,273 |
| Checkmate leaf visits | 8 |
| Exact node visits | 16 |
| Certified dominated choices below root | 242 |
| Certified ignored-edge fraction below root | 242 / 206,583 = 0.1171% |
| Certified dominated root choices | 0 |
| Root interval | $[-1,+1]$ |

The node identity

\[
1+20+400+8{,}902+197{,}281=206{,}604
\tag{AG.2}
\]

and terminal identity

\[
197{,}273+8=197{,}281
\tag{AG.3}
\]

give simple internal cross-checks. The checkmate leaves lie below the initial
moves f2-f3, f2-f4, and g2-g4. They do not make those moves proved losses,
because other continuations remain possible. All twenty root children retain
$[-1,+1]$.

### AG.4. Forcing dominance formalizes ignored legal moves

The number of legal alternatives a path can ignore is useful only after it is
made subordinate to correct play. Let $P$ be the player whose strategy is being
proved and let $u_P(a)\in\{-1,0,+1\}$ be the outcome after action $a$, ordered
from $P$'s perspective. When the outcome is decisive, define mate efficiency

\[
m_P(a)=
\begin{cases}
-d(a),&u_P(a)=+1,\\
+d(a),&u_P(a)=-1,\\
0,&u_P(a)=0,
\end{cases}
\tag{AG.4}
\]

where $d(a)$ is exact distance to mate. Thus a player prefers a shorter forced
win and a longer forced loss. For a proof strategy $\Sigma$, let
$e_\Sigma(a)$ count sibling legal choices removed by stored interval or
lexicographic-bound certificates. The forcing-dominance value is

\[
\operatorname{FD}_P(a)=
\bigl(u_P(a),m_P(a),e_\Sigma(a)\bigr),
\tag{AG.5}
\]

ordered lexicographically. The first coordinate chooses correct
game-theoretic play, the second chooses mate efficiency, and only the third
prefers the proof that ignores the most certified-irrelevant moves.

For a strategy DAG rather than one line, raw repeated-node counts can be
manipulated by cycles. C3 therefore counts eliminations at unique complete
states and records the witnessing inequality. A future aggregate should report
both

\[
E(\Sigma)=\sum_{s\in R_\Sigma}
\bigl(|A(s)|-|K_\Sigma(s)|\bigr)
\quad\text{and}\quad
\rho(\Sigma)=
\frac{E(\Sigma)}{\sum_{s\in R_\Sigma}|A(s)|},
\tag{AG.6}
\]

where $R_\Sigma$ is the set of unique reachable controlled states, $A(s)$ all
legal actions, and $K_\Sigma(s)$ the actions that a verifier must retain.
Neither $E$ nor $\rho$ can authorize a deletion; they measure compression after
the deletion has been proved safe.

The current 242 certificates are WDL-interval certificates below the root, not
complete distance-to-mate certificates. Zero root choices are eliminated.
Therefore shallow forcing-dominance does not yet select an initial move.

### AG.5. Updated wall and claim gate

C3 falsifies the idea that full move generation or shallow forced-mate
detection is the missing variable. The active computational wall is a source of
strong, cheap, sound bounds before minimax values are known. Candidate
relational coordinates must be tested table-hidden on solved endgames: they
must recover WDL and distance-to-mate choices without reading those labels,
then emit certificates checked by an independent implementation.

“Full chess solved” remains prohibited until:

1. the complete frozen terminal semantics are implemented;
2. the initial interval collapses to one value;
3. a repetition-safe strategy DAG covers every required adversarial reply;
4. an independent verifier accepts every transition, bound, dominance
   deletion, outcome, and mate-distance claim; and
5. construction, verification, memory, and certificate sizes are reported.

The exact C3 claim is narrower: a tested complete-move state engine reaches the
real initial position, and a proof-only depth-four expansion certifies that the
root is still unresolved. This advances the computational program without
claiming a result that the evidence does not contain.

## Appendix AH. Dimensional Chess C4: constraint waves and relational phase

### AH.1. Human search is selective constraint propagation

A legal move list is not a value ordering. Human players instead track attacked
squares, defended pieces, king safety, mobility, confinement, threats, and the
opponent's strongest reply. Let $\mathcal A_P(s)$ be player $P$'s attack field
and $\mathcal L_P(s)$ the legal actions in state $s$. Action $a$ produces

\[
s'=T(s,a).
\tag{AH.1}
\]

For kings, an attacked destination is illegal. For other pieces it is not
automatically irrational: a sacrifice may force mate or gain a larger
advantage. The exact disqualification rule therefore remains interval
dominance, not membership in an attack map.

Terminal and draw facts act as boundary values in the alternating Bellman
operator:

\[
\begin{aligned}
D^-_{k+1}(s)&=\max_aD^-_k(T(s,a)),
&
D^+_{k+1}(s)&=\max_aD^+_k(T(s,a))
&&(\tau=\mathrm W),\\
D^-_{k+1}(s)&=\min_aD^-_k(T(s,a)),
&
D^+_{k+1}(s)&=\min_aD^+_k(T(s,a))
&&(\tau=\mathrm B).
\end{aligned}
\tag{AH.2}
\]

Repeated application is the precise constraint wave: forced facts propagate
backward through squares, moves, and adversarial replies. Removing material
usually reduces interactions and branching, which explains why the field
becomes easier to resolve in endgames without asserting that every individual
material removal makes a position computationally simpler.

### AH.2. Table-hidden geometric language

C4 freezes five variants across the complete KRK and KQK universes. The bound
phase sees legal transitions, terminal rules, the theorem that a lone black
king cannot mate, and declared coordinates only. Exact tablebase WDL and DTM
labels are computed after all variants and intervals are frozen.

The distance base uses side to move, terminal kind, edge/corner confinement,
check and capture-draw status, legal-move count, king and piece distances, and
piece protection. Three isolated lifts add:

1. relative dot product and absolute cross product of the white-king and
   long-range-piece vectors about the black king;
2. the size of the black king's connected static safe region;
3. own/opponent mobility and edge-direction counts.

The combined constraint wave includes all three and the remaining axis
distances. Transition profiles are deduplicated inside each signature block, so
their count measures operational graph compression rather than only label
compression.

### AH.3. Exact table-hidden results

| Universe and signature | Exact DTM | Certified WDL+DTM eliminations | State compression | Profile compression |
|---|---:|---:|---:|---:|
| KRK distance base | 3,911 / 50,015 = 7.82% | 21.21% | 1.7608× | 1.1000× |
| KRK + orientation | 13,775 / 50,015 = 27.54% | 74.25% | 1.0700× | 1.0005× |
| KRK + safe region | 6,759 / 50,015 = 13.51% | 60.27% | 1.3621× | 1.0811× |
| KRK + mobility | 8,791 / 50,015 = 17.58% | 70.56% | 1.4164× | 1.0937× |
| **KRK combined** | **44,633 / 50,015 = 89.24%** | **506,548 / 558,816 = 90.65%** | **1.0067×** | **1.0001×** |
| KQK distance base | 6,959 / 46,137 = 15.08% | 54.31% | 1.6326× | 1.0800× |
| KQK + orientation | 30,554 / 46,137 = 66.22% | 90.02% | 1.0660× | 1.0011× |
| KQK + safe region | 22,008 / 46,137 = 47.70% | 87.84% | 1.3205× | 1.0556× |
| KQK + mobility | 21,496 / 46,137 = 46.59% | 86.47% | 1.2828× | 1.0745× |
| **KQK combined** | **45,276 / 46,137 = 98.13%** | **563,800 / 609,873 = 92.45%** | **1.0071×** | **1.0003×** |

Every variant proves every WDL value in both universes. Mate distance is the
harder coordinate: equal WDL does not imply equal fastest-mate or
longest-resistance time. Relative orientation produces the largest isolated
DTM improvement, showing that separate scalar distances erase a genuine
relational phase. Safe-region and mobility coordinates also carry information.

The state certificate contains 96,152 rows. A verifier independent of feature
construction and interval propagation compares every row with the previously
verified exact tables, recomputes legal-successor dominance, reproduces both
elimination totals, and reports zero WDL violations, zero DTM violations, and
zero summary mismatches.

### AH.4. Predictive success does not equal scaling

The combined KRK signature has 49,681 blocks for 50,015 states and 50,010
successor profiles. The combined KQK signature has 45,813 blocks for 46,137
states and 46,123 successor profiles. Thus it recovers most exact mate
distances by becoming almost injective. It still enumerates the concrete graph
to construct the signatures.

This is a critical falsification. High DTM coverage and more than 90% certified
move elimination do not imply a small proof when the certificates depend on
almost one transition profile per state. The absolute constraint field is
informative but is not yet the scalable missing variable.

### AH.5. Constraint-delta lift

Human analysis often represents what a move changes rather than the entire
absolute attack field. Define

\[
C(s)=
\bigl(
\text{safe mobility},
\text{opponent mobility},
\text{confinement},
\text{attack dependencies},
\text{forcing rights}
\bigr)
\tag{AH.3}
\]

and

\[
\Delta C(s,a)=C(T(s,a))-C(s).
\tag{AH.4}
\]

The C5 candidate groups actions by equivalent constraint deltas: opened and
closed lines, pins created or broken, regions split or joined, defenders added
or removed, and adversarial replies eliminated. A reusable delta law could
describe many absolute boards with one transition type.

The proposal survives only if it:

1. is constructed without solved labels;
2. emits sound WDL and DTM bounds;
3. transfers across rook, queen, bishop, knight, pawn, and mixed-material
   strata;
4. counts recognition, construction, refinement, and certificate costs; and
5. retains substantial successor-profile compression.

The exact C4 result is therefore two-sided: the proposed restriction wave is a
verified source of mate information, while its present absolute encoding is
rejected as a full-chess shortcut. The next missing variable is a compact
canonical law for move-induced constraint change.

## Appendix AI. Dimensional Chess C5: local tension and tactical entry

### AI.1. Attack multiplicity is a field, not a verdict

For state $s$, colour $c$, and square $q$, define the geometric attack
multiplicity

\[
A_c(q;s)=\sum_{p\in P_c(s)}\mathbf 1\{p\leadsto q\}.
\tag{AI.1}
\]

Sliding attacks stop at the first occupied square. Pawns contribute their
diagonal attacks, kings their adjacent squares, and pinned pieces retain their
geometric pressure. This field is deliberately separated from legal move
generation: a geometric dependency may exist even when moving the attacking
piece would expose its king.

Define local tension, signed control, and total load by

\[
\tau(q;s)=A_W(q;s)A_B(q;s),\qquad
\nu(q;s)=A_W(q;s)-A_B(q;s),\qquad
\lambda(q;s)=A_W(q;s)+A_B(q;s).
\tag{AI.2}
\]

Thus $\tau$ is positive exactly on mutually attacked squares, $\nu$ records the
local numerical imbalance, and $\lambda$ records activity even when only one
side attacks. For an occupied square, raw fragility is

\[
F(q;s)=\max\{0,\operatorname{attackers}(q;s)
                  -\operatorname{defenders}(q;s)\}.
\tag{AI.3}
\]

This is a search feature, not a theorem that the occupant is lost. Checks,
pins, sacrifices, intermediate moves, and mating threats can dominate the raw
count.

### AI.2. Tension-breaking entry

Let legal action $a:q_0\to q_1$ produce $s'=T(s,a)$. From the mover's
perspective define

\[
E_A(s,a)=A_{\rm mover}(q_1;s),\qquad
E_D(s,a)=A_{\rm opponent}(q_1;s),
\tag{AI.4}
\]

and the destination's local tension release

\[
R_\tau(s,a)=\max\{0,\tau(q_1;s)-\tau(q_1;s')\}.
\tag{AI.5}
\]

C5 presently calls $a$ a tension-breaking entry when it captures an enemy
occupant on an already contested destination, $E_A>E_D$, and $R_\tau>0$. The
condition operationalizes the proposal to spend computation first where a
move converts interacting attack dependencies into a favorable local
imbalance.

### AI.3. Adversarial ordering

For each legal action, C5 enumerates the opponent's legal replies and keeps the
worst conventional static reply vector

\[
V=(\text{terminal},\text{material},\text{pressure},
   \text{control},\text{mobility})
\tag{AI.6}
\]

from the mover's perspective. It then orders computation by this worst reply,
immediate mate, captured material, the tension-entry coordinates, control
flips, and replies eliminated. This is a move-ordering law only. Sound removal
still requires the interval-dominance inequality of Appendices AF and AG.

### AI.4. Executable fixture and independent audit

In the declared Scholar's-mate state

```text
r1bqkb1r/pppp1ppp/2n2n2/4p2Q/2B1P3/8/PPPP1PPP/RNB1K1NR w KQkq - 4 4
```

there are 43 legal moves. The `f7` pawn is attacked from `c4` and `h5` and
defended from `e8`, giving

\[
(A_W,A_B,\tau,\nu,F)=(2,1,2,1,1).
\tag{AI.7}
\]

The C5 ordering ranks `Qxf7#` first. Its entry coordinates are

\[
(E_A,E_D,E_A-E_D,R_\tau)=(2,1,1,1),
\tag{AI.8}
\]

and it reduces Black's legal replies from 28 to zero. `Bxf7+` reaches the same
target but leaves one legal reply. An independent verifier does not import the
tension implementation: it reconstructs both colours' attack origins, checks
the formulas for all 64 squares, recounts the 43 legal moves, and verifies both
reply facts. All checks pass.

### AI.5. Counterexample and boundary

Before `Qxf7#`, total board tension is 16. After checkmate it is 28. The losing
side nevertheless has zero legal moves. Consequently total tension is not a
monotone evaluation function and cannot be optimized in isolation. It marks
where dependencies are concentrated; terminal semantics and adversarial reply
search determine the value.

The same boundary appears in the geometric hanging count: the White queen on
`h5` is attacked and geometrically undefended before the move, yet White has a
forced immediate mate. This directly falsifies any rule that discards every
move because its moving piece or destination looks locally unsafe.

C5 therefore contributes a verified computation-allocation coordinate, not a
solution of chess. The next experiment must freeze a tactically diverse corpus
and compare proof nodes at equal budgets against standard ordering baselines.
The stronger candidate is a dependency graph for legal exchange sequences,
because raw attacker counts omit piece value, pin legality, tempo, and recapture
order.

## Appendix AJ. Dimensional Chess C6: legal exchange dependencies

### AJ.1. From simultaneous counts to ordered interactions

C5 counts which pieces geometrically attack a square but not the order in which
they can legally arrive. C6 constructs a local capture–recapture graph. Let
$a_0:q_0\to q$ be a legal capture by player $P$, let $s_1=T(s,a_0)$, and let
$\mathcal X_q(s)$ be the legal captures in state $s$ that land on the same
square $q$. If $M_P$ is material balance from $P$'s perspective, define

\[
\Delta M_P(s_i)=M_P(s_i)-M_P(s).
\tag{AJ.1}
\]

The restricted optional-exchange recurrence is

\[
E_P(s_i,q)=
\begin{cases}
\displaystyle
\max\left(\{\Delta M_P(s_i)\}\cup
\{E_P(T(s_i,a),q):a\in\mathcal X_q(s_i)\}\right),
&\operatorname{turn}(s_i)=P,\\[6pt]
\displaystyle
\min\left(\{\Delta M_P(s_i)\}\cup
\{E_P(T(s_i,a),q):a\in\mathcal X_q(s_i)\}\right),
&\operatorname{turn}(s_i)\ne P.
\end{cases}
\tag{AJ.2}
\]

The current material delta is the option to stop exchanging on $q$. Because
each graph edge comes from full legal move generation, pins, illegal king
captures, and revealed slider rays affect the sequence automatically.

This is not a full-position minimax value. It omits captures elsewhere,
zwischenzugs, quiet threats, and most off-square check evasions. A recorded
graph node may be in check even though the abstract stop option is not a legal
full-game action. C6 therefore uses $E_P$ only as an ordering coordinate.

### AJ.2. Exact bounded proof layer

The benchmark surrounds the heuristic ordering with an exact finite AND/OR
mate predicate. For attacker $P$,

\[
B_P(s,d)=
\begin{cases}
1,&s\text{ is checkmate won by }P,\\
0,&d=0\text{ or }s\text{ is a draw or other terminal},\\
\bigvee_{a\in\mathcal L(s)}B_P(T(s,a),d-1),
&\operatorname{turn}(s)=P,\\
\bigwedge_{a\in\mathcal L(s)}B_P(T(s,a),d-1),
&\operatorname{turn}(s)\ne P.
\end{cases}
\tag{AJ.3}
\]

All orderings traverse the same legal successor relation. They may short-circuit
at different times, but they cannot change a proved Boolean result.

### AJ.3. Deterministic recapture corpus

Using seed `20260915`, the generator accepts the first eight distinct states
that contain a legal capture followed by a legal same-square recapture and have
a forced mate of minimal depth three plies. It consumes 258 candidate states.
Admission uses only the Boolean mate property; no comparative node count or
runtime is read. The selected root exchange graphs reach two or three capture
plies and contain two to four nodes.

| Ordering | Proof nodes | Reduction vs lexical | Measured time |
|---|---:|---:|---:|
| Lexical UCI | 10,394 | 0.00% | 2.014 s |
| Checks/captures forcing | 699 | 93.27% | 0.579 s |
| C5 tension | 809 | 92.22% | 0.700 s |
| **C6 dependency** | **668** | **93.57%** | **0.638 s** |

C6 uses 17.43% fewer nodes than C5 and 4.43% fewer than forcing in aggregate.
Graph construction nevertheless makes it 10.22% slower than forcing in this
run. Per position it beats forcing three times, ties once, and loses four times.
The aggregate advantage is consequently a narrow small-corpus result, not a
universal ordering theorem.

### AJ.4. Independent audit

The independent C6 verifier imports no C6 code. It re-enumerates all root moves,
proves the absence of mate in one, and constructs the three-ply AND/OR
certificate for every position. It reproduces every reported winning root
move, covering 33 defender reply branches with 53 mating continuations. It also
reconstructs the largest declared root exchange graph in each position and
matches every value, node count, maximum capture depth, and checked-node count.
All eight audits pass, and the four aggregate proof-node totals are reproduced.

### AJ.5. Consequence for the full-chess program

C6 supplies evidence that legal recapture order contains information not
present in simultaneous tension counts. It does not produce a sound new bound,
does not prune a legal move, and does not narrow the initial-position interval.
The result therefore improves computation allocation without crossing the
solution boundary.

The next candidate is a proof-carrying forcing-dependency graph that includes
checks, mandatory evasions, off-square captures, promotions, and quiet forcing
threats. Its nodes must carry sound lower and upper game-value intervals. It
must also show substantial compression after the cost of graph recognition and
construction is counted. Absent that bound or a complete strategy certificate
from the orthodox initial position, chess remains unsolved.

## Appendix AK. Initial-position mate frontier R1

### AK.1. Direct run on the orthodox root

C6 is next applied to the normal initial state

```text
rnbqkbnr/pppppppp/8/8/8/8/PPPPPPPP/RNBQKBNR w KQkq - 0 1
```

using the exact bounded predicate from Equation AJ.3. At seven plies the
dependency-ordered runs return

\[
B_W(s_0,7)=0,
\qquad
B_B(s_0,7)=0.
\tag{AK.1}
\]

Thus neither side can force checkmate within seven plies. Equation AK.1 is not
a WDL value: a mate outside the horizon and every drawing continuation remain
unclassified.

### AK.2. Measured frontier

| Attacker | C6 nodes | Generated edges | Horizon nodes | Exchange graphs | Time |
|---|---:|---:|---:|---:|---:|
| White | 128,724 | 251,799 | 78,842 | 5,370 | 105.820 s |
| Black | 14,846 | 143,274 | 4,046 | 3,127 | 53.319 s |

For White as attacker, the C6 node sequence is 21, 486, 7,536, and 128,724 at
one, three, five, and seven plies. The last extension multiplies nodes by about
17.08. The current ordering has not altered exponential frontier growth.

### AK.3. Independent result and ordering falsification

A verifier that imports no C6 module recomputes the predicate with lexical UCI
ordering. It also obtains Equation AK.1, using 106,322 nodes for White as
attacker and 15,082 for Black. The independent result is `PASS`.

C6 uses 21.07% more nodes than lexical order for the White-attacker proof and
1.56% fewer for the Black-attacker proof. Combined, C6 uses 143,570 nodes
versus 121,404, an increase of 18.26%. Its aggregate advantage on the C6
synthetic tactical corpus therefore does not transfer to this initial-position
negative-mate task.

The result does not establish that lexical order is intrinsically meaningful.
It encounters counterexample branches earlier by accident in this finite run.
It does establish that a single attractive-move order is not the missing
general solution coordinate.

### AK.4. Dual proof direction

At an attacker node, a positive mate proof needs one successful action while a
negative proof must reject every action. At a defender node, a positive proof
must cover every reply while a negative proof needs one escape. Search priority
therefore depends on both whose turn it is and which side of the interval is
being certified.

C5 heat and C6 exchanges allocate work but do not describe unexpanded
successors. They cannot narrow the root WDL interval, which remains

\[
I(s_0)=[-1,+1].
\tag{AK.2}
\]

The next stage must build forcing-dependency graph nodes with sound lower and
upper bounds on all represented completions. Checks, mandatory evasions,
off-square captures, promotions, quiet threats, repetition, and draw claims
must be included or conservatively widened. Only interval dominance may turn
that abstraction into safe pruning. Without such bounds, increasing the C6
horizon merely moves the exponential frontier.

## Appendix AL. Dimensional Chess C7: horizon-safe projection

### AL.1. Repetition coordinates have a finite activation horizon

Let the repetition-key history along a path be

\[
h=(p_0,p_1,\ldots,p_t).
\tag{AL.1}
\]

An identical full chess position has the same side to move, so recurrence
intervals are even. It cannot recur after two plies: the opponent cannot both
undo the first player's move and undo their own required move in a single
action without changing placement, material, castling rights, or en-passant
state. The minimum recurrence interval is therefore four plies.

If a position first appears at ply \(i\), its third occurrence cannot be earlier
than ply \(i+8\). At total horizon \(D\), only repetition candidates satisfying

\[
i\le D-8
\tag{AL.2}
\]

can enable a threefold claim. C7 retains multiplicities only for those
candidates. Fullmove number is always discarded. Halfmove clock is discarded
only if the fifty-move threshold is globally unreachable inside the horizon.
This is an exact quotient for the bounded predicate, not a heuristic collision.

At \(D=9\), Equation AL.2 retains only the initial position and the position
after the first move. Histories that differ only after that prefix cannot differ
on threefold availability before the horizon.

### AL.2. Polarity-aware ordering test

At five plies with White designated as attacker, lexical defender ordering
uses 7,480 proof calls. C6 dependency ordering uses 7,536 and a checks/captures
ordering uses 9,992. Changing attacker ordering while fixing the defender order
does not change these call counts in this negative proof. This confirms the
logical polarity diagnosed in Appendix AK: every attacking candidate must be
rejected, while finding one defender escape short-circuits its line.

A richer escape heuristic takes 95.13 seconds and 9,926 calls. A naive
proof-number implementation expands 40,316 nodes and generates 743,471
children in 32.44 seconds. Both are rejected for this task. Their failure does
not make lexical notation intrinsically informative; it only identifies the
best tested traversal for the present certificate.

### AL.3. Seven-ply quotient check

Below eight plies, threefold repetition and the fifty-move threshold are both
unreachable from the initial state. The projected lexical computation obtains

\[
B_W(s_0,7)=B_B(s_0,7)=0
\tag{AL.3}
\]

using 95,718 calls for White and 10,551 for Black. The earlier unprojected
lexical verifier used 106,322 and 15,082. Combined calls fall from 121,404 to
106,269, a 12.47% reduction, without changing the predicate.

### AL.4. Independently verified nine-ply frontier

C7 then computes

\[
B_W(s_0,9)=B_B(s_0,9)=0.
\tag{AL.4}
\]

| Attacker | Proof calls | Generated edges | Cache hits | Horizon nodes | Time |
|---|---:|---:|---:|---:|---:|
| White | 2,951,587 | 5,259,337 | 1,437,821 | 1,308,014 | 575.877 s |
| Black | 127,991 | 736,248 | 59,263 | 30,841 | 23.080 s |

A verifier that imports no C7 code separately reconstructs the candidate
prefix, draw actions, and AND/OR recurrence. It exactly reproduces all reported
Boolean results and every node, edge, cache, horizon, and terminal count at
seven and nine plies. It checks 400 two-ply paths from the root and 8,902 from
all first-ply states, finding no two-ply recurrence. Explicit legal knight
cycles reach the root's third occurrence at ply eight and the third occurrence
of a post-first-move position at ply nine. The audit returns `PASS`.

### AL.5. Boundary of the result

The White proof-call frontier grows by a factor of 30.84 from seven to nine
plies; Black grows by 12.13. Projection removes coordinates that cannot affect
the finite theorem, but it does not eliminate the remaining adversarial tree.
Equation AL.4 is not a WDL value, supplies no unbounded optimal strategy, and
does not solve chess. The root interval remains

\[
I(s_0)=[-1,+1].
\tag{AL.5}
\]

C7 nevertheless sharpens the framework's computational principle: a hidden
coordinate belongs in a finite proof state exactly when it can cross a terminal
or transition threshold within that proof's horizon. The next stage requires a
proof-carrying abstraction with sound lower and upper completion bounds. Only
such bounds can merge tactically different continuations and narrow Equation
AL.5 without enumerating them.

## Appendix AM. Dimensional Chess C8: movement-frequency relations

### AM.1. Relational move phase

C8 maps legal move \(a:q\to q'\) in state \(s\) to

\[
\phi(s,a)=
(p,c,\Delta x,\Delta y,\chi,
 \operatorname{sgn}\Delta d_{K_s},
 \operatorname{sgn}\Delta d_{K_o},
 g_s,g_o,\Delta r,h(q),h(q')).
\tag{AM.1}
\]

The fields record moving and captured piece types, color-oriented displacement,
destination parity, Chebyshev-distance changes to both kings, geometric
relations to both kings, edge-ring change, and home-relative rank phases. Eight
deterministic phase-shuffled five-ply searches record defender cutoffs. With
depth-squared success and failure weights \(S_u,F_u\), class \(u\) receives

\[
q_u=\frac{S_u+1}{S_u+F_u+2}.
\tag{AM.2}
\]

The weights are then frozen. Defender moves are sorted by Equation AM.2;
attacker moves remain lexical. No move is removed, and only the exact bounded
predicate from Appendix AJ determines value.

### AM.2. Learned pattern

The White-attacker book contains 102 classes. Ten of its top twenty are quiet
pawn advances and four are quiet knight developments; all twenty are
non-captures. Nineteen move closer in Chebyshev distance to the opposing king.
The highest-scoring class is a one-square pawn move from home-relative phase
one to phase two that preserves distance to the mover's king and reduces
distance to the opposing king. It records 296 training cutoffs and zero
failures. The corresponding two-square advance records 286 and zero.

These counts characterize early counterexamples to White's short-horizon mate
claims. They are not chess-value labels for the moves.

### AM.3. Transfer to seven and nine plies

At seven plies, the relation ordering generates 162,511 edges versus 172,970
for lexical order, a 6.05% reduction, while calls rise from 95,718 to 98,352.
Three relation repeats reproduce both counts exactly and take a median 13.417
seconds. Three lexical repeats reproduce their counts and take a median 15.272
seconds, a 12.15% relation-time reduction.

At nine plies with White as attacker, the exact results are

| Method | Calls | Generated edges | Horizon nodes | Proof time |
|---|---:|---:|---:|---:|
| C7 lexical | 2,951,587 | 5,259,337 | 1,308,014 | 575.877 s |
| C8 relation | 2,750,245 | 4,299,169 | 930,982 | 375.703 s |

C8 reduces calls by 6.82%, edges by 18.26%, horizon nodes by 28.83%, and proof
time by 34.76%. Training costs 31.087 seconds; the 406.790-second combined time
is 29.36% below the C7 proof time. Both computations obtain the already
independently verified value

\[
B_W(s_0,9)=0.
\tag{AM.3}
\]

### AM.4. Counterexamples to stronger frequency claims

Absolute oriented-square classes increase seven-ply calls by 2.39%. Directly
conditioning response classes on the previous move expands the relation
vocabulary from 102 to 1,264 classes: piece-, vector-, and relation-pair variants
increase calls by 75.50%, 80.75%, and 49.18%. Updating relation weights online
increases calls by 52.53%. These failures show that added context can destroy
reuse faster than it adds information.

The result is also role-asymmetric. For Black as designated attacker, its own
239-class relation model raises nine-ply calls from 127,991 to 136,288, edges
from 736,248 to 1,070,334, and proof time from 23.080 to 34.733 seconds. Applying
the White-trained color-oriented book also loses. Equation AM.2 is not a
universal color-normalized chess law.

### AM.5. Audit and boundary

The C8 Boolean results equal the separately implemented C7 certificate. A
structural audit checks seven schemes across 960 deterministically reachable
states and 168,882 legal-move instances. Every ordered list is exactly a
permutation of the legal list; 6,720 state/scheme cases genuinely reorder it.
The audit returns `PASS`.

C8 validates only a role-specific computation-allocation pattern. It neither
compresses exact state nor supplies sound interval bounds for unexpanded
continuations. Thus

\[
I(s_0)=[-1,+1]
\tag{AM.4}
\]

remains unchanged. The next admissible lift must attach a proof obligation or
sound completion interval to a relational class. Without that certificate,
frequency may prioritize a branch but never discard it.

## Appendix AN. Dependent-sign evidence under a declared Markov null

### AN.1. The bounded question

Appendix AA showed that marginally balanced signs can be strongly dependent:
a stationary chain that repeats its previous sign with probability (0.90)
caused the iid conditional-sign process to cross in 3,772 of 5,000 synthetic
histories. The unresolved question was not whether one statistic can be valid
under arbitrary dependence. It was whether an anytime-valid process can be
constructed after the dependence law is stated explicitly.

Let (X_k\in\{-1,+1\}) be the sequence of nonzero projected-residual signs in
nonzero-event time. Conditional on (X_1), the null in this revision is the
fixed first-order homogeneous Markov law

\[
P_0(X_k=X_{k-1}\mid\mathcal F_{k-1})=0.90,
\qquad
P_0(X_k=-X_{k-1}\mid\mathcal F_{k-1})=0.10.
\tag{AN.1}
\]

The initial sign may be arbitrary. Exact zeros are skipped, so Equation AN.1
must hold for the next nonzero sign conditional on the full past. This is a
model-class assumption, not a consequence of marginal 50/50 signs.

### AN.2. Predictive likelihood-ratio martingale

Before transition (k), let (N_{ij}(k-1)) count earlier transitions from
state (i) to state (j). The Krichevsky--Trofimov predictor is

\[
\widehat Q_k(j\mid i)=
\frac{N_{ij}(k-1)+\tfrac12}{N_{i\cdot}(k-1)+1}.
\tag{AN.2}
\]

Define

\[
L_n=\prod_{k=2}^{n}
\frac{\widehat Q_k(X_k\mid X_{k-1})}
     {P_0(X_k\mid X_{k-1})}.
\tag{AN.3}
\]

Because (widehat Q_k) is determined by the past and is a probability
distribution,

\[
\mathbb E_0[L_k\mid\mathcal F_{k-1}]
=L_{k-1}\sum_j
P_0(j\mid X_{k-1})
\frac{\widehat Q_k(j\mid X_{k-1})}{P_0(j\mid X_{k-1})}
=L_{k-1}.
\tag{AN.4}
\]

Thus (L_n) is a nonnegative null martingale with (L_1=1), and Ville's
inequality gives

\[
P_0\!\left(\sup_n L_n\ge 1/\alpha\right)\le\alpha.
\tag{AN.5}
\]

This is a written probability proof for the declared model. The construction
is an application of predictive Markov likelihood ratios, consistent with
Fields, Javidi, and Shekhar's 22 January 2025 sequential Markov-chain test
(https://arxiv.org/abs/2501.13187), not a new theorem about dependence.

### AN.3. Fresh finite synthetic validation

The protocol fixed (alpha=0.01), 5,000 histories, 400 looks, the null matrix,
the add-one-half predictor, a fresh seed, six scenarios, and source hashes
before outcome generation. The results were

| Sign process | Markov KT crossings | Old iid-sign crossings |
|---|---:|---:|
| Declared (p_{\rm stay}=0.90) null | 39 / 5,000 | 4,134 / 5,000 |
| Same null, every sign inverted | 39 / 5,000 | 4,134 / 5,000 |
| Same null, fixed positive initial sign | 36 / 5,000 | 4,158 / 5,000 |
| More persistent, (p_{\rm stay}=0.97) | 4,629 / 5,000 | 4,977 / 5,000 |
| Less persistent, (p_{\rm stay}=0.75) | 4,999 / 5,000 | 1,448 / 5,000 |
| Balanced iid, (p_{\rm stay}=0.50) | 5,000 / 5,000 | 32 / 5,000 |

The paired sign inversion produced exactly identical decisions. A separate
implementation regenerated every chain, recomputed both statistics, and
matched 24,000,000 stored decision states. The initial scalar audit exceeded
the bounded runtime; it was mechanically vectorized after outcomes without
changing the seed, scenarios, scoring implementation, formulas, thresholds,
saved paths, or summaries. This provenance is retained in the protocol.

These counts are finite synthetic computation. Equation AN.5, not the observed
39 or 36 null crossings, supplies the anytime guarantee. The alternative
counts measure power only for these frozen simulations.

### AN.4. Counterexamples and boundary

The old iid-sign process is not repaired by marginal balance: it rejected
4,134 of 5,000 valid (p_{\rm stay}=0.90) Markov-null histories because its
conditional-balance assumption was false. Conversely, the new process would
be misspecified if 0.90 were estimated from the same scored stream, changed
over time, or represented only an average transition frequency. A first-order
matrix also does not cover higher-order or non-Markov sign dependence.

The engine discards magnitudes and does not test scale, tail thickness, or a
unique physical mechanism. It is standalone and does not share its 1% budget
with another detector. No measured physical evidence was collected, and a
crossing cannot identify a theological cause.

The next bounded test is a composite-null construction that remains anytime
valid for every prespecified persistence value in an interval such as
([0.88,0.92]). It must be challenged at both interval boundaries and against
non-Markov chains before any replacement of the point-null engine.

## Appendix AO. Hydrogen spectroscopy clock and tangent-rotor holdout

### AO.1. What real data can calibrate

The previous hydrogen construction separated a standard time-dependent
\(1s+2p_z\) reference from a synthetic pair of tangent 3-spheres rotating in
four spatial coordinates. The bounded question in this revision is whether the
synthetic family can predict the reference after fitting only selected phases,
while the reference clock is tied to observational spectroscopy.

The NIST Hydrogen strong-lines table lists two vacuum Lyman-alpha components,

\[
\lambda_1=121.566824\ {\rm nm},\qquad
\lambda_2=121.567364\ {\rm nm},
\tag{AO.1}
\]

with listed relative intensities 1000 and 500. Their intensity-weighted
centroid is \(121.567004\) nm and its optical period is \(405.503877\) as. The
frequency separation is \(10.954243544\) GHz. These values constrain the
energy/time scale. The listed intensities are not treated as prepared-state
populations and Equation AO.1 does not provide 3D density voxels.

The release-0.2 fixed-proton nonrelativistic Coulomb prediction was
\(121.502273411\) nm. Its centroid-relative error is therefore

\[
\frac{121.502273411-121.567004}{121.567004}
=-5.324684\times10^{-4}.
\tag{AO.2}
\]

Release 0.3 retains the ideal spatial basis but replaces the displayed phase
timestamps with the NIST centroid. Calling this a spectroscopy-clocked proxy is
more accurate than calling it a measured movie of an atom.

### AO.2. Frozen spatial discrimination protocol

The protocol fixed 24 phases, a \(10a_0\) half-width, eight training phases,
sixteen disjoint holdout phases, the parameter bounds, optimizer seed, and five
acceptance gates before fitting. The rotor family retained two equal tangent
3-spheres in \(\mathbb R^4\), one rotating center axis, perspective projection,
two Gaussian shell envelopes, and their complex interference. Its projected
axis was aligned to the known \(2p_z\) symmetry axis; this coordinate choice
was not scored as a discovery.

The six fitted quantities were radius, camera-to-radius ratio, projection
scale, wave number, shell width, and phase offset. The objective was mean
squared Hellinger distance on the eight training phases only. A frozen
training-phase mean and its spherical radialization were declared as baselines.

### AO.3. Held-out result and counterexample

After 1,862 training-objective evaluations, the frozen \(33^3\)-voxel holdout
scores were

| Predictor | Mean squared Hellinger | Mean total variation | Mean relative \(L^2\) |
|---|---:|---:|---:|
| Tangent-rotor family | 0.172620216 | 0.432929726 | 0.821171870 |
| Training-phase mean | 0.022196552 | 0.131144467 | 0.188746349 |
| Spherical radial baseline | 0.072178684 | 0.259993679 | 0.225927917 |

The candidate is worse than both baselines on every aggregate metric. Its
held-out axial-centroid correlation is \(-0.086074\), opposite-phase dynamic
nodal leakage is \(0.057650\), and mirror total-variation error is \(0.053451\).
These fail thresholds of \(0.95\), \(0.02\), and \(0.02\). Only exact-cycle
repeatability passes, at \(4.34\times10^{-19}\) maximum probability error.

This supplies a counterexample to the inference that tangency plus periodic
interference is sufficient for hydrogen-like evolution. The geometry can be
internally consistent and periodic while predicting the wrong spatial mass and
wrong phase transport.

### AO.4. Verification and evidence boundary

All 23 unit-test groups pass. A separately written replay imports none of the
production calibration, quantum, rotor, or metric functions. It reconstructs
the wavelengths with decimal arithmetic, regenerates both voxel sequences,
recomputes the phase split and scores, and reproduces all five decisions and
the rejection.

Equation AO.2 is a finite comparison with a critically compiled spectral
reference. The voxel holdout is a synthetic model-discrimination experiment,
not a laboratory measurement. Photoionization microscopy has directly observed
nodal structure for externally driven hydrogen Stark states (Stodolna et al.,
Phys. Rev. Lett. 110, 213001, 2013), but that preparation is not silently
substituted for the field-free \(1s+2p_z\) reference used here.

The exact projected-envelope rotor family is rejected as an adequate generator
of the declared reference. This does not reject every higher-dimensional model,
establish a fourth spatial dimension, identify an unconstrained residual with
God, or alter the earlier closure, collision, angular-impulse, and finite-prefix
corrections.

The next admissible candidate must be derived from a Hamiltonian or continuity
equation before comparison. Its spatial basis must be frozen before new data are
used, and it must address both Lyman-alpha components plus a spatial observable
from a matching experimental preparation. Importing the known hydrogen basis
by construction would reproduce the target but would not explain it.

## Appendix AP. Temperature, phase direction, and bounded missing components

### AP.1. Why a failed model does not select one cause

Appendix AO rejected one specified tangent-rotor envelope. A mismatch can arise
from omitted state variables, but also from an incorrect primitive geometry,
dynamics, projection, observable map, calibration, or optimizer. Therefore the
exclusive inference

\[
\text{failed fit}\Longrightarrow\text{missing rotors are the only cause}
\tag{AP.1}
\]

is invalid. A broader family can be tested only after its new variables and
failure conditions are fixed. Allowing arbitrary per-voxel corrections or an
unbounded number of components would interpolate finite data without identifying
an explanatory law.

### AP.2. Frozen latent-component rescue

The bounded rescue freezes all six release-0.3 geometric parameters. It permits
one through four copies of the same rotor, each selected from twelve phase offsets
spaced by \(\pi/6\), with one global nonnegative simplex weight per component.
Greedy selection and weight fitting use the eight old training phases only. The
four-component model is the sole acceptance candidate; smaller models are nested
diagnostics. The sixteen previously untouched phases are scored once.

| Components | Training \(H^2\) | Holdout \(H^2\) |
|---:|---:|---:|
| 1 | 0.204041584 | 0.172620216 |
| 2 | 0.203640581 | 0.171868367 |
| 3 | 0.203586339 | 0.172032715 |
| 4 | 0.203586339 | 0.172032715 |

The fourth component receives zero weight. The rescue improves holdout error by
only 0.3403%, below the predeclared 20% minimum, and remains worse than the
time-mean baseline \(H^2=0.022196552\). Its held-out axial-centroid correlation
is \(-0.058760\), dynamic nodal leakage is \(0.058831\), and opposite-phase mirror
TV is \(0.043823\). Five of six gates fail. Thus the same-geometry phase-mixture
rescue is rejected. Other explicit latent-variable mechanisms remain open to a
fresh test; Equation AP.1 does not become true.

### AP.3. Temperature does not reverse the phase clock

Let a declared two-level Hamiltonian be

\[
H=\operatorname{diag}(0,\Delta E),\qquad
\rho_\beta=\frac{e^{-\beta H}}{\operatorname{tr}(e^{-\beta H})}.
\tag{AP.2}
\]

Because \(\rho_\beta\) is a function of \(H\),

\[
[\rho_\beta,H]=0,
\qquad
e^{-iHt/\hbar}\rho_\beta e^{iHt/\hbar}=\rho_\beta.
\tag{AP.3}
\]

This written result holds for positive or negative \(\beta\) whenever the Gibbs
state exists. With the NIST-clocked \(10.198836391\) eV gap, finite calculations
give excited-to-ground ratios of \(4.64611\times10^{-172}\) at 300 K,
\(7.24457\times10^{-6}\) at 10,000 K, and \(1.38034\times10^5\) at -10,000 K.
The negative-temperature row is meaningful only in this upper-bounded two-level
truncation. It reverses population ordering, not \(H\), \(t\), or the unitary
phase direction. A coherent control has phase slope
\(+1.5494760135\times10^{16}\) rad/s; replacing \(t\) by \(-t\) changes its sign.

Positive temperature has no finite maximum Kelvin endpoint: as
\(T\to+\infty\), \(\beta=1/(k_BT)\to0^+\). Negative-temperature equilibrium is
a special population-inverted regime for an upper-bounded spectrum, not a
colder continuation past 0 K. The complete hydrogen spectrum includes an
unbounded continuum, so the two-level negative-temperature calculation cannot
be promoted to a global equilibrium state of free hydrogen.

### AP.4. Melting delay and evidence boundary

A frozen sample can require more supplied energy and time because its temperature
must rise and its phase change absorbs latent heat. Heat-transfer rate, geometry,
pressure, heat capacity, and material state also matter. None of those macroscopic
facts alone identifies a clockwise or counterclockwise microscopic rotation.

All 29 unit-test groups pass. A separately written audit reconstructs the
spectroscopic energy and period, the three population ratios, the time-reversal
sign, the saved mixture metrics and weights, and all six decisions; 11 of 11
checks pass. These are mathematics, finite computation, and synthetic tests.
Only the NIST spectral clock is measurement-derived. No measured time-resolved
3D hydrogen density, fourth spatial dimension, unrestricted hidden-variable
family, or theological cause follows.

The next admissible model must state a new interaction law, preferably as a
Hamiltonian or continuity equation, and a fixed parameter budget. It must be
frozen before using a new experimental preparation with a matching spatial
observable. The old holdout is now consumed by this prespecified extension and
must not be used again to tune a replacement.

## Appendix AQ. Electromagnetic correction ladder before higher-dimensional forces

### AQ.1. Correction: electromagnetism was already present

The release-0.3 ideal hydrogen Hamiltonian was incomplete, but it did not omit
electromagnetism. Its potential

\[
V(r)=-\frac{e^2}{4\pi\epsilon_0 r}
\tag{AQ.1}
\]

is the nonrelativistic electromagnetic interaction between a point proton and
an electron. The fixed-proton approximation omitted the proton's motion; the
Schrodinger treatment also omitted relativistic spin structure and radiative
quantum-electrodynamic corrections. “Missing quantum electromagnetism” is
therefore replaced by the narrower statement “missing precision two-body,
relativistic, and QED structure.”

This matters methodologically. If a deliberately incomplete standard baseline
misses a measured value, the residual cannot be assigned to a new force until
the known corrections have been included or bounded.

### AQ.2. Frozen H0/H1/H2 ladder

NIST gives neutral-hydrogen levels

\[
\tilde\nu(2p\,{}^2P^\circ_{1/2})=82258.9191\ \mathrm{cm}^{-1},
\qquad
\tilde\nu(2p\,{}^2P^\circ_{3/2})=82259.2850\ \mathrm{cm}^{-1}.
\tag{AQ.2}
\]

Their vacuum wavelengths are (121.567364480) nm and
(121.566823733) nm. Weighting them by the NIST strong-line ratio (500:1000)
gives the frozen comparison centroid (121.567003982) nm. Three nested models
were evaluated without fitting:

1. **H0:** fixed-proton nonrelativistic Coulomb spectrum;
2. **H1:** H0 with the electron-proton reduced mass
   \(\mu=m_em_p/(m_e+m_p)\);
3. **H2:** point-nucleus Dirac-Coulomb energies with the same reduced mass
   substituted.

For H1, Coulomb transition energies scale by (mu/m_e), so wavelengths scale
by (m_e/\mu=1+m_e/m_p). With the 2022 CODATA mass ratio,

\[
\frac{\mu}{m_e}=0.999455679424762.
\tag{AQ.3}
\]

The point-Coulomb Dirac binding energy used for H2 is

\[
E_{nj}=\mu c^2\left[
1+\frac{\alpha^2}{
\left(n-j-\tfrac12+\sqrt{(j+\tfrac12)^2-\alpha^2}\right)^2}
\right]^{-1/2}-\mu c^2.
\tag{AQ.4}
\]

It is a controlled intermediate model, not a complete bound-state-QED result.

### AQ.3. Result

| Model | Predicted centroid (nm) | Centroid error (ppm) |
|---|---:|---:|
| H0 fixed-proton Schrodinger | 121.502273411 | -532.4683 |
| H1 reduced-mass Schrodinger | 121.568445617 | +11.8588 |
| H2 reduced-mass Dirac | 121.566602395 | -3.3034 |

Reduced mass alone removes (97.7729\%\) of H0's absolute centroid error.
H2 removes (99.3796\%\), predicts two distinct lines at
(121.566962046) nm and (121.566422570) nm, and predicts a wavelength
fine-structure split of (0.000539476) nm. The NIST level split is
(0.000540747) nm, so the declared H2 split error is (-0.2351\%\).

All 35 unit tests pass. A separately written 50-digit Decimal audit imports no
production functions and reproduces the six central observed/model wavelengths
and centroids within (2\times10^{-8}) nm. The NIST levels are measured or
critically compiled evidence; Equations AQ.3--AQ.4 and the comparisons are
written mathematics plus deterministic finite computation. No new hydrogen
measurement was performed.

### AQ.4. Residual budget and higher-dimensional admission rule

H2 still omits complete relativistic recoil, radiative self-energy and vacuum
polarization (the Lamb/QED contribution), finite proton charge radius,
hyperfine structure, radiative linewidth, and external-field effects. The
remaining (-3.3034) ppm centroid residual is therefore first a budget for
omitted standard terms. It does not identify a unique new force, a string or
rotor ontology, a fourth spatial coordinate, or a theological cause.

A higher-dimensional proposal becomes testable only after specifying

\[
H_{\mathrm{candidate}}=H_{\mathrm{standard}}+\lambda\mathcal O_4,
\tag{AQ.5}
\]

where the extra coordinate and units, the field equation or action deriving
(\mathcal O_4), its coupling to charge/current, its gauge or conservation law,
and a finite parameter budget are declared before fitting. It must then predict
a fresh state-dependent pattern that cannot be absorbed by standard corrections
or refitted constants. No such operator is specified in this revision, so no
higher-dimensional fit is performed and the physical origin remains unresolved.

The next test is to complete and independently validate the standard precision
ladder across several hydrogen transitions--not only Lyman-alpha--before freezing
one explicit higher-dimensional operator. The earlier rotor rejection, closure
idempotence, integer-tick collision boundary, angular-impulse correction, and
finite-prefix generator limitation remain unchanged.

## Appendix AR. Fixed-grid and fixed-tick falsification engine

### AR.1. Two hypotheses that must not be conflated

A fixed spacetime lattice asserts coordinate labels such as

\[
x_i=n_i\Delta x,\qquad t=k\Delta t,
\quad n_i,k\in\mathbb Z.
\tag{AR.1}
\]

That does **not** by itself imply that every field amplitude, energy eigenvalue,
frequency, wavelength, temperature, or other derived observable is an integer
multiple of \(\Delta x\) or \(\Delta t\). Lattice field theories routinely put
coordinates on a grid while fields remain real-, complex-, group-, or
operator-valued. This revision therefore distinguishes:

- **Lattice-coordinate hypothesis:** positions and times lie on fixed coordinate
  increments; and
- **Direct-multiple hypothesis:** each declared direct measurement satisfies
  \(y_i=n_i\Delta\) for an integer \(n_i\).

The second is a stronger, separately falsifiable hypothesis. Failure of one
proposed direct tick does not logically refute every lattice dynamics, but a
theory that explicitly requires every observable to be a direct multiple must
survive that stronger test.

### AR.2. Interval decision and non-identification theorem

For a reported value \(y_i\) with declared tolerance \(\epsilon_i>0\), a proposed
tick \(\Delta>0\) passes exactly when

\[
\min_{n\in\mathbb Z}|y_i-n\Delta|\leq\epsilon_i
\tag{AR.2}
\]

for every measurement. One incompatible interval rejects that proposed tick
under the direct-multiple hypothesis.

**Theorem AR.1 (finite-precision non-identification).** For any finite set of
measurements with positive tolerances, every positive tick satisfying

\[
\Delta\leq 2\min_i\epsilon_i
\tag{AR.3}
\]

passes Equation AR.2.

**Proof.** Choose \(n_i\) by rounding \(y_i/\Delta\) to its nearest integer. Then
\(|y_i-n_i\Delta|\leq\Delta/2\leq\min_j\epsilon_j\leq\epsilon_i\) for every
\(i\). \(\square\)

Thus finite-precision data alone cannot discover or confirm an arbitrarily
small universal tick. Identification additionally requires a lower bound on the
tick, a restricted integer-count law, a complexity penalty, or new predictions
whose resolution scales appropriately. This is a written mathematical result,
not a statistical tendency.

### AR.3. Frozen hydrogen-spectroscopy example

The engine uses the two NIST strong neutral-hydrogen Lyman-alpha wavelengths
printed as \(121.566824\) nm and \(121.567364\) nm and converts them to vacuum
light-travel times with exact \(c=299,792,458\) m/s. Half the last printed digit,
\(5\times10^{-7}\) nm, is treated only as a digit-resolution interval; it is not
claimed to be NIST's complete experimental uncertainty. This gives

\[
\epsilon_t=1.6678204759907602\times10^{-24}\ \mathrm{s},
\qquad
2\epsilon_t=3.3356409519815205\times10^{-24}\ \mathrm{s}.
\tag{AR.4}
\]

| Proposed direct time tick | Frozen result | Largest residual / digit tolerance |
|---|---:|---:|
| \(10^{-18}\) s | rejected | 297,827.896 |
| \(10^{-21}\) s | rejected | 165.807252 |
| \(10^{-22}\) s | rejected | 14.0682228 |
| CODATA-2022 Planck time \(5.391247\times10^{-44}\) s | compatible | \(1.2853\times10^{-20}\) |
| Printed-wavelength quantum \((10^{-15}\ \mathrm m)/c\) | compatible by construction | \(<6.0\times10^{-82}\) in Decimal computation |

The last row is exactly divisible in rational arithmetic because each printed
wavelength is an integer number of \(10^{-15}\) metres. It is a reporting-grid
artifact, not a discovery about nature. The Planck-time candidate also passes
automatically because it is about twenty orders of magnitude below even the
nearest-multiple automatic-fit floor. Its pass is consistency only and supplies
no positive evidence for that tick.

### AR.4. Exact four-coordinate discrete string

To make the proposed hidden-motion geometry concrete without pretending it is
already hydrogen, the lab defines four integer spatial coordinate fields
\(U_A(s,k)\), \(A\in\{x,y,z,w\}\), on periodic string sites \(s\) and integer
time ticks \(k\):

\[
U_A(s,k)=F_A((s-k)\bmod N)+G_A((s+k)\bmod N).
\tag{AR.5}
\]

For integer profiles \(F_A,G_A\), every state is integer-valued and Equation
AR.5 obeys the exact Courant-one discrete wave equation

\[
U_A(s,k+1)=U_A(s-1,k)+U_A(s+1,k)-U_A(s,k-1).
\tag{AR.6}
\]

The same equation solved for \(U_A(s,k-1)\) gives exact one-tick reversal. The
implemented 64-site profile is bounded and periodic after 64 ticks. A 3D image
is the occupancy obtained by dropping \(w\). Changing only \(w\) produces a
different full state with the identical 3D image, giving a constructive proof
that the projection is non-injective. Therefore visible 3D waves do not by
themselves reconstruct the hidden coordinate or establish that it is physical.

Across 128 generated ticks, production and independent scalar implementations
both check 32,256 wave-recurrence values and 32,256 reverse-recurrence values.
The production verifier also checks 16,384 periodicity values. All 13 unit tests
pass, and the independent audit reproduces all five tick decisions plus the
projection collision. These are exact finite computations for one toy rule.

### AR.5. Failure criterion and next physical gate

The executable theory now has a real failure condition: any proposed direct
tick is rejected when one verified direct measurement interval contains no
integer multiple. A universal claim over “all observable values ever” cannot be
confirmed by a finite list, and shrinking \(\Delta\) always makes finite
interval fitting easier. The current engine therefore rejects three coarse
candidate ticks but neither identifies a fundamental tick nor refutes every
possible lattice-coordinate theory.

The next admissible physical test must add a derived discrete hydrogen
Hamiltonian or update law on the same declared \((\Delta x,\Delta t)\), specify
how charge and electromagnetic gauge structure enter, freeze its parameters,
and predict multiple hydrogen transitions before comparison. A continuum-limit
convergence study must separate lattice discretization error from any claimed
new physics. Until then, a fifth-dimensional string-particle interpretation is
a hypothesis; the simulation is a consistent kinematic example, not measured
evidence for it. All v1.14 and v1.15 corrections, including the restricted
\(2K\)-sample recurrence guarantee, remain unchanged.

## Appendix AS. Geometry, phase boundaries, and conductivity collisions

### AS.1. Scope and type discipline

Revision 1.44 asks one bounded question: can a fixed, elementary molecular
geometry coordinate predict a held-out phase boundary after composition is
held constant?  It does not infer microscopic strings from a successful
correlation.  The input temperatures and resistivities are measured/reference
data; graph indices, interpolation, classifications, and bounds are finite
computations; the representation-collision inequalities are written
mathematics; a higher-dimensional cause remains a hypothesis.

A phase is not an intrinsic label of an isolated formula.  For a pure material
in equilibrium, the stable phase at specified temperature and pressure is the
one with the least appropriate free energy.  The executable classification in
this appendix is restricted to approximately one atmosphere and uses reported
fusion and normal-boiling boundaries.  It is not a universal equation of
state.

### AS.2. A representation-collision bound

Suppose a representation maps two materials to the same feature vector and a
deterministic predictor consequently assigns both the same scalar value \(p\).
For observed properties \(a<b\), the triangle inequality gives

\[
\max\{|p-a|,|p-b|\}\geq \frac{b-a}{2}.
\tag{AS.1}
\]

For positive properties evaluated multiplicatively, the best common
prediction is their geometric mean and its unavoidable worst factor is

\[
\inf_{p>0}\max\left\{\frac{p}{a},\frac{a}{p},
\frac{p}{b},\frac{b}{p}\right\}=\sqrt{\frac{b}{a}}.
\tag{AS.2}
\]

These are obstructions to an insufficient representation, not claims that a
particular richer representation is true.

### AS.3. Frozen same-formula molecular test

NIST Chemistry WebBook SRD 69 lists the following average phase-change values
for three isomers with the same formula \(\mathrm{C_5H_{12}}\) and molecular
mass 72.1488 g/mol.  The lab computes the Wiener index \(W\), the sum of all
unordered pairwise graph distances in the five-carbon skeleton.

| Isomer | Carbon-graph degree sequence | \(W\) | Fusion point (K) | Normal boiling point (K) | Phase at 293.15 K, about 1 atm |
|---|---|---:|---:|---:|---|
| neopentane | 4,1,1,1,1 | 16 | 255 | 282.6 | gas |
| isopentane | 3,2,1,1,1 | 18 | 113 | 301.1 | liquid |
| n-pentane | 2,2,2,1,1 | 20 | 143.4 | 309.2 | liquid |

The protocol chooses the minimum- and maximum-\(W\) structures without reading
their temperatures, fits the unique affine rule through them, and predicts the
middle structure once:

\[
\widehat T_b(W)=6.65W+176.2\ \mathrm{K}.
\tag{AS.3}
\]

For held-out isopentane, Equation AS.3 predicts 295.9 K versus the NIST average
301.1 K, an absolute error of 5.2 K.  The prediction is above 293.15 K and thus
places the molecule on the condensed side of its boiling boundary; the
separately measured 113 K fusion point identifies the observed equilibrium
phase there as liquid.  This phase statement is conditional on the supplied
fusion datum and approximately one-atmosphere equilibrium, not an independent
three-phase prediction.

Composition alone cannot distinguish these rows.  Equation AS.1 and the 26.6 K
boiling range imply that any one-value formula-only predictor has worst error
at least 13.3 K.  The held-out computation therefore supports a narrow claim:
connectivity supplies predictive information that atom counts omit.

### AS.4. Counterexamples to one-dimensional harmony

The same coordinate does not order the fusion temperatures: neopentane has the
smallest \(W\) and lowest boiling point but by far the highest fusion point of
the three.  Molecular symmetry, three-dimensional conformation, crystal
packing, entropy, and intermolecular interactions are separate variables.  A
single graph scalar can therefore capture one trend while failing another
property of the same molecules.

The elemental control has the same form.  Matula's recommended total
resistivities at 300 K are \(1.629\times10^{-8}\ \Omega\,\mathrm m\) for silver
and \(10.54\times10^{-8}\ \Omega\,\mathrm m\) for palladium.  Both are
period-5, fcc, d10 metals under the lab's deliberately coarse signature, yet
their resistivity ratio is 6.470.  Equation AS.2 says that any one resistivity
assigned to that shared signature has unavoidable worst multiplicative error
at least 2.544.  The differing valence occupancy already signals missing
information; quantitative conduction additionally depends on band dispersion,
Fermi-surface geometry, carrier velocities, phonons, defects, and scattering
lifetimes.

### AS.5. Verification, interpretation, and next test

All 12 unit tests pass.  A separately implemented Floyd--Warshall/scalar audit
reconstructs the three graphs, their distance sums and degree sequences, all
three phase rows, the two collision bounds, the single held-out prediction,
and the Ag/Pd ratio.  The observed numbers are not new measurements and the
small three-isomer test does not establish generalization to arbitrary
molecules.

The constructive result is that a geometry-first program is viable only as a
typed multivariable predictor: composition, connectivity, conformation,
symmetry, temperature, pressure, and interaction/electronic variables must be
declared before testing.  Failure cannot be automatically assigned to an
unseen string, rotation, fifth dimension, or theological cause.  Such a new
coordinate earns physical standing only if one shared, independently motivated
coupling improves fresh held-out predictions over standard quantum-chemical or
solid-state baselines.

The next bounded test is a preregistered alkane-isomer study with substantially
more molecules.  It should freeze graph and three-dimensional descriptors,
train on one subset, predict both boiling and fusion temperatures on a sealed
subset, and compare against a composition-only baseline plus a standard
quantitative structure--property baseline.  A separate conductivity test must
use measured band/transport descriptors and temperature-dependent data; it
must not infer a hidden dimension from the residual.  All v1.14 and v1.15
corrections, including idempotent complete deterministic closure, sampled
non-crossing limits, the angular-impulse formula, and the restricted
\(2K\)-sample recurrence guarantee, remain unchanged.

## Appendix AT. Property-specific transfer across alkane isomers

### AT.1. Scope and frozen split

Revision 1.45 executes Appendix AS's reserved larger test without changing the
question after seeing a score: can elementary carbon-skeleton topology improve
prediction over composition alone when the carbon count itself moves outside
the fitted range?  The data are NIST Chemistry WebBook SRD 69 phase-change
records for every constitutional isomer of pentane, hexane, and heptane.

The training block contains all three \(\mathrm{C_5H_{12}}\) and all five
\(\mathrm{C_6H_{14}}\) isomers.  The holdout block contains all nine
\(\mathrm{C_7H_{16}}\) isomers.  No C7 target is used to fit, select, clip, or
recalibrate a coefficient.  This is a frozen-fit retrospective reference-data
test, not a cryptographically hidden prospective experiment or a new physical
measurement.

Two ordinary least-squares representations are fitted separately for normal
boiling temperature and fusion temperature.  The composition baseline is

\[
\widehat T=\beta_0+\beta_1 n,
\tag{AT.1}
\]

where \(n\) is the carbon count.  The topology model is

\[
\widehat T=\gamma_0+\gamma_1 n+\gamma_2 W+\gamma_3 M_1,
\qquad
W=\sum_{i<j}d(i,j),\quad M_1=\sum_i \deg(i)^2.
\tag{AT.2}
\]

Here \(W\) is the Wiener distance sum and \(M_1\) is the first Zagreb index of
the hydrogen-suppressed carbon tree.  Their historical use as chemical-graph
descriptors motivates the coordinates; it does not guarantee a physical
mechanism or out-of-range transfer.

### AT.2. Held-out boiling result

The C7 boiling scores are:

| Frozen model | MAE (K) | RMSE (K) | Maximum absolute error (K) | Mean signed error (K) |
|---|---:|---:|---:|---:|
| carbon count only | 8.554 | 10.107 | 16.627 | +7.982 |
| carbon count + \(W+M_1\) | 4.500 | 5.382 | 9.131 | -4.059 |

Thus the topology model improves MAE by 4.054 K and RMSE by 4.725 K on all nine
held-out structures.  This is finite computation applied to compiled measured
or critically evaluated inputs.  It supports the narrow inference that these
two connectivity statistics carry transferable information about boiling
within the declared C5--C7 alkane family.  It does not show that the statistics
are microscopic causes, that they remain adequate for other chemical classes,
or that a four- or five-dimensional motion has been observed.

### AT.3. Fusion counterexample

Using the identical representations and split for fusion gives the opposite
result:

| Frozen model | MAE (K) | RMSE (K) | Maximum absolute error (K) | Mean signed error (K) |
|---|---:|---:|---:|---:|
| carbon count only | 27.574 | 41.839 | 110.807 | -27.574 |
| carbon count + \(W+M_1\) | 135.934 | 148.789 | 227.863 | +135.934 |

The topology extrapolation predicts an unphysical 410.46 K fusion temperature
for n-heptane versus the selected 182.6 K reference value.  The prediction is
reported without clipping.  Even the composition baseline misses the highly
symmetric 2,2,3-trimethylbutane value of 248 K by 110.81 K.  The failure is
therefore not a minor coefficient correction: the declared two-dimensional
carbon graph is not an adequate melting representation across this split.

One NIST page lists two fusion values for 2,3-dimethylpentane, 149.36 K and
139.85 K.  The first is frozen in the scored table and the alternate is retained
in the data ledger.  Their 9.51 K difference cannot rescue a topology MAE of
135.934 K, so the principal conclusion is insensitive to this selection.

### AT.4. What a residual can and cannot establish

A failed prediction can be caused by omitted variables, a misspecified
functional form, distribution shift, noisy reference data, or an incorrect
physical hypothesis.  It does not by itself prove that the engine is correct
and only an unseen string coordinate is missing.  In the present problem,
ordinary omitted quantities already include three-dimensional conformation,
molecular symmetry beyond two scalar indices, polymorphism, crystal packing,
and intermolecular contributions to free energy.  A fifth-dimensional string
model would have to specify a new coordinate and coupling before looking at a
fresh target block, then beat these ordinary alternatives on that block.

The distinction is now explicit:

- NIST temperatures are measured or critically evaluated reference inputs.
- Graph indices, least-squares coefficients, predictions, and scores are
  finite computations.
- Equations AT.1--AT.2 define empirical models, not fundamental laws.
- The statement that packing-related variables are missing is a supported
  diagnosis of representation insufficiency, not a unique causal proof.
- Higher-dimensional strings, universal grid ticks, and theological meaning
  remain hypotheses or interpretations, not outputs of this experiment.

### AT.5. Verification and next test

All 17 unit tests pass.  A separately written audit reconstructs all 17 carbon
graphs by Floyd--Warshall, solves the four normal-equation systems with an
independent Gaussian-elimination routine, reproduces 36 held-out scalar
predictions, checks 28 coefficients and metrics, and reruns the preserved
v0.1.0 pentane and Ag/Pd controls.  This verifies implementation consistency,
not physical truth.

The next test may not repair Equation AT.2 against the C7 target and then reuse
those targets as confirmation.  It should define crystal-sensitive
three-dimensional descriptors using the C5--C7 development data, freeze
separate boiling and fusion models, and evaluate once on all 18 octane isomers.
The present topology model remains a required baseline.  Existing v1.14 and
v1.15 corrections remain unchanged: fixed-family complete closure is
idempotent; sampled integer-tick capture is not continuous non-crossing;
angular impulse has no extra time factor; arbitrary generators are not
identified by a finite prefix; and the \(2K\)-sample continuation guarantee
still assumes a noiseless homogeneous constant-coefficient recurrence of order
at most \(K\) from the initial state.

## Appendix AU. Latent rotation matrices and the inverse-projection boundary

### AU.1. Question and evidence types

This appendix answers a narrower form of the proposed reverse-engineering
program: can reference melting and boiling data be placed into explicit
higher-dimensional motion matrices, and do those data determine the hidden
motion? The first answer is yes as a mathematical construction. The second is
no without additional, independently constrained observables.

The types must remain separate:

- The boiling and fusion temperatures are measured or critically evaluated
  reference inputs compiled in Matter Harmony Lab 0.2.0.
- The graph-distance embedding, hidden coordinates, generator, projector,
  frequency encoder, and phase kernel are constructed coordinates.
- Norm preservation and the three non-identifiability results below are
  written mathematics.
- The 17 molecular matrices, 65-frame trajectory, tests, and independent audit
  are finite computations.
- The statement that any one generator describes physical molecular motion in
  a fourth or fifth spatial dimension is a hypothesis.
- No theological interpretation follows from a fitted or unfitted residual.

### AU.2. Explicit five-coordinate dynamics and 3D projection

For a molecule with \(N\) carbon vertices, let \(X_5(0)\in\mathbb R^{N\times5}\)
be a row-state matrix. Its first three columns are a classical
multidimensional-scaling embedding of carbon-graph shortest-path distances.
The fourth and fifth columns are centered vertex degree and eccentricity,
scaled to 35% of the visible root-mean-square radius. These choices make a
reproducible synthetic geometry; they are not experimental conformers,
orbitals, or probability amplitudes.

Let

\[
P=\begin{bmatrix}1&0&0&0&0\\0&1&0&0&0\\0&0&1&0&0\end{bmatrix},
\qquad \Omega^T=-\Omega.
\tag{AU.1}
\]

The latent state, visible position, and visible velocity are

\[
X_5(t)=X_5(0)e^{t\Omega^T},\qquad
X_3(t)=X_5(t)P^T,\qquad
V_3(t)=X_5(t)\Omega^TP^T.
\tag{AU.2}
\]

Because \(\Omega\) is skew-symmetric, \(R(t)=e^{t\Omega}\) is orthogonal:
\(R(t)^TR(t)=I\). Hence each row norm of \(X_5(t)\) is constant. This is an
exact rotation law in the declared Euclidean latent space.

The implemented generator couples visible coordinate 1 to hidden coordinate
4 at angular rate \(\omega_4\), visible coordinate 2 to hidden coordinate 5 at
\(\omega_5\), visible coordinate 3 to hidden coordinate 5 at \(\omega_3\),
and the two hidden coordinates to each other at \(\omega_h\). For the displayed
n-heptane candidate, the rounded matrix is

\[
\Omega=\begin{bmatrix}
0&0&0&-1.6000&0\\
0&0&0&0&-0.9882\\
0&0&0&0&-1.2941\\
1.6000&0&0&0&-0.3059\\
0&0.9882&1.2941&0.3059&0
\end{bmatrix}.
\tag{AU.3}
\]

Equation AU.2 generates an explicit \(N\times3\) position matrix and an
\(N\times3\) velocity matrix at every time. A pairwise display kernel is

\[
K_{ij}(t)=\cos(\phi_i(t)-\phi_j(t))
\exp\!\left[-\frac{\|x_i(t)-x_j(t)\|^2}{2\sigma(t)^2}\right],
\tag{AU.4}
\]

where \(\phi_i\) is the polar angle of the two hidden coordinates and
\(\sigma\) is the median nonzero visible separation. Equation AU.4 visualizes
phase alignment and spatial overlap. It is not a derived electromagnetic,
quantum, or intermolecular force.

### AU.3. Property coordinates and the relabeling counterexample

For a scalar property \(y\in[y_{\min},y_{\max}]\), the engine defines

\[
\omega(y)=\omega_{\min}+
(\omega_{\max}-\omega_{\min})
\frac{y-y_{\min}}{y_{\max}-y_{\min}},
\qquad (\omega_{\min},\omega_{\max})=(0.4,1.6).
\tag{AU.5}
\]

Boiling and fusion values receive separate coordinates \(\omega_4\) and
\(\omega_5\). Equation AU.5 is bijective on the finite interval, so inverse
decoding must reproduce every source value. The 34 values for 17 molecules
decode with maximum numerical error \(2.8422\times10^{-14}\) K.

That exact agreement is not evidence for the latent mechanism. Let \(\pi\) be
any permutation of a finite target table. Encoding \(y_{\pi(i)}\) by Equation
AU.5 and immediately applying its inverse also returns \(y_{\pi(i)}\) exactly.
The executable shuffled-label control does so with the same
\(2.8422\times10^{-14}\) K maximum error. Thus a target-derived rotation label
is a lossless reparameterization, not a prediction or causal explanation.

### AU.4. Two independent non-identifiability theorems

**Hidden-kernel theorem.** Let
\(H(t)=\operatorname{diag}(I_3,Q(t))\), where \(Q(t)\) is any two-dimensional
rotation. Since \(PH(t)=P\),

\[
X_5(0)H(t)^TP^T=X_5(0)P^T.
\tag{AU.6}
\]

Therefore every real hidden-plane angular speed produces the identical 3D
trajectory when that plane is uncoupled from visible coordinates. The finite
test finds zero projected difference over 101 sampled times, but Equation AU.6
is the proof for all real times.

**Fixed-tick alias theorem.** In any rotation plane sampled only at
\(t_n=n\Delta\), define \(\omega_k=\omega+2\pi k/\Delta\) for integer \(k\).
Then

\[
R(\omega_k n\Delta)=R(\omega n\Delta+2\pi kn)=R(\omega n\Delta).
\tag{AU.7}
\]

Hence an exact fixed tick does not by itself identify the continuous angular
rate. The implementation checks 205 alias pairs independently; the largest
floating-point discrepancy is \(4.931\times10^{-11}\) at deliberately large
alias frequencies. This is numerical roundoff around the exact equality. As
already required by revision 1.15, finite sampled prefixes do not identify an
arbitrary generator.

### AU.5. Result, limitations, and next test

Matter Harmony Lab 0.3.0 creates matrices for all 17 frozen alkane graphs and a
65-frame n-heptane animation. All 30 unit tests pass: the 17 preserved property
tests and 13 new projection tests. A separately written audit imports no
production module and reproduces 68 property/frequency checks, 2,730 projected
position and velocity scalars, 124 hidden-invariance states, and 205
integer-tick alias pairs.

The result establishes an internally consistent candidate family and a sharp
inverse-problem boundary. It does not establish that temperature is an angular
speed, that melting reverses rotation, that the displayed graph embedding is a
real molecular shape, or that a hidden dimension caused any property. Static
scalar values cannot distinguish the infinitely many constructions in
Equations AU.6--AU.7. Standard alternatives—including conformer geometry,
crystal packing, electronic structure, pressure, intermolecular forces, and
model misspecification—remain live.

The next admissible experiment must define one shared descriptor-to-generator
map without reading the property being predicted. It should be frozen using
C5--C7 development molecules and evaluated once on reserved C8 structures or,
preferably, on a genuinely time-dependent or spectroscopic observable with a
declared observation operator. The topology baseline and ordinary physical
descriptors must remain in the comparison. Static target values may score the
prediction but may not define its latent frequency.

Revision 1.14's idempotent complete deterministic closure remains unchanged.
Revision 1.15's sampled non-crossing limit, angular-impulse formula, arbitrary-
generator non-identifiability, and the recurrence learner's assumption also
remain unchanged: the \(2K\)-sample continuation guarantee applies only to a
noiseless homogeneous constant-coefficient linear recurrence of order at most
\(K\) from the initial state. No seven-dimensional, P=NP, physical-hidden-
string, or theological claim is introduced by this appendix.

## Appendix AV. Water harmonics as a coupled projection experiment

### AV.1. Scope and calibrated three-dimensional inputs

The bounded task is to animate many connected water molecules while making the
proposed higher-coordinate movement mathematically explicit. The construction
starts with ordinary quantities rather than inferring them from latent motion:

1. \(N=64\) water molecules;
2. molar mass \(M=18.0153\,\mathrm{g\,mol^{-1}}\);
3. density \(\rho=997.047\,\mathrm{kg\,m^{-3}}\) at \(298.15\,\mathrm K\);
4. rigid three-site geometry \(r_{\mathrm{OH}}=0.9572\) angstrom and
   \(\angle\mathrm{HOH}=104.52^\circ\).

The volume per molecule is

\[
v_1=\frac{M}{N_A\rho},                                                  \tag{AV.1}
\]

and the conventional diamond-cell length containing eight sites is

\[
a=(8v_1)^{1/3}=6.214721\ \text{angstrom}.                              \tag{AV.2}
\]

Two cells in each direction give a \(12.429442\)-angstrom periodic box. Its
128 nearest-neighbor edges have degree four and a common minimum-image
oxygen--oxygen distance \(a\sqrt3/4=2.691053\) angstrom. This is an ordered
tetrahedral test scaffold chosen for a compact connected control. Matching
density and rigid geometry is therefore by construction, not a prediction, and
the scaffold is not identified with an instantaneous configuration of liquid
water.

### AV.2. Five coordinates per molecule and a norm-preserving coupling

Let

\[
q\in\mathbb R^{3N},\qquad h\in\mathbb R^{2N},\qquad
z=\begin{bmatrix}q\\h\end{bmatrix}.                                    \tag{AV.3}
\]

The visible vector \(q\) contains three dimensionless displacement coordinates
per molecule. The vector \(h\) contains two declared hidden coordinates per
molecule. A coordinate- and graph-derived matrix

\[
C\in\mathbb R^{3N\times2N}                                              \tag{AV.4}
\]

couples each hidden pair to two local visible directions, using 58% local and
42% nearest-neighbor weight before spectral normalization. These percentages
are frozen construction choices, not fitted physical constants. Define

\[
G=\begin{bmatrix}0&C\\-C^T&0\end{bmatrix},\qquad
\frac{dz}{d\tau}=Gz.                                                    \tag{AV.5}
\]

Then \(G^T=-G\), and the written invariant is

\[
\frac{d}{d\tau}\|z\|_2^2=z^T(G+G^T)z=0.                               \tag{AV.6}
\]

The initial state is a localized two-phase hidden pulse with zero visible
displacement. At the dimensionless simulator ticks

\[
\tau_n=n\Delta\tau,\qquad \Delta\tau=0.125,                             \tag{AV.7}
\]

the engine evaluates \(z(\tau_n)=e^{\tau_nG}z(0)\) by a Hermitian spectral
decomposition. This fixed tick is a reproducible numerical sampling choice. It
is not a discovered universal physical increment.

### AV.3. Three-dimensional rendering and the hidden-null control

For molecule \(i\), the rendered oxygen center is

\[
R_i(\tau)=R_i^{(0)}+s q_i(\tau),\qquad s=0.34\ \text{angstrom},          \tag{AV.8}
\]

after which the two hydrogen centers are reconstructed at the fixed reference
bond length and angle. Translucent hue and radius encode

\[
\phi_i=\operatorname{atan2}(h_{i,2},h_{i,1}),\qquad
A_i=\|h_i\|_2,                                                          \tag{AV.9}
\]

and candidate-link appearance depends on

\[
I_{ij}=\tfrac12[1+\cos(\phi_i-\phi_j)].                                 \tag{AV.10}
\]

Equations AV.9--AV.10 are an observation convention that lets a viewer see the
latent phase relation. They are not charge density, electric field, magnetic
field, bond energy, or a new force law.

The mandatory counterexample separates hidden motion from visible motion. Set
\(C=0\) and rotate each hidden pair arbitrarily. Then \(q(\tau)=q(0)\) for all
times. The executed trajectory begins with \(q(0)=0\) and reports exactly zero
projected motion under this control. Thus a fifth coordinate does not create
an observable wave unless an explicit coupling is supplied. Moreover, the
integer-tick alias from Appendix AU remains: a two-plane frequency \(\omega\)
and \(\omega+2\pi k/\Delta\tau\) agree at all sampled ticks.

### AV.4. Executed evidence and its limits

The 64-frame run gives:

- maximum generator skew error \(0\);
- state-norm drift \(2.220\times10^{-16}\);
- maximum O--H error \(1.332\times10^{-15}\) angstrom;
- maximum H--O--H error \(1.137\times10^{-13}\) degrees;
- hidden-only projection error \(0\);
- integer-tick alias error \(5.519\times10^{-14}\);
- peak visible RMS state \(0.120060\) in its declared dimensionless units.

All 41 tests in Matter Harmony Lab 0.4.0 pass, including molecule-order
equivariance and equality of coarse and refined integrations at shared sample
times. A separately written program imports no production module and checks
36,864 rendered atomic-coordinate scalars, the density arithmetic, all 128
periodic links, rigid geometry, norm preservation, the hidden-null control, and
the tick alias. These are written mathematics and finite computations. No new
physical measurement was made.

The release does not implement a molecular force field, thermal ensemble,
diffusion, viscosity, dielectric response, magnetism, hydrodynamics, quantum
electrodynamics, or a physical higher-dimensional observation operator. It
therefore cannot yet explain water flow, electricity, or magnetism and cannot
validate fourth- or fifth-dimensional motion. The next admissible experiment
is to freeze an ordinary electrostatic water baseline with explicit partial
charges and compare a time-dependent dielectric or structure-factor observable
against held-out reference data. A latent coupling may be evaluated only as an
additional prespecified model, not as the default meaning of its residual.

Revision 1.14's idempotent complete deterministic closure remains unchanged.
Revision 1.15's sampled non-crossing limit, angular-impulse formula,
arbitrary-generator non-identifiability, and recurrence learner assumption
also remain unchanged: the \(2K\)-sample continuation guarantee applies only
to a noiseless homogeneous constant-coefficient linear recurrence of order at
most \(K\) from the initial state. This appendix neither establishes seven
physical dimensions nor assumes \(P=NP\), and no unconstrained residual is
given a theological interpretation.

## Appendix AW. Water origami and the oscilloscope observation boundary

### AW.1. Ordinary charge and dielectric baseline

The bounded question is whether a time-dependent electric measurement can be
turned into a visible many-water animation without treating the animation as a
unique microscopic reconstruction. Release 0.5.0 first adds ordinary partial
charges to the rigid geometry retained from Appendix AV:

\[
q_{\mathrm O}=-0.834e,\qquad q_{\mathrm H}=+0.417e.                  \tag{AW.1}
\]

Thus \(q_{\mathrm O}+2q_{\mathrm H}=0\). With
\(r_{\mathrm{OH}}=0.9572\) angstrom and
\(\theta_{\mathrm{HOH}}=104.52^\circ\), the charge-model molecular dipole is

\[
\mu=2q_{\mathrm H}r_{\mathrm{OH}}\cos(\theta_{\mathrm{HOH}}/2)
    =2.346971\ \mathrm D.                                             \tag{AW.2}
\]

This is a property of the declared TIP3P-like geometry, not a fresh measured
dipole. The macroscopic reference is the single-Debye linear response

\[
\tau_D\frac{dP}{dt}+P=E.                                               \tag{AW.3}
\]

The published 25-degree-C primary relaxation frequency
\(f_D=18.56\ \mathrm{GHz}\) is frozen as an input, giving

\[
\tau_D=(2\pi f_D)^{-1}=8.575159\ \mathrm{ps}.                          \tag{AW.4}
\]

For \(E(t)=\sin(\omega t)\), the steady response is

\[
P(t)=\frac{1}{\sqrt{1+(\omega\tau_D)^2}}
\sin\!\left(\omega t-\arctan(\omega\tau_D)\right).                  \tag{AW.5}
\]

At \(\omega\tau_D=1\), the gain is \(1/\sqrt2\) and the lag is
\(45^\circ\). A separately stepped Runge--Kutta calculation agrees with
Equation AW.5 to \(3.529\times10^{-10}\) over the executed interval. The
frequency agreement is calibration, not an out-of-sample prediction.

### AW.2. Origami as a many-to-one observation map

Let \(d_i(t)\in\mathbb R^3\) be the unit dipole direction of molecule \(i\),
and concatenate the \(N\) directions as \(x(t)\in\mathbb R^{3N}\). The
oscilloscope in this experiment reports only their vector mean:

\[
y(t)=Ax(t),\qquad
A=\frac1N\begin{bmatrix}I_3&I_3&\cdots&I_3\end{bmatrix}.              \tag{AW.6}
\]

For \(N=64\), \(A\) has rank three and ambient nullity \(3N-3=189\).
Calling this map an origami fold is an analogy with precise content: many
different high-dimensional states occupy the same observed point after the
fold.

The release constructs that collision without changing dipole lengths. Pair
the molecules. For one desired measured x component \(p(t)\), define each
pair by

\[
d_{2j}(t)=\begin{bmatrix}p(t)\\s(t)\cos\phi_j(t)\\s(t)\sin\phi_j(t)\end{bmatrix},
\qquad
d_{2j+1}(t)=\begin{bmatrix}p(t)\\-s(t)\cos\phi_j(t)\\-s(t)\sin\phi_j(t)\end{bmatrix},
\quad s(t)=\sqrt{1-p(t)^2}.                                            \tag{AW.7}
\]

Every vector in Equation AW.7 has unit norm, while every pair sums to
\((2p(t),0,0)^T\) independently of \(\phi_j(t)\). Therefore any two distinct
phase histories \(\phi_j^{(A)}(t)\) and \(\phi_j^{(B)}(t)\) give

\[
Ax_A(t)=Ax_B(t)=\begin{bmatrix}p(t)&0&0\end{bmatrix}^T                \tag{AW.8}
\]

for every real time at which the construction is defined. This is a written
non-identifiability result, not merely an animation coincidence.

### AW.3. Executed oscilloscope and counterexamples

The executable uses 96 stored frames over two periods. It maps the normalized
Debye response into \(|p(t)|<1\), chooses two different transverse phase
families, constructs both rigid three-site molecular movies, and computes the
scope trace again from the explicit charges. The results are:

- orientation-fold difference exactly zero in the stored arrays;
- explicit-charge trace difference \(4.441\times10^{-16}\);
- observation-kernel residual \(3.469\times10^{-18}\);
- microscopic RMS separation from \(0.688568\) to \(0.820871\);
- unit-dipole error \(2.220\times10^{-16}\);
- positive normalized cycle dissipation \(0.0294141\ \mathrm{ps}^{-1}\).

The 3D panels show the two microscopic folds, and the oscilloscope panel shows
their exactly overlapping polarization traces beside the applied field. Thus
the visualization exposes rather than hides the inverse problem: even a
noise-free waveform does not select one of these molecular realizations.

A second counterexample retains the fixed-tick boundary. If samples occur at
\(t_n=n\Delta t\), then sinusoids at \(f\) and \(f+k/\Delta t\) agree at every
sample for integer \(k\). The executed floating discrepancy is
\(8.399\times10^{-14}\), roundoff around the exact identity. A continuous
bandwidth restriction or another measurement is needed to remove these
aliases.

### AW.4. Evidence type, limitations, and next test

All 52 package tests pass. An independent implementation imports no production
module and checks the analytic response, charge arithmetic, rigid geometry,
92,160 stored atomic-coordinate scalars, equal folded traces, distinct
microstates, positive loss, and the alias control.

The following distinctions are mandatory:

1. Equations AW.1--AW.5 are ordinary model mathematics with published inputs.
2. Equations AW.6--AW.8 are written linear algebra and a constructed
   non-identifiability proof.
3. The 96-frame run is finite synthetic computation.
4. No new physical measurement was performed.
5. The microscopic orientation histories, finite scaffold, display scale, and
   camera are constructed and are not inferred from the dielectric trace.

Consequently the release is not liquid-water molecular dynamics and does not
establish a four- or five-dimensional generator, a universal spatial grid, a
universal physical tick, magnetism, quantum electrodynamics, or a theological
interpretation. A proposed higher-dimensional law must specify its forward
coupling before seeing the test result and predict a fresh observable better
than this ordinary baseline. The next bounded experiment is a frozen
multi-frequency complex-permittivity curve tested on held-out measurements.
If polarization leaves microscopic models equivalent, a structural observable
such as a scattering-derived correlation must be added rather than assigning
the unresolved degrees of freedom to a preferred ontology.

Revision 1.14's complete deterministic closure remains idempotent. Revision
1.15's integer-tick non-crossing limitation, angular-impulse formula, and
finite-prefix generator non-identifiability remain unchanged. The recurrence
learner's \(2K\)-sample continuation guarantee still assumes a noiseless
homogeneous constant-coefficient linear recurrence of order at most \(K\) from
the initial state.

## Appendix AX. Sunlight, moving water, and a finite visual observer

### AX.1. Bounded question and ordinary optical baseline

The bounded question is whether a fixed source, a moving interface, and a
finite visual observer can produce the requested underwater light-bending
image without treating a rendered movie as a unique reconstruction of reality.
Matter Harmony Lab 0.6.0 fixes the source direction and uses a nominal 5772 K
Planck spectrum. The source is distant enough that its incident rays are
parallel over the four-metre displayed surface.

The constructed water height is a three-component finite-depth linear wave,

\[
h(x,y,t)=\sum_{j=1}^{3} A_j
\sin(\mathbf k_j\!\cdot\!\mathbf r-\omega_jt+\phi_j),\qquad
\omega_j^2=g|\mathbf k_j|\tanh(|\mathbf k_j|H).                       \tag{AX.1}
\]

The amplitudes, wavelengths, directions, phases, and depth are frozen in the
release protocol. They are not recovered from a measured water surface. The
analytic normal is

\[
\hat{\mathbf n}=\frac{(-\partial_xh,-\partial_yh,1)}
{\sqrt{1+(\partial_xh)^2+(\partial_yh)^2}}.                            \tag{AX.2}
\]

For every spectral bin, the IAPWS R9-97 formulation supplies
\(n_w(\lambda,T,\rho)\). The executable then applies vector Snell refraction
and unpolarized Fresnel reflection. In scalar form,

\[
n_a\sin\theta_i=n_w(\lambda)\sin\theta_t,
\qquad R_F=\tfrac12(R_s+R_p),\qquad T_F=1-R_F.                         \tag{AX.3}
\]

This partitions interface energy in the no-absorption baseline. Intersecting
each transmitted ray with a bottom plane produces an expected-energy density,
not a list of measured individual photon paths. Local ray concentration
generates the moving caustic.

At 298.15 K and 997.047 kg/m3 the executed refractive indices are 1.345110691
at 380 nm, 1.332858051 at 589.3 nm, and 1.328482654 at 780 nm. The
normal-incidence 589.3 nm Fresnel reflectance is 0.02035830. Across the moving
surface, mean reflectance is 0.02557769 and mean transmittance is 0.97442231.
These are calculations from frozen reference inputs, not predictions of a
higher-dimensional theory.

### AX.2. Spectral observation and the metamer counterexample

The eye surrogate receives the spectral caustic and applies a three-coordinate
standard-color fold,

\[
\mathbf y(x,y,t)=
\int_{380\,\mathrm{nm}}^{780\,\mathrm{nm}}
L(x,y,t,\lambda)
\begin{bmatrix}\bar x(\lambda)\\\bar y(\lambda)\\\bar z(\lambda)\end{bmatrix}
\,d\lambda.                                                          \tag{AX.4}
\]

The functions in Equation AX.4 are a declared analytic approximation to the
CIE 1931 two-degree standard observer. This is a population colorimetric
standard, not an individual retina, optic nerve, or visual cortex.

Equation AX.4 cannot be inverted on arbitrary spectra. On the 41-bin control
grid its matrix has rank three and sampled nullity 38. The executable
constructs two nonnegative spectra \(s_A\ne s_B\) in the same observation
fiber:

\[
C s_A=C s_B,
\qquad
\frac{\|s_A-s_B\|_2}{\|s_A\|_2}=0.709286,                             \tag{AX.5}
\]

while the maximum stored XYZ difference is
\(1.421\times10^{-14}\). Thus even a noise-free standard-color observation
does not contain every visible spectrum. A claim that a human-eye observer
can measure everything humans have observed is therefore false for this
declared map: metameric inputs already collide before adaptation, eye motion,
retinal sampling, or neural processing are added.

### AX.3. Why the animation is not a hologram

The optical carrier frequencies corresponding to 780 and 380 nm are about 384
and 789 THz. The executable does not step those oscillations. It analytically
propagates time-averaged spectral irradiance and averages three surface states
over a 1/120-second display exposure. The movie contains 150 output frames at
60 frames per second.

Smooth playback can reduce visible frame discontinuity under some viewing
conditions, but no universal flicker threshold is assumed. More importantly,
temporal fusion cannot restore information removed by the observer. A
hologram requires coherent complex-field propagation and wavefront
reconstruction. This ray engine discards optical phase. For example,

\[
E_A=e^{i\varphi},\qquad E_B=e^{i(\varphi+\pi/2)},
\qquad |E_A|^2=|E_B|^2=1.                                             \tag{AX.6}
\]

The executed intensity difference in Equation AX.6 is below
\(6.7\times10^{-16}\). The displayed caustic is therefore a radiance
projection with three-dimensional geometric cues, not a holographic
reconstruction and not evidence that the displayed motion originates in a
fourth or fifth spatial plane.

### AX.4. Executed evidence, limits, and next test

The 0.6.0 run executes 48,988,800 ray–wavelength–exposure samples and renders
9,331,200 RGB channel values. Median caustic luminance contrast is 1.326639.
All 65 package tests pass. A separately written audit imports no production
module and re-evaluates the IAPWS index, spectral observer, energy partition,
metamer collision, phase-loss control, fixed-tick alias, frame arrays, and
60 fps media metadata.

The evidence types remain distinct:

1. IAPWS refractive index, the CIE standard observer, and nominal solar
   temperature are reference inputs.
2. Snell/Fresnel relations and the metamer and phase collisions are written
   mathematics.
3. The ray bundle, caustic, and movie are finite synthetic computation.
4. No new water-surface or optical measurement was made.
5. A physical higher-dimensional generator remains a hypothesis requiring a
   specified forward law and a fresh prediction beyond this baseline.

The engine omits water absorption, volumetric scattering, polarization-
resolved transport, diffraction, coherence, eye accommodation, retinal cone
mosaics, eye movements, neural processing, and observer variation. A failure
against data would not uniquely identify a missing string, rotation, or
dimension; surface error and these ordinary omissions remain live alternatives.

The next bounded test is to freeze a measured water-surface height movie,
depth, source spectrum, and camera geometry, then score predicted caustic ridge
positions and chromatic offsets on held-out frames without refitting. A
separate coherent wave-optics branch must first pass analytic phase-grating
tests before any output is called holographic.

Revision 1.14's complete deterministic closure remains idempotent. Revision
1.15's integer-tick non-crossing limitation, angular-impulse formula, and
finite-prefix generator non-identifiability remain unchanged. The recurrence
learner's \(2K\)-sample continuation guarantee still assumes a noiseless
homogeneous constant-coefficient linear recurrence of order at most \(K\) from
the initial state. This appendix neither establishes seven physical
dimensions nor assumes \(P=NP\), and it assigns no residual to a theological
cause.


## Appendix AY. Water-drop acoustics and the observer-rate boundary

### AY.1. Bounded question and declared impact

The bounded question is whether a water-drop impact can be connected to a
surface ripple, an airborne pressure waveform, and synchronized media without
mistaking a plausible synthesis for a unique reconstruction of a real event.
Matter Harmony Lab 0.7.0 fixes a 4.0 mm water drop, 0.08224 m vacuum fall,
0.080 m pool depth, and a nominal room-temperature water/air baseline. The
calculated impact speed is

\[
v_i=\sqrt{2gh}=1.2700385\ {\rm m\,s^{-1}}.                         \tag{AY.1}
\]

The corresponding Weber and Bond numbers are 89.3836 and 2.17373. These
dimensionless numbers characterize the declared baseline; they do not by
themselves determine splash topology or whether a bubble is entrained. Those
outcomes require a nonlinear two-phase free-surface calculation or
measurement. The executable therefore declares one bubble after impact rather
than claiming to predict it.

The radial display packet has a 0.025 m carrier and uses the ordinary
finite-depth capillary--gravity dispersion relation,

\[
\omega^2(k)=
\left(gk+\frac{\sigma}{\rho_w}k^3\right)\tanh(kh).              \tag{AY.2}
\]

Its frequency is 9.56335 Hz, phase speed is 0.239084 m/s, and group speed is
0.195422 m/s. A Gaussian radial envelope and decaying central cavity make a
finite visualization. The carrier obeys Equation AY.2, but the envelope is a
declared narrow-band approximation rather than a solved impact flow.

### AY.2. Entrained-bubble pitch and airborne pressure

Phillips, Agarwal, and Jordan used synchronized high-speed video and acoustic
measurements to show that the characteristic dripping-tap sound is driven by
resonant oscillation of an entrained bubble. The release therefore uses the
ideal Minnaert relation for one declared spherical bubble,

\[
f_M=\frac{1}{2\pi R_b}
\sqrt{\frac{3\gamma P_b}{\rho_w}}.                               \tag{AY.3}
\]

For the frozen radius \(R_b=0.800\) mm and hydrostatic pressure at 4 mm depth,
Equation AY.3 gives 4110.9205 Hz. The stored 48 kHz pressure trace has a
4116.67 Hz peak in its finite 60 ms FFT window, 5.75 Hz from the analytic
carrier.

The above-water field reported in the 2018 experiment is not adequately
described as the underwater field simply crossing the interface. Release
0.7.0 instead uses a deliberately bounded surface-coupled compact-source
surrogate,

\[
p(r,t)=\frac{\rho_a}{4\pi r}
\frac{d^2V_{\rm eff}}{dt^2}\left(t-\frac{r}{c_a}\right),         \tag{AY.4}
\]

where \(V_{\rm eff}\) is a damped sinusoidal effective volume. The 3% volume
amplitude, 18 ms decay, 6 ms entrainment delay, and impact-pulse coefficient
are constructions. Equation AY.4 supplies a physically scaled candidate for
pitch, decay, retarded arrival, and inverse-distance spreading; it does not
supply a calibrated absolute sound level. Playback is peak-normalized.

At \(r=0.750\) m and \(c_a=346.13\) m/s, the calculated air transit is
2.16682 ms. The impact pulse arrives at 0.4021668 s and the declared bubble
tone at 0.4081668 s. The delivered MP4 therefore combines 180 picture frames
at 60 fps with 144,000 mono audio samples at 48 kHz. The visual stream cannot
resolve a 2.17 ms transit, whereas the audio stream can.

### AY.3. Why neither frame rate nor pitch is a unique inverse

A human eye is not a camera with one universal frame rate. Temporal visibility
varies with luminance, contrast, spatial structure, eye movement, retinal
location, and observer. Davis, Hsieh, and Lee reported artifacts above 500 Hz
for spatially structured modulation even though conventional uniform-field
measurements often fall near 50--90 Hz. Sixty fps is therefore only the output
media rate in this release. It is not asserted as a biological tick or a
guarantee of invisibility.

Acoustic pitch is also non-injective. From Equation AY.3,

\[
f_M(R,P)=f_M(2R,4P).                                                  \tag{AY.5}
\]

The executable checks this collision to machine precision. Consequently one
ideal tone cannot independently identify both radius and pressure. A stronger
ambiguity remains because the same synthetic surface packet can be paired with
different declared bubbles: the release does not solve the cavity dynamics
that would link the displayed ripple to a unique bubble distribution.

These are observation fibers in the same precise sense used elsewhere in this
paper. A lower-dimensional measurement may constrain a declared forward model
without uniquely determining its source state. Calling the unresolved part a
fifth-dimensional rotation would add an interpretation, not information.

### AY.4. Executed evidence, limits, and next test

All 79 package tests pass. A separately written audit imports no production
module and recomputes 1,658,880 surface-height scalars and 144,000 pressure
samples. It independently checks free fall, Weber and Bond numbers,
capillary--gravity phase and group speed, ideal Minnaert pitch, the pitch
collision, retarded inverse-distance pressure, exact PCM samples, and the
three-second 60 fps/48 kHz media metadata.

The evidence types remain distinct:

1. Water/air properties and the published bubble-sound mechanism are reference
   inputs and primary-research provenance.
2. Equations AY.1--AY.5 and the pitch collision are written conditional
   mathematics.
3. The ripple, pressure waveform, WAV, and movie are finite synthetic
   computation.
4. No drop, bubble, microphone, or room response was measured in this release.
5. A physical higher-dimensional generator remains a hypothesis and is not
   needed to produce the bounded sound.

The engine omits aerodynamic drag, nonlinear cavity formation, splash
droplets, multiple bubbles, viscous and thermal bubble corrections, walls,
reverberation, atmospheric attenuation, head-related transfer, and hearing.
Its constructed active-window SPL is not a measurement. Failure against real
data could arise from any of these ordinary omissions and would not uniquely
identify a missing string, rotation, or dimension.

The next test is empirical and precommitted: measure drop diameter and impact
speed; infer bubble radius from synchronized high-speed video; calibrate
microphone distance and room impulse response; then score held-out ripple-front
position, tonal peak, decay constant, arrival time, and absolute pressure
without refitting.

Revision 1.14's complete deterministic closure remains idempotent. Revision
1.15's integer-tick non-crossing limitation, angular-impulse formula, and
finite-prefix generator non-identifiability remain unchanged. The recurrence
learner's \(2K\)-sample continuation guarantee still assumes a noiseless
homogeneous constant-coefficient linear recurrence of order at most \(K\) from
the initial state. This appendix neither establishes seven physical dimensions
nor assumes \(P=NP\), and it assigns no residual to a theological cause.


## Appendix AZ. Bubble hidden-variable inverse and observation fibers

### AZ.1. Bounded question and correction of scope

The bounded question is whether the complete damped pressure trace of the
single-bubble model uniquely determines bubble radius. Revision 1.50 already
showed that one ideal Minnaert tone cannot identify radius and pressure
simultaneously. This appendix makes a more realistic ambiguity explicit by
adding ordinary thermodynamic and interfacial nuisance variables.

The revision-1.50 relation

\[
f_M=\frac{1}{2\pi R}\sqrt{\frac{3\gamma p_\infty}{\rho}}
\tag{AZ.1}
\]

is retained as an ideal formula, not retracted. For a small spherical bubble,
linearizing Rayleigh--Plesset dynamics while including vapor pressure and
surface tension gives the declared stiffness model

\[
\omega_0^2=
\frac{3\kappa(p_\infty-p_v)+2(3\kappa-1)\sigma/R}
{\rho R^2}.
\tag{AZ.2}
\]

Here \(\kappa\) is an effective polytropic exponent. Prosperetti's primary
analysis shows why it cannot silently be equated with one universal constant:
the effective exponent and thermal damping depend on frequency, radius, and
gas/liquid properties [Prosperetti (1977)](https://doi.org/10.1121/1.381252).
Equation AZ.2 is therefore used only for a bounded sensitivity study. It is
not presented as the full complex thermal model.

At 298.15 K the executable uses \(p_v=3169.824486\) Pa from the IAPWS reference
formulation, \(\rho=997.047\ {\rm kg\,m^{-3}}\),
\(\sigma=0.07197\ {\rm N\,m^{-1}}\), and atmospheric plus hydrostatic pressure
at 4 mm depth. For \(R=0.800\) mm, Equation AZ.2 gives

\[
f_0(\kappa=1)=3421.6939\ {\rm Hz},\qquad
f_0(\kappa=1.4)=4048.9557\ {\rm Hz}.
\tag{AZ.3}
\]

The latter lies 61.9648 Hz, or 1.5073%, below the ideal value 4110.9205 Hz.
This is a checked scope correction: 4110.9205 Hz is the ideal-Minnaert output,
not a unique physical prediction after ordinary corrections are admitted.

### AZ.2. Exact radius--thermal-response observation fiber

For fixed positive \(\kappa\), Equation AZ.2 is monotone on the declared
positive-radius branch and can be inverted numerically. Holding the previous
4110.9205 Hz carrier fixed while scanning

\[
1\le\kappa\le1.4
\tag{AZ.4}
\]

gives

\[
0.6659558\ {\rm mm}\le R\le0.7879498\ {\rm mm}.
\tag{AZ.5}
\]

The 18.3186% span relative to the smaller endpoint is not a statistical
confidence interval. It is the exact image of the declared sensitivity
interval through Equation AZ.2. The endpoints are retained rather than
collapsed into a false point estimate.

The pressure observer used in the bounded sound branch depends on the second
derivative of

\[
V(t)=V_s e^{-t/\tau}\sin(2\pi f_0t),\qquad
V_s=\frac{4\pi}{3}R^3\varepsilon,
\tag{AZ.6}
\]

where \(\varepsilon\) is the fractional volume amplitude. Consequently two
states have the same complete pressure trace whenever they share \(f_0\),
\(\tau\), arrival, propagation distance, and \(R^3\varepsilon\). Release
0.8.0 constructs 401 such states. Its endpoints are

\[
(\kappa,R,\varepsilon)
=(1.0,0.6659558\ {\rm mm},0.0496913)
\tag{AZ.7}
\]

and

\[
(\kappa,R,\varepsilon)
=(1.4,0.7879498\ {\rm mm},0.0300000).
\tag{AZ.8}
\]

Both use the same 18 ms decay, 6 ms local arrival, and 0.75 m observer
distance. Their generated 48 kHz traces agree to

\[
\max_t|p_1(t)-p_2(t)|=3.667\times10^{-16}\ {\rm Pa}
\tag{AZ.9}
\]

over 3,840 samples per endpoint. This is an exact constructive
non-identifiability result inside the declared equations, not a claim that
either endpoint is the true bubble.

### AZ.3. Rank statement and what measurement must add

Use log parameters

\[
\theta=(\log R,\log\kappa,\log\varepsilon,\log\tau)
\tag{AZ.10}
\]

and log observables

\[
y=(\log f_0,\log V_s,\log\tau).
\tag{AZ.11}
\]

The executed finite-difference Jacobian \(\partial y/\partial\theta\) is
3-by-4, has numerical rank three, and therefore has local nullity one. The
null direction is the differential form of the exact family in Equations
AZ.7--AZ.9. More precise audio does not remove a structural nullspace when
the observation operator itself folds the source state.

An additional independent observable is required. Synchronized high-speed
imaging of radius and depth would intersect the acoustic fiber. A calibrated
underwater hydrophone would constrain the near-field radiation, while an
airborne microphone would test the surface-transfer model. Temperature,
liquid properties, sensor positions, and boundary response must be frozen
before fitting. Oguz and Prosperetti showed that proximity to a free surface
changes bubble oscillation and radiation
[Oguz and Prosperetti (1990)](https://doi.org/10.1121/1.399560), so that effect
cannot be hidden inside an unconstrained amplitude. Their rain-noise model
also predicted bubbles somewhat larger than observed and consequently shifted
the spectral peak, a direct historical warning that agreement in sound level
does not validate source geometry
[Oguz and Prosperetti (1991)](https://doi.org/10.1017/S0022112091002768).

### AZ.4. Evidence ledger, limitations, and next test

The evidence types remain separate:

1. The IAPWS water properties and cited bubble-acoustics studies are reference
   inputs and primary-research provenance.
2. Equations AZ.2, AZ.6, the equivalence construction, and the rank ceiling are
   written conditional mathematics.
3. The 401-state family, 7,680 endpoint pressure samples, bisection inverses,
   and finite-difference Jacobian are finite synthetic computation.
4. No bubble radius, pressure waveform, effective \(\kappa\), or free-surface
   correction was measured here.
5. A hidden-dimensional source remains a hypothesis with no additional
   support from this ambiguity; an ordinary classical family already produces
   it exactly.
6. Theological interpretation is outside the observation map.

All 88 package tests pass. A separately written audit imports no production
module and reproduces all 401 corrected-frequency states, 7,680 endpoint
pressure samples, constant oscillating volume, the rank-3/4 local inverse, and
the waveform collision.

The model still omits the full frequency-dependent thermal law, viscous and
acoustic-radiation damping, free-surface proximity, nonspherical modes,
multiple bubbles, nonlinear entrainment, calibrated underwater-to-air
transfer, walls, and sensor response. These are specific candidate variables,
not a license to add arbitrary degrees of freedom after failure. Each addition
must reduce a predeclared residual on fresh data without destroying earlier
predictions.

The next bounded empirical test is to measure radius and depth from
synchronized high-speed video and record calibrated underwater and airborne
pressure for repeated impacts. Fit the thermal/damping branch on one subset,
then score held-out pitch, decay, phase, and amplitude without refitting. If
the observation fiber remains, add a measurement that cuts it; do not select a
preferred hidden state by interpretation alone.

Revision 1.14's complete deterministic closure remains idempotent. Revision
1.15's integer-tick non-crossing limitation, angular-impulse formula, and
finite-prefix generator non-identifiability remain unchanged. The recurrence
learner's \(2K\)-sample continuation guarantee still assumes a noiseless
homogeneous constant-coefficient linear recurrence of order at most \(K\) from
the initial state. This appendix neither establishes seven physical
dimensions nor assumes \(P=NP\), and it assigns no residual to God or another
theological cause.


## Appendix BA. Collective vibrational families and thermodynamic phase

### BA.1. The bounded question and an exact counterexample

The question is not whether vibration matters. It is whether a vector of
isolated-molecule vibration frequencies, by itself, determines whether matter
is solid, liquid, or gas. Release 0.9.0 freezes the NIST water fundamentals

\[
q_{\rm H_2O}=(1595,3657,3756)\ {\rm cm}^{-1}
\tag{BA.1}
\]

as a deliberately phase-blind molecular-identity descriptor
([NIST Chemistry WebBook](https://webbook.nist.gov/cgi/cbook.cgi?ID=C7732185&Mask=800)).
The same vector is paired with three externally specified H2O controls at
101325 Pa: 260 K solid, 298.15 K liquid, and 400 K vapor. These phase labels
come from ordinary thermodynamic state information, not from Equation BA.1;
the water equation-of-state provenance is the
[IAPWS-95 release](https://iapws.org/technical-guidance/release/IAPWS-95).

Let \(h(q)\) be any deterministic classifier using only this descriptor.
All three inputs to \(h\) are identical, so all three outputs are identical.
On the balanced three-control set, at most one label is correct. Therefore

\[
\max_h {\rm accuracy}(h)=\frac13,
\qquad
\min_h {\rm error}(h)=\frac23.
\tag{BA.2}
\]

This is a written proof for the stated information restriction, not a fitted
failure and not a claim that condensed-phase water retains unshifted gas-phase
normal modes. It proves that phase-blind molecular frequencies cannot be a
phase oracle. Intermolecular organization, temperature, pressure, composition,
and system size must enter the observation map.

### BA.2. A collective spectrum that can express both diffusion and vibration

Rahman's molecular-dynamics study of liquid argon calculated the
velocity-autocorrelation function and its broad frequency spectrum while also
checking structural and self-diffusion observables
([Rahman 1964](https://doi.org/10.1103/PhysRev.136.A405)). The two-phase
thermodynamics construction later decomposed a liquid density of states into
gas-like diffusive and solid-like vibrational parts
([Lin, Blanco, and Goddard 2003](https://doi.org/10.1063/1.1624057)); a revised
treatment makes the assumptions and fluidicity correction explicit
([Sun et al. 2017](https://doi.org/10.1063/1.5001798)). These are methods for
collective trajectories, not licenses to label a phase from one bond frequency.

The new executable uses the bounded surrogate

\[
F(\omega;f_g)=f_gF_g(\omega)+(1-f_g)F_s(\omega),
\qquad 0\le f_g\le1,
\tag{BA.3}
\]

where the frozen solid-like basis satisfies \(F_s(0)=0\) and the gas-like
basis satisfies \(F_g(0)>0\). Hence within this declared two-basis family,

\[
f_g=\frac{F(0)}{F_g(0)}.
\tag{BA.4}
\]

Equation BA.4 is exact conditional mathematics for the surrogate. It is not a
general estimator for arbitrary molecular data. Finite execution recovers all
101 fractions from zero to one with maximum absolute error
\(1.1102230246251565\times10^{-16}\). The integrated low-frequency mass rises
monotonically from \(8.9987\times10^{-6}\) to 0.4599443.

Three declared reference profiles use \(f_g=0\), 0.35, and 1 for solid-like,
liquid-like, and gas-like controls. Their pairwise squared Hellinger distances
are

| Pair | \(H^2\) |
|---|---:|
| solid-like / liquid-like | 0.3077968075 |
| solid-like / gas-like | 0.7512211852 |
| liquid-like / gas-like | 0.1832326308 |

The positive distances show that this particular collective descriptor can
separate the three frozen profiles. The full 101-state path simultaneously
supplies the counterexample to any stronger claim: it is continuous, so nature
does not have to form three universally separated vibration families. A phase
boundary requires thermodynamic stability and finite-condition conventions;
it cannot be read from a universal spectral threshold alone.

### BA.3. What was checked and what remains empirical

The executable retains two argon provenance checks. Rahman's published state
maps to reduced values \(T^*=0.78797997\) and \(\rho^*=0.81769990\) under the
declared Lennard-Jones constants. The representative liquid-argon state in the
revised two-phase treatment maps to 131.78 K and 1428.27 kg/m3 and supplies the
published \(f_g=0.35\) anchor. The plotted spectral line shapes, widths, and
collision scale are nevertheless declared synthetic choices. They were not
extracted from either publication.

The evidence types are therefore:

1. NIST frequencies, IAPWS thermodynamics, and the cited argon studies are
   reference inputs and research provenance.
2. Equations BA.2 and BA.4 are written conditional mathematics.
3. The 2,401-point spectra, 101-state continuum, distances, and figure are
   finite synthetic computations.
4. No condensed-phase trajectory or spectrum was measured in this release.
5. Hidden rotations or higher-dimensional particles remain hypotheses; these
   results neither require nor identify them.
6. Theological interpretation is outside the phase classifier.

All 98 package tests pass. A separately implemented audit reproduces the
2,401-frequency grid, 101 continuum states, all three phase-control labels,
all three spectral distances, and the exact inversion. The correction is
actionable: phase work must begin with a many-particle trajectory or measured
collective response, then test whether a frozen descriptor transfers across
conditions and chemical families.

The next test is to generate velocity-autocorrelation spectra with one
validated intermolecular potential across externally labeled solid, liquid,
and vapor states. Freeze all line-shape choices and any classifier on a
training set, then score held-out temperatures, pressures, densities, and
molecules without refitting. Diffusion, structure factor, and enthalpy should
be simultaneous controls. If the spectrum fails, first test the potential,
finite-size effects, sampling, and omitted ordinary interactions; failure does
not uniquely select an extra dimension.

Revision 1.14's complete deterministic closure remains idempotent. Revision
1.15's integer-tick non-crossing limitation, angular-impulse formula, and
finite-prefix generator non-identifiability remain unchanged. The recurrence
learner's \(2K\)-sample continuation guarantee still assumes a noiseless
homogeneous constant-coefficient linear recurrence of order at most \(K\) from
the initial state. This appendix establishes neither seven physical dimensions
nor \(P=NP\), and it assigns no unconstrained residual to God.


## Appendix BB. Five-dimensional harmonies and a finite three-dimensional observer

### BB.1. Bounded model and complete linear harmony theorem

The bounded question is: if a latent state has five real coordinates and
moves by one fixed norm-preserving linear law, what can a three-coordinate
observer see? Define

\[
\dot z(t)=\Omega z(t),\qquad \Omega^T=-\Omega,
\qquad y(t)=Pz(t),\quad P=[I_3\;0].
\tag{BB.1}
\]

This is a theoretical candidate family, not an assertion about physical
spacetime. Because \(\Omega\) is real and skew-symmetric, an orthogonal change
of basis gives

\[
Q^T\Omega Q=
\operatorname{diag}\!\left(J(\omega_1),J(\omega_2),0\right),
\quad
J(\omega)=
\begin{bmatrix}0&-\omega\\ \omega&0\end{bmatrix},
\tag{BB.2}
\]

for \(\omega_1,\omega_2\ge0\). Exponentiating each block proves

\[
\boxed{
y(t)=c+\sum_{j=1}^{2}
\left[a_j\cos(\omega_jt)+b_j\sin(\omega_jt)\right].
}
\tag{BB.3}
\]

Therefore a fixed five-coordinate skew-linear model produces one stationary
component and at most two elliptical 3D harmonies. Every projected coordinate
also satisfies

\[
y^{(5)}+(\omega_1^2+\omega_2^2)y^{(3)}
+\omega_1^2\omega_2^2y'=0.
\tag{BB.4}
\]

Equations BB.2--BB.4 are written mathematics. If a measured signal requires
three independent frequencies, drift, damping, harmonics, or mode coupling,
then this exact model class is incomplete. Adding arbitrary hidden variables
afterward would not rescue its original prediction.

### BB.2. A product taxonomy rather than one “harmony” label

The engine uses six independent classification axes:

1. **Spectrum:** static, single-tone, equal-frequency, commensurate within a
   declared rational search, or no small rational match.
2. **Instantaneous coupling:** rank zero, one, or two for the visible-hidden
   block \(B=\Omega_{1:3,4:5}\).
3. **Known-generator observability:** rank of
   \(\mathcal O=[P;P\Omega;\ldots;P\Omega^4]\).
4. **One-history order:** rank of a finite block-Hankel matrix built from one
   noiseless 3D trajectory.
5. **Projected geometry:** dimension of the centered 3D trajectory span.
6. **Observer response:** exposure zeros, sampling aliases, and gauge
   collisions.

The known-system rank criterion follows the observability framework introduced
by [Kalman (1960)](https://doi.org/10.1016/S1474-6670(17)70094-8). The history
rank is motivated by output-only modal realization, especially the
eigensystem realization method of
[Juang and Pappa (1985)](https://ntrs.nasa.gov/citations/19850022899). Release
1.0 computes only finite noiseless ranks; it does not claim a noise-robust
physical system identification.

| Reference class | coupling rank | known \(\mathcal O\) rank | one-history rank | dynamic 3D span |
|---|---:|---:|---:|---:|
| static | 0 | 3 | 1 | 0 |
| silent hidden plane | 0 | 3 | 3 | 2 |
| rank-one, hidden unmixed | 1 | 4 | 4 | 3 |
| rank-one, hidden mixing | 1 | 5 | 5 | 3 |
| rank-two generic | 2 | 5 | 5 | 3 |
| commensurate 3:2 | 2 | 5 | 5 | 3 |
| equal frequency | 2 | 5 | 3 | 2 |
| exact \(1:\sqrt2\) reference | 2 | 5 | 5 | 3 |

Two counterexamples are decisive. The two rank-one cases have different
observability because hidden-hidden mixing carries the second hidden coordinate
through the same visible gate. Coupling rank alone is not a complete harmony
class. Conversely, the equal-frequency system is fully observable when its
generator is known, yet one autonomous history has rank three: a stationary
term plus only one temporal sine/cosine pair. Known-system state observability
does not imply identification of an unknown generator or physical dimension.

### BB.3. What a finite observer does

An instantaneous observation already loses a two-dimensional kernel because
\(P\) has rank three. Dynamics may reveal some or all of those coordinates over
time if \(\Omega\) is known. A camera, eye, or detector also integrates over a
nonzero exposure. For a centered boxcar exposure of length \(\tau\), each
modal amplitude is multiplied by

\[
g_\tau(\omega)=\operatorname{sinc}(\omega\tau/2).
\tag{BB.5}
\]

Thus \(\tau=2\pi/\omega\) erases that mode exactly in the declared noiseless
model. For the exact \((1,\sqrt2)\) reference, \(\tau=2\pi\) lowers both
one-history and effective observability rank from five to three, while the
second mode retains amplitude 0.2169542944. For the commensurate 3:2 reference,
the common-period exposure \(\tau=4\pi\) erases both rotations and lowers both
ranks to one.

This is information loss, not physical collapse. The current observer has no
back-action term: it changes \(y\), not \(z\). A quantum measurement model would
require a state space, interaction, measurement operators, outcome rule, and
validated probabilities not supplied by Equation BB.1.

Fixed ticks add a different collision. In a sampled rotation plane,

\[
\omega' = \omega+\frac{2\pi k}{\Delta}
\tag{BB.6}
\]

gives the same integer-tick positions. This retains revision 1.15's warning:
integer samples do not uniquely identify continuous behavior.

### BB.4. Hidden-basis gauge and the inverse boundary

Let

\[
G_H=\operatorname{diag}(I_3,H),\qquad H\in O(2).
\tag{BB.7}
\]

Because \(PG_H=P\), the transformed pair

\[
\Omega'=G_H\Omega G_H^T,\qquad z_0'=G_Hz_0
\tag{BB.8}
\]

obeys

\[
Pe^{t\Omega'}z_0'=Pe^{t\Omega}z_0
\quad\text{for every }t.
\tag{BB.9}
\]

Release 1.0 executes 101 hidden-plane angles and 64 times per angle. The
largest Frobenius displacement between transformed and baseline generators is
4.0735906631, while all 6,464 projected states agree within
\(5.17\times10^{-14}\). The orientation of the hidden axes is therefore a
coordinate gauge under this observer. A unique physical interpretation cannot
be selected from those outputs.

### BB.5. Finite validation and the next falsification

Four frozen spectral families receive 128 independently rotated generators
each. With the matrix supplied, the classifier recovers all 512 bounded labels
and both canonical frequencies with maximum error
\(1.11\times10^{-15}\). “No small rational match” means only that no ratio with
denominator at most 16 meets the \(10^{-9}\) tolerance. It does not prove an
arbitrary measured ratio irrational. The \(\sqrt2\) reference is known to be
irrational because it was constructed symbolically that way, not because the
finite search discovered irrationality.

The evidence types are:

1. Equations BB.2--BB.9 are written conditional mathematics.
2. The eight archetypes, 512 matrix-supplied classifications, finite ranks,
   6,464 gauge-time states, and exposure tests are finite synthetic computation.
3. Kalman observability and ERA supply primary-method provenance.
4. No physical five-dimensional trajectory, molecular spectrum, or quantum
   measurement was observed.
5. A physical fifth dimension remains a hypothesis and is not identified by a
   compatible linear realization.
6. Theological interpretation remains outside this observation map.

All 114 package tests pass. A separate implementation performs 48 catalog
checks, 2,048 randomized checks, all 6,464 gauge-time comparisons, two exposure
erasures, and 81 alias comparisons without importing the production module.

The next test must remove the supplied-generator advantage. Freeze a
two-tone/noise inference protocol using only 3D trajectories, then score it on
held-out missing-frame, noisy, drifting, nonlinear, and deliberately
three-tone controls. It must return “outside model class” rather than assigning
every unexplained residual to a fifth-dimensional particle.

Revision 1.14's complete deterministic closure remains idempotent. Revision
1.15's integer-tick non-crossing limitation, angular-impulse formula, and
finite-prefix generator non-identifiability remain unchanged. The recurrence
learner's \(2K\)-sample continuation guarantee still assumes a noiseless
homogeneous constant-coefficient linear recurrence of order at most \(K\) from
the initial state. This appendix establishes neither seven physical dimensions
nor \(P=NP\), and it assigns no residual to God or another theological cause.

## Appendix BC — Hydrogen spectroscopy gate for the two-harmony observer (revision 1.54)

Appendix BB proves that a fixed real skew-symmetric five-coordinate generator
has at most two nonnegative rotation frequencies. Under a linear scalar or
vector observation, the positive-frequency support therefore has cardinality
at most

\[
N_+(d)=\lfloor d/2\rfloor,
\qquad N_+(5)=2.
\tag{BC.1}
\]

The NIST H I strong-line table resolves four visible Balmer features. This
revision freezes one representative strongest listed component for each:

| Feature | Transition | Air wavelength (nm) | Listed relative intensity |
|---|---:|---:|---:|
| H-delta | 6 to 2 | 410.17400 | 15 |
| H-gamma | 5 to 2 | 434.04620 | 30 |
| H-beta | 4 to 2 | 486.13615 | 60 |
| H-alpha | 3 to 2 | 656.28518 | 180 |

The table is reference evidence, not a new measurement. Because these are
four distinct resolved features, the current fixed linear five-coordinate
observer fails the necessary cardinality gate

\[
4\le N_+(5)=2.
\tag{BC.2}
\]

The conclusion does not depend on an optimizer. Two free frequencies can fit
any selected pair exactly and necessarily leave two features unexplained.
For the visualized restricted ratios, a one-scale calibration searches both
ways of anchoring either mode to every measured line. At a declared 1%
relative wavelength tolerance, the 3:2 family can match two features, while
the tested locked and \(1:\sqrt2\) families cover fewer. Those are finite
calibrated comparisons, not predictions or evidence for a 3:2 mechanism.

A centered exposure multiplies existing mode amplitudes by
\(\operatorname{sinc}(\omega\tau/2)\). It cannot create the missing spectral
lines. A model that attempts the full spectrum must therefore declare a new
mechanism, such as validated energy-state transitions, a nonlinear observation
with explicit combination frequencies, a time-varying/interacting generator,
or a larger linear state. Each alternative changes the model class and must
be frozen before new validation.

Equations BC.1--BC.2 are written linear algebra. The wavelength comparison is
a finite calculation using external reference data. No time-resolved 5D
motion was measured, and the result proves neither a physical fifth dimension
nor its absence. It specifically rejects the current two-frequency linear
observer as a complete generator of the four-feature Balmer spectrum.

## Appendix BD — Multidimensional vector observers (revision 1.55)

Let the declared latent state be (x(t)\in\mathbb R^d) with

\[
\dot x(t)=A x(t),\qquad A^T=-A.
\tag{BD.1}
\]

For each calibrated three-dimensional observer (j), the engine reports both
projected position and projected tangent velocity,

\[
y_j(t)=P_jx(t),\qquad \dot y_j(t)=P_jAx(t),
\quad P_j\in\mathbb R^{3\times d}.
\tag{BD.2}
\]

These equations make the proposed “string movement” visible as vectors without
claiming that the vectors are measured physical strings. The observer is
passive; no back-action term is present.

### BD.1. What one observer cannot see

Rank-nullity gives

\[
\dim\ker P_j=d-\operatorname{rank}(P_j)\ge d-3.
\tag{BD.3}
\]

Therefore any (h\in\ker P_j) generates an exact instantaneous collision,

\[
P_j(x+h)=P_jx.
\tag{BD.4}
\]

The executable constructs a five-dimensional pair separated by one latent
unit with zero projected separation under its first observer. A single 3D
view therefore cannot identify a generic 4D or 5D state.

### BD.2. When several observers recover one common state

For simultaneous calibrated observations, stack the projectors,

\[
M=\begin{bmatrix}P_1\\P_2\\\vdots\\P_k\end{bmatrix},
\qquad Y=Mx.
\tag{BD.5}
\]

The common state is uniquely identifiable exactly when

\[
\operatorname{rank}(M)=d.
\tag{BD.6}
\]

This is an if-and-only-if statement: full column rank supplies the left
pseudoinverse; deficient rank supplies a nonzero collision vector in
(ker M). The frozen coordinate banks have full rank and reconstruct the
synthetic states in dimensions 4, 5, 6, and 8 to numerical precision. The
result assumes known matrices, aligned coordinates, simultaneity, and noiseless
data. It does not license stacking uncalibrated or asynchronous measurements.

### BD.3. Observers do not manufacture harmonies

Every real skew-symmetric (d\times d) matrix is orthogonally equivalent to
at most (lfloor d/2\rfloor) planar rotation blocks and, for odd (d), at
least one zero direction. Consequently

\[
N_+(d)=\lfloor d/2\rfloor.
\tag{BD.7}
\]

Changing (P_j) can reveal, attenuate, erase, or mix those modes in the output,
but cannot change the eigenfrequencies of (A). The tested capacity table is:

| Latent dimension | Positive-mode capacity | Single-view nullity | Four-feature count gate |
|---:|---:|---:|---|
| 4 | 2 | 1 | fail |
| 5 | 2 | 2 | fail |
| 6 | 3 | 3 | fail |
| 7 | 3 | 4 | fail |
| 8 | 4 | 5 | pass count only |
| 9 | 4 | 6 | pass count only |

The passage at dimension eight is not a hydrogen prediction. Four free modes
can merely provide four adjustable line locations. A physical model must also
derive wavelengths, intensities, degeneracies, selection rules, and state
transitions with frozen parameters and held-out tests. Dimension must not be
selected after seeing the number of target features.

### BD.4. Evidence boundary

Equations BD.3, BD.6, and BD.7 are written finite-dimensional linear algebra.
The reconstruction errors, collision, trajectories, and capacity table are
finite synthetic computations. The NIST Balmer count remains external
reference evidence; no new spectroscopy was performed. No physical 4D, 5D,
8D, or 9D vector was observed, and no residual is assigned to a string
particle, God, or another theological cause.

Revision 1.14's complete deterministic closure remains idempotent. Revision
1.15's integer-tick non-crossing limitation, angular-impulse formula, and
finite-prefix generator non-identifiability remain unchanged. The recurrence
learner's (2K)-sample continuation guarantee still assumes a noiseless
homogeneous constant-coefficient linear recurrence of order at most (K)
from the initial state. Nothing here proves seven physical dimensions or
(P=NP).

## Appendix BE — Hydrogen transition holdout (revision 1.56)

The request to prove quantum physics or physical dimensions is replaced by a
falsifiable requirement: a proposed law must predict measurements that did not
set its parameters. Quantum mechanics already has extensive experimental
support; one retrospective spectral fit cannot prove the whole theory, and an
extra coordinate cannot be inferred merely because a flexible term reduces a
residual.

### BE.1. Frozen data split and conversion

The NIST H I strong-line table reports the visible features used here as air
wavelengths. Before fitting, release 1.3.0 solves

\[
\lambda_{\rm vac}=n_{\rm air}(\lambda_{\rm vac})\lambda_{\rm air}
\tag{BE.1}
\]

with the Peck--Reeder standard-dry-air relation at its stated reference
conditions. The converted wavenumbers for H-alpha, H-beta, and H-gamma agree
with independent differences in the NIST persistent-line energy table within
0.0011 cm^-1. This cross-check validates the conversion implementation; it is
not a new measurement.

The protocol was fixed as follows:

| Role | Features |
|---|---|
| calibration | H-alpha (3 to 2), H-beta (4 to 2) |
| untouched primary holdout | H-gamma (5 to 2), H-delta (6 to 2) |
| secondary extrapolation | NIST listed-intensity-weighted Lyman-alpha centroid |

The displayed-decimal half-step is retained as a reporting-precision
sensitivity check. It is not promoted to a complete spectroscopic uncertainty
model.

### BE.2. Two bounded laws

For upper and lower principal indices (n_u>n_l), define

\[
\Delta_k(n_u,n_l)=n_l^{-k}-n_u^{-k}.
\tag{BE.2}
\]

The one-parameter baseline and two-parameter candidate are

\[
M_0:\quad \widetilde\nu=R\Delta_2,
\qquad
M_1:\quad \widetilde\nu=R\Delta_2+C\Delta_4.
\tag{BE.3}
\]

(M_0) is a Rydberg-form baseline, fitted by least squares on the two calibration
lines. (M_1) uses its two parameters to interpolate those lines exactly. The
(C\Delta_4) term is deliberately labeled phenomenological. It has not been
derived from an extra-dimensional Hamiltonian, action, field equation,
coupling to charge or current, gauge symmetry, or conservation law. Similar
principal-index dependence can arise from ordinary precision corrections, so
even a successful fit would not uniquely identify dimensionality.

### BE.3. Held-out result

The primary score is vacuum-wavelength root-mean-square error in parts per
million on the two untouched Balmer features:

| Model | Calibration RMSE (ppm) | Holdout RMSE (ppm) | Worst holdout (ppm) |
|---|---:|---:|---:|
| (M_0) Rydberg form | 0.522008 | 6.030008 | 6.909563 |
| (M_1) plus (Delta_4) | approximately 0 | 6.948567 | 7.728623 |

The added term worsens the frozen score by 15.2331%. It is also worse on each
holdout feature at both endpoints generated by the source table's displayed
wavelength precision. On the separate Lyman-alpha extrapolation, (M_0) has
signed error (+7.7269) ppm and (M_1) has (-11.2483) ppm. Therefore (M_1) is
rejected as an improvement. The exact fitted coefficient and its failure are
preserved rather than reinterpreted as a hidden variable.

This comparison does not establish that the fitted (M_0) is complete quantum
electrodynamics. Appendix AQ already records omitted recoil, radiative,
finite-size, hyperfine, linewidth, and external-field contributions. The few
ppm residuals remain a budget for model and reference-feature mismatch, not a
unique signal from another dimension.

### BE.4. Exact finite-prefix counterexample

Let (x(n)=\Delta_2(n,2)) and define

\[
q(n)=[x(n)-x(3)][x(n)-x(4)].
\tag{BE.4}
\]

For every real (D), the family

\[
\widetilde\nu_D(n)=R\Delta_2+C\Delta_4+Dq(n)
\tag{BE.5}
\]

has exactly the same H-alpha and H-beta values because (q(3)=q(4)=0). Yet
(q(5)) and (q(6)) are nonzero, so its continuation changes with (D). The
executed coefficient (D=111.2553101) cm^-1 forces H-gamma exactly but leaves a
(-7.1389) ppm wavelength error at H-delta.

This is a written non-identifiability result, not merely an optimizer failure.
It retains revision 1.15's broader correction: a finite observed prefix does
not identify an arbitrary generator. More calibration values can constrain a
declared finite model class; they cannot select an unrestricted continuation.

### BE.5. Evidence boundary and next gate

1. Equations BE.2--BE.5 and the invisible-term construction are written
   mathematics.
2. The standard-air conversion, parameter estimates, eight Balmer predictions,
   rounding endpoints, and two Lyman extrapolations are finite computation.
3. The NIST wavelengths and energy levels are externally compiled measured
   reference evidence; no new spectrum was recorded.
4. The (Delta_4) candidate is a rejected phenomenological hypothesis, not a
   physical extra dimension.
5. Theological interpretation is not inside the observation model and no
   residual is identified with God.

All 143 package tests pass. A separate 50-digit Decimal implementation imports
no production transition functions and reproduces all eight Balmer prediction
errors, the model decision, and the training-invisible coefficient.

The next admissible dimensional test must derive one explicit operator from a
declared higher-dimensional dynamics, include the standard precision terms or
bound their contribution, freeze its coupling on one transition family, and
predict a different state-dependent pattern before reading those targets. A
candidate that merely adds adjustable powers of (1/n) does not meet that gate.

Revision 1.14's complete deterministic closure remains idempotent. Revision
1.15's integer-tick non-crossing limitation, angular-impulse formula, and
finite-prefix generator non-identifiability remain unchanged. The recurrence
learner's (2K)-sample continuation guarantee still assumes a noiseless
homogeneous constant-coefficient linear recurrence of order at most (K) from
the initial state. Nothing in this appendix proves seven physical dimensions,
(P=NP), or a universal grid and tick.

## Appendix BF — Hydrogen isotope correction ladder (revision 1.57)

Revision 1.56 rejected an arbitrary \(\Delta_4\) fit but left the next valid
question open: can a concrete finite-range operator explain an isotope-sensitive
residual after ordinary precision physics is included? This appendix answers
that bounded question for the direct deuterium-minus-hydrogen 1S--2S shift.

### BF.1. Frozen measured target and ordinary reference

Parthey et al. report the direct optical result

\[
\Delta f_{D-H}^{\rm obs}=670\,994\,334.606(15)\ {\rm kHz}.
\tag{BF.1}
\]

The updated pionless-EFT/QED compilation of Lensky, Hagelstein, and
Pascalutsa gives, with its very small contact term fixed to zero,

\[
\Delta f_{D-H}^{\rm std}=
671\,000\,534.811(924)
-1369.346\left(\frac{r_d}{\rm fm}\right)^2\ {\rm kHz}.
\tag{BF.2}
\]

The contact coefficient is not fitted here because that work reports its
effect on this observable as negligible at the present uncertainty. Inserting
their muonic-deuterium extraction \(r_d=2.12763(78)\) fm gives

\[
\Delta f_{D-H}^{\rm std}=670\,994\,336.042632\ {\rm kHz}.
\tag{BF.3}
\]

The signed error is \(+1.436632\) kHz. Combining the 0.924 kHz theory term,
the radius propagation, and the 0.015 kHz measurement uncertainty in
quadrature gives 4.637998 kHz, so the error is only \(+0.309753\) standard
deviations. This is not a perfectly blind comparison: the muonic-radius
extraction has a weak shared-theory dependence through a low-energy constant
informed by the isotope shift, which the source reports as negligible at this
precision. It is a cross-system consistency check rather than an
assumption-free proof.

For comparison, the nonrelativistic reduced-mass-only expression

\[
\Delta f_{\mu}=\frac34 cR_\infty
\left(\frac{\mu_D}{m_e}-\frac{\mu_H}{m_e}\right)
\tag{BF.4}
\]

evaluated with the frozen 2022 CODATA constants gives
\(671007030.506398\) kHz, an error of \(+12695.900398\) kHz. The modern
ordinary ladder reduces the absolute error by a factor of 8837.3. A residual
left by the deliberately incomplete rung is therefore not evidence for a
missing dimension.

### BF.2. Explicit finite-range proxy

The tested operator is

\[
\frac{\delta V(r)}h=g\frac{e^{-r/\lambda}}r,
\qquad \lambda>0,
\tag{BF.5}
\]

where \(g\) has units Hz m. This Yukawa-like form is an explicit radial
operator, but it is not derived from a fifth-dimensional action,
compactification, gauge symmetry, charge assignment, or conserved current.
It is a phenomenological proxy and may not be relabeled as a physical
dimension even if it fits.

For a hydrogenic \(nS\) state with reduced-mass Bohr radius \(a\), define

\[
Y_n(a,\lambda)=
\left\langle nS\left|\frac{e^{-r/\lambda}}r\right|nS\right\rangle .
\tag{BF.6}
\]

The executable expands \(L_{n-1}^{1}(2r/na)\) and evaluates BF.6 as a finite
sum of gamma-function moments; no spatial grid or numerical eigenstate fit is
used. The isotope sensitivity of the 1S--\(nS\) transition is

\[
S_n(\lambda)=
[Y_n(a_D,\lambda)-Y_1(a_D,\lambda)]
-[Y_n(a_H,\lambda)-Y_1(a_H,\lambda)].
\tag{BF.7}
\]

### BF.3. Exact range/coupling non-identifiability

For every positive range at which \(S_2(\lambda)\ne0\), choosing

\[
g(\lambda)=
\frac{\Delta f_{D-H}^{\rm obs}-\Delta f_{D-H}^{\rm std}}
{S_2(\lambda)}
\tag{BF.8}
\]

fits the same central residual exactly. Equation BF.8 constructs a continuum
of collisions: one scalar datum supplies one constraint on two continuous
parameters.

The finite scan checks 61 logarithmic ranges from 1 fm through 10 nm. Every
pair reproduces the \(-1436.632276\) Hz central residual to machine precision.
Yet the 1 fm and 10 nm pairs give prospective 1S--3S isotope corrections of
\(-1581.055626\) and \(-1702.839834\) Hz, respectively, and their
unrecalibrated absolute hydrogen 1S--2S shifts differ by 3.519441 MHz. Thus
the calibration datum cannot identify the interaction range, coupling, or
later state dependence.

### BF.4. Decision, limitations, and next test

The frozen rule required a standard-model residual beyond three combined
standard deviations before an extra operator could be considered necessary.
The observed 0.309753-standard-deviation discrepancy fails that gate. The
operator is not accepted, and its exact-fit family is retained as a negative
result rather than tuned into an ontological claim.

Equations BF.5--BF.8 and the collision family are written mathematics. The
correction ladder, uncertainty propagation, analytic moments, and 61-point
scan are finite computation. Equations BF.1--BF.3 use published measured and
theory inputs; no new spectroscopy was performed. The 1S--3S values are
prospective discriminants, not measured validation. A future gate must freeze
two independently measured isotope-sensitive transitions and the full
ordinary-theory covariance, use one to set a coupling at a preregistered
range, and score the other without refitting.

Revision 1.14's complete deterministic closure remains idempotent. Revision
1.15's integer-tick non-crossing limitation, angular-impulse formula, and
finite-prefix generator non-identifiability remain unchanged. The recurrence
learner's \(2K\)-sample continuation guarantee still assumes a noiseless
homogeneous constant-coefficient linear recurrence of order at most \(K\)
from the initial state. Nothing in this appendix proves seven physical
dimensions, \(P=NP\), a universal physical grid/tick, a string particle, or a
theological cause.

## Appendix BG — Frozen hydrogen-isotope holdout (revision 1.58)

Revision 1.57 left a continuum of finite-range central operators that all fit
one H--D 1S--2S central residual. This appendix freezes one member and asks a
strictly smaller question: does it improve a different isotope-sensitive
transition without refitting?

### BG.1. Frozen operator and historical target

The range is fixed at the electron reduced Compton wavelength,

\[
\lambda=3.8615926796\times10^{-13}\ {\rm m},
\tag{BG.1}
\]

before scoring the target. This is a named atomic scale, not a value derived
from a fifth-dimensional action. The coupling is set only by BF.8 and remains
fixed. For isotope sensitivity \(S_n(\lambda)\) defined by BF.7, the predicted
H--D 2S--8S correction is

\[
\delta f_{2S-8S}=g\,[S_8(\lambda)-S_2(\lambda)]
=-201.996425\ {\rm Hz}.
\tag{BG.2}
\]

The separately published transition frequencies are

\[
f_H=770649350012.0(8.6)\ {\rm kHz},\qquad
f_D=770859041245.7(6.9)\ {\rm kHz},
\tag{BG.3}
\]

so their printed central difference is
\(209691233.7\) kHz. CODATA 2010 reports experimental correlation
\(\rho_{HD}=0.1225\). Consequently,

\[
\sigma_{\rm exp}
=\sqrt{6.9^2+8.6^2-2(0.1225)(6.9)(8.6)}
=10.345613\ {\rm kHz}.
\tag{BG.4}
\]

These values have been public since 1997. The comparison is a historical
holdout, not a blind prospective experiment.

### BG.2. Ordinary reference and correlated uncertainty

For an \(nS\) level, the published parameterization has the form

\[
\frac{E_{nS}}h=-\frac{cR_\infty}{n^2}\frac{m_r}{m_e}
+\frac{E_{\rm NS}}{n^3}+\Delta(nS).
\tag{BG.5}
\]

Using the tabulated H/D finite-size coefficients and additive remainders gives
the following D-minus-H 2S--8S ladder:

\[
209689697.033249
-736.037122
+2265.440000
=209691226.436128\ {\rm kHz}.
\tag{BG.6}
\]

The first term is reduced mass, the second finite nuclear size, and the third
the difference of the four additive theory remainders. The CODATA covariance
is propagated with coefficient vector \((1,-1,-1,1)\) in the order H2S, H8S,
D2S, D8S. Its standard uncertainty is 0.068444 kHz. External radius
uncertainties contribute 0.672174 kHz. Combining these with BG.4 gives

\[
\sigma_{\rm combined}=10.367652\ {\rm kHz}.
\tag{BG.7}
\]

Uncertainties of the frozen Rydberg and mass-ratio inputs contribute less than
0.01 kHz and are bounded rather than displayed in BG.7. The calculation is a
published precision approximation, not a new complete bound-state-QED
calculation or an independent remeasurement.

### BG.3. Decision and resolution limit

The ordinary prediction has signed error

\[
z_{\rm ordinary}=-0.700628.
\tag{BG.8}
\]

Adding BG.2 produces \(209691226.234131\) kHz and

\[
z_{\rm operator}=-0.720112.
\tag{BG.9}
\]

The frozen interaction therefore makes the central agreement slightly worse
and earns no predictive credit under the predeclared improvement rule. It is
not excluded: its full effect is only 0.019483 combined standard deviations,
and the uncertainty is 51.326 times larger. Non-rejection is ambiguity, not
confirmation.

Equations BG.1--BG.9 are written conditional mathematics. Their numerical
evaluation is finite computation. BG.3 uses published measured inputs; no new
spectroscopy was performed. An independent executable imports neither isotope
module, evaluates the radial integrals by direct quadrature, and reproduces
the correction and covariance.

The next gate must first rank unscored transitions by the absolute frozen
operator contrast divided by full correlated uncertainty. A target should be
frozen and scored only if it has material power to distinguish the models.
Failure, success, or an unresolved residual cannot by itself identify a
physical dimension, universal grid/tick, string particle, or theological
cause. The v1.14 closure correction, all v1.15 corrections, and the recurrence
learner's noiseless homogeneous constant-coefficient order-at-most-K
initial-state assumption remain unchanged.

## Appendix BH — Conserved quantum flow and observable harmonics (revision 1.59)

The hydrogen animation already used Coulomb Schrödinger eigenstates and their
relative phase. It did not yet expose the conserved spatial current or ask
which interference harmonics a declared measurement can actually see. This
appendix supplies that missing bridge without adding an extra coordinate.

### BH.1. Electromagnetic evolution and continuity

For a particle of charge \(q\), scalar potential \(\phi\), and vector potential
\(\mathbf A\), minimal coupling gives

\[
i\hbar\frac{\partial\Psi}{\partial t}=
\left[\frac{(-i\hbar\nabla-q\mathbf A)^2}{2m}+q\phi\right]\Psi.
\tag{BH.1}
\]

The conserved position probability obeys

\[
\frac{\partial\rho}{\partial t}+\nabla\!\cdot\mathbf j=0,
\qquad
\mathbf j=\frac{\hbar}{2mi}
(\Psi^*\nabla\Psi-\Psi\nabla\Psi^*)-
\frac{q}{m}\mathbf A|\Psi|^2.
\tag{BH.2}
\]

The previous fixed-proton engine already contained the electromagnetic
Coulomb scalar potential. Release 0.6 implements the field-free
\(\mathbf A=0\) specialization of BH.2; it does not yet simulate a driven
external field.

For the equal coherent packet

\[
\Psi(\mathbf r,t)=\frac{1}{\sqrt2}
\left[\phi_{1s}(\mathbf r)e^{-iE_1t/\hbar}
+\phi_{2p_z}(\mathbf r)e^{-iE_2t/\hbar}\right]
\tag{BH.3}
\]

with real spatial eigenfunctions and
\(\theta=(E_2-E_1)t/\hbar\), the current in atomic units is

\[
\mathbf j=\frac{\sin\theta}{2}
(\phi_{2p_z}\nabla\phi_{1s}-\phi_{1s}\nabla\phi_{2p_z}).
\tag{BH.4}
\]

Taking the divergence and substituting the two Coulomb eigenvalue equations
gives

\[
\nabla\!\cdot\mathbf j=(E_2-E_1)\phi_{1s}\phi_{2p_z}\sin\theta
=-\frac{\partial\rho}{\partial t},
\tag{BH.5}
\]

so the continuity equation holds identically. Equation BH.5 is a written
mathematical result. The arrows rendered by the executable are probability
flow; they are not trajectories of a localized electron.

### BH.2. Dipole harmonic and ideal rate

Light does not observe the density alone. In the electric-dipole
approximation the relevant observable is

\[
\mathbf d(t)=-e\langle\Psi(t)|\mathbf r|\Psi(t)\rangle.
\tag{BH.6}
\]

Direct integration gives

\[
\langle1s|z|2p_z\rangle
=\frac{256}{243\sqrt2}a_0
=0.744935539028a_0.
\tag{BH.7}
\]

Consequently the equal packet has an oscillating \(z\) dipole of amplitude
\(6.3158269\times10^{-30}\) C m, or \(1.8934378\) D. The ideal fixed-proton
electric-dipole formula gives

\[
A_{2p_z\rightarrow1s}
=\frac{\omega^3|e\langle1s|z|2p_z\rangle|^2}
{3\pi\epsilon_0\hbar c^3}
=6.26831503\times10^8\ {\rm s}^{-1},
\tag{BH.8}
\]

corresponding to \(1.59532505\) ns. This is an ideal nonrelativistic E1
calculation, not a precision bound-state-QED lifetime or a new measurement.

The production \(81^3\) grid reproduces BH.7 as \(0.744932784051a_0\), a
relative error \(3.70\times10^{-6}\). Its volume-integrated current matches
the derivative of the dipole with relative error \(5.35\times10^{-4}\). A
separate implementation imports none of the production module, evaluates
BH.7 by adaptive radial quadrature, and differentiates the current on a
\(129^3\) Cartesian grid. Outside the declared \(r<0.5a_0\) nuclear-cusp
region its fourth-order continuity residual is \(0.0128041\) in relative
\(L^2\). The full-interior value \(0.116031\) is retained because a Cartesian
stencil converges slowly across the Coulomb cusp; it is not relabeled as a
successful uniform numerical derivative.

### BH.3. Selection rules filter raw harmonics

An equal `1s + 2p_z + 3s` packet contains three pairwise density frequencies:

| Pair | Gap (Hartree) | Ideal wavelength (nm) | Density | \(z\)-dipole |
|---|---:|---:|---|---|
| \(2p_z\leftrightarrow3s\) | \(5/72\) | 656.112276 | present | present |
| \(1s\leftrightarrow2p_z\) | \(3/8\) | 121.502273 | present | present |
| \(1s\leftrightarrow3s\) | \(4/9\) | 102.517543 | present | absent |

The last matrix element vanishes because both \(s\) states have even parity.
The separate `1s + 2s` control likewise has a changing local density but
\(\langle1s|z|2s\rangle=0\). Thus a list of energy differences overpredicts
the optical signal unless it is filtered through the measurement operator.
The standard E1 angular rule is \(\Delta l=\pm1\), with
\(\Delta m=0\) for \(z\) polarization. Other multipoles, external-field
mixing, and two-photon processes are different declared measurement models;
E1 silence does not mean that a state or density beat is nonexistent.

This operator gate is also a correction to the framework's use of the word
“harmony.” A harmonic belongs jointly to a prepared state, Hamiltonian, and
observable. It is not an ontological object inferred from visual rhythm
alone.

### BH.4. Evidence classes, decision, and next gate

Equations BH.1--BH.8 and the parity cancellation are written standard quantum
mechanics. The voxel fields, matrix-element checks, ideal rate, and 42 passing
tests are finite deterministic computation. The independent audit passes all
seven frozen checks. No new measured hydrogen data were collected, and the
rendered slices are model output rather than a camera image of an electron.
Attosecond wave-packet experiments demonstrate that prepared electronic motion
can be probed, but they do not validate this particular computed hydrogen
packet or any extra-dimensional generator.

The bounded decision is therefore: Schrödinger evolution was not forgotten;
the explicit continuity/current equation and observable-selection gate were.
Adding them yields two dipole-visible harmonics from three density harmonics in
the declared three-state control. It supplies no evidence for a fourth or
fifth spatial dimension, a universal grid/tick, a hidden localized orbit, a
string particle, or a theological cause.

The next gate is a frozen weak-pulse time-dependent Schrödinger calculation
that implements BH.1 in both length and velocity gauges. It must require
gauge agreement, predict the allowed one-photon `1s -> 2p` response and the
suppressed `1s -> 2s` control before consulting compiled transition
probabilities or lifetimes.

Method context: NIST relates spectral-line strengths to the appropriate
multipole transition matrix element
(https://physics.nist.gov/Pubs/AtSpec/node17.html). Kübel et al. report a
primary experiment imaging prepared ultrafast valence-electron wave-packet
motion (https://arxiv.org/abs/1903.03739), while Rossi et al. give a 2026
primary method for polarization-controlled attosecond preparation and
difference-density analysis (https://arxiv.org/abs/2606.22102). These sources
motivate the observable gate; they do not validate this engine's hydrogen
packet or any extra-dimensional interpretation.

Revision 1.14's complete deterministic closure remains idempotent. Revision
1.15's integer-tick non-crossing limitation, angular-impulse formula, and
finite-prefix generator non-identifiability remain unchanged. The recurrence
learner's \(2K\)-sample continuation guarantee still assumes a noiseless
homogeneous constant-coefficient linear recurrence of order at most \(K\)
from the initial state.

## Appendix BI — Golden-ratio exactness and coincidence audit (revision 1.60)

The phrase “golden alignment” was previously undefined in the framework. This
appendix gives it an exact algebraic gate, audits the current hydrogen and
fixed-grid models, and separates derivation from pattern matching. It does not
insert a golden parameter into the dynamics.

### BI.1. Canonical golden quantities

Let

\[
\varphi=\frac{1+\sqrt5}{2}.
\tag{BI.1}
\]

Then

\[
\varphi^2=\varphi+1,\qquad
\varphi^{-1}=\varphi-1,\qquad
1-\varphi=-\varphi^{-1}.
\tag{BI.2}
\]

Every integer power is exhausted by

\[
\varphi^n=F_n\varphi+F_{n-1},
\tag{BI.3}
\]

where \(F_n\) is the Fibonacci sequence. The golden-turn fraction and the
usual smaller golden angle are

\[
\alpha_g=\varphi^{-2}=2-\varphi=\frac{3-\sqrt5}{2},
\qquad
\theta_g=2\pi\alpha_g=137.507764050\ldots{}^\circ.
\tag{BI.4}
\]

The complementary directed rotation is
\(2\pi-\theta_g=222.492235950\ldots{}^\circ\). These are the same unoriented
placement rule traversed in opposite directions, not two new constants.

For a positive dimensionless candidate \(x\), exact golden-ratio alignment
means

\[
x^2-x-1=0,
\tag{BI.5}
\]

which selects \(x=\varphi\). A small decimal residual in BI.5 is only an
approximation until the declared generator implies the polynomial. A useful
reported diagnostic is

\[
\epsilon_\varphi(x)=\frac{|x^2-x-1|}{1+x^2},
\tag{BI.6}
\]

but no threshold in finite precision can prove irrational equality.

### BI.2. Where \(\varphi\) is exact

The Fibonacci companion matrix

\[
Q=\begin{pmatrix}1&1\\1&0\end{pmatrix}
\tag{BI.7}
\]

has characteristic polynomial \(\lambda^2-\lambda-1\), eigenvalues
\(\varphi\) and \(-1/\varphi\), and

\[
Q^n=\begin{pmatrix}F_{n+1}&F_n\\F_n&F_{n-1}\end{pmatrix}.
\tag{BI.8}
\]

This is an intrinsic algebraic manifestation. It is relevant to a declared
Fibonacci recurrence in an integer or real recurrence engine. It does not
silently extend the v1.15 continuation guarantee: that guarantee remains
conditional on a noiseless homogeneous constant-coefficient linear recurrence
of order at most \(K\), from the initial state, observed for at least \(2K\)
samples.

Standard icosahedral coordinates are the permutations of
\((0,\pm1,\pm\varphi)\). Their edge and next-separation ratio is exactly
\(\varphi\). This is exact geometry after choosing an icosahedron; it is not
evidence that an unrelated atom selected those coordinates. Likewise, a
golden-angle point cloud is construction-imposed when Equation BI.4 is passed
to the sampler.

The other common names close under the same algebra rather than introducing
new constants. A golden rectangle has side ratio \(\varphi\) by definition. A
regular pentagon has diagonal-to-side ratio
\(2\cos(\pi/5)=\varphi\), derived from its symmetry. A logarithmic spiral
\(r(\theta)=ae^{b\theta}\) is a golden spiral only after choosing

\[
b=\frac{2\log\varphi}{\pi},
\]

which makes every quarter turn multiply the radius by \(\varphi\). Thus the
pentagon relation is derived from declared geometry, whereas the rectangle and
spiral labels impose the golden scale in their definitions.

In the integrable \(E_8\) spectrum, the first two masses obey

\[
\frac{m_2}{m_1}=2\cos\frac{\pi}{5}=\varphi.
\tag{BI.9}
\]

Coldea et al. reported neutron-scattering evidence consistent with that
prediction in the near-critical Ising-chain material
\(\mathrm{CoNb_2O_6}\). This is a measured quantum manifestation tied to an
emergent \(E_8\) model, not a universal feature of all quantum systems.

### BI.3. Exact exclusion from the ideal hydrogen gap ratios

The fixed-proton nonrelativistic Coulomb energies are

\[
E_n=-\frac{1}{2n^2}\ {\rm Hartree}.
\tag{BI.10}
\]

For integer \(n\), every \(E_n\) is rational in Hartree. Therefore every
nonzero gap \(E_b-E_a\) is rational, and every ratio of two nonzero gaps is
rational. Because \(\varphi\) is irrational,

\[
\boxed{\frac{E_b-E_a}{E_d-E_c}\ne\varphi}
\tag{BI.11}
\]

for all ideal Coulomb bound-state transitions. BI.11 is an all-level written
proof. It is stronger than any finite numerical search and does not apply
unchanged to relativistic, radiative, external-field, molecular, nuclear, or
many-body spectra.

The frozen finite control enumerates all 780 transitions with
\(1\le n<m\le40\) and all 303,809 distinct gap-pair comparisons. Its closest
ratio is

\[
\frac{\Delta E_{19\to30}}{\Delta E_{23\to34}}
=\frac{7491169}{4629825}
=1.61802422337777\ldots,
\tag{BI.12}
\]

whose absolute error from \(\varphi\) is
\(9.7653721\times10^{-6}\). This close rational is preserved as the required
counterexample to visual/numerical overinterpretation: 303,809 comparisons can
produce a compelling-looking match without an exact golden law.

### BI.4. Fixed grids cannot contain the exact golden turn

One angular increment on a finite \(N\)-state grid is \(k/N\) turns and is
rational. Equation BI.4 is irrational. Hence no integers \(k,N>0\) satisfy

\[
\frac{k}{N}=\frac1{\varphi^2}.
\tag{BI.13}
\]

The exact golden rotation has no finite recurrence period. Fibonacci
convergents provide controlled periodic approximations; for example,

\[
\frac{987}{2584}
\tag{BI.14}
\]

misses the golden turn by \(6.69777\times10^{-8}\), or
\(2.41120\times10^{-5}\) degrees per step. It nevertheless repeats after a
finite rational period and is not the exact irrational rotation.

This yields a correction for the fixed-grid/fixed-tick hypothesis. An engine
may use a finite angular grid, or it may carry an exact algebraic irrational
state such as \(a+b\sqrt5\), but a finite \(N\)-state angular grid cannot do
both exactly. If continuous golden phase is fundamental, the state
representation and update rule must say how the algebraic number is stored and
how observation maps it to finite data.

### BI.5. Measured manifestations do not transfer automatically

Douady and Couder obtained Fibonacci phyllotactic order in a physical
laboratory experiment and numerical simulation, describing convergence toward
the golden mean through an iterative self-organizing process. Shechtman et al.
measured metallic matter with long-range icosahedral orientational order and
no ordinary translational lattice. These results establish that specific
dynamics and symmetries can select golden structures. They do not show that
every spiral, harmony, atom, molecule, or projection is golden.

The evidence classes for this revision are:

1. Equations BI.1--BI.11 and BI.13 are written mathematics.
2. The 303,809 comparisons, Fibonacci convergence table, icosahedral distance
   audit, and 144-point golden-angle prefix are finite computation.
3. A separate executable imports none of the production module and reproduces
   the exhaustive search plus eight other checks.
4. The cited \(E_8\), quasicrystal, and phyllotaxis papers contain external
   measured evidence for their own systems. No new measurement was performed.
5. A hidden-dimensional golden generator remains a hypothesis. It has no
   declared Hamiltonian, coupling, units, observable, or fresh prediction here.
6. No golden residual identifies God or another theological cause.

Primary sources: Douady and Couder, *Physical Review Letters* 68, 2098 (1992),
https://doi.org/10.1103/PhysRevLett.68.2098; Shechtman, Blech, Gratias, and
Cahn, *Physical Review Letters* 53, 1951 (1984),
https://doi.org/10.1103/PhysRevLett.53.1951; Coldea et al., *Science* 327, 177
(2010), https://doi.org/10.1126/science.1180085.

All 51 Hydrogen Matrix package tests pass. The independent golden-ratio audit
passes nine checks. Earlier hydrogen, rotor, temperature, spectroscopy, and
probability-current results are unchanged.

The next admissible test must derive one dimensionless golden prediction from
a declared Hamiltonian or projection operator before consulting the target
data. The observable, uncertainty model, non-golden baseline, and rejection
threshold must be frozen. Searching many ratios and reporting the closest one
is not a prediction.

Revision 1.14's complete deterministic closure remains idempotent. Revision
1.15's integer-tick non-crossing limitation, angular-impulse formula, and
finite-prefix generator non-identifiability remain unchanged. This appendix
does not establish seven physical dimensions, \(P=NP\), or a theological
identification.

## Appendix BJ. Forward observable compiler and the quantum light boundary

### BJ.1. From “display everything” to a typed forward map

No finite engine can establish that it calculates every property of reality by
declaring unimplemented effects to be hidden. The executable replacement is a
typed contract

\[
X\xrightarrow{\;U_t\;}X(t)
\xrightarrow{\;\Pi\;}\psi(\mathbf r,t)
\xrightarrow{\;\mathcal O_k\;}y_k(t),
\tag{BJ.1}
\]

where the state \(X\), evolution \(U_t\), projection \(\Pi\), and every observer
operator \(\mathcal O_k\) are explicit. A requested channel is `generated` only
when all four objects exist. An approximation is marked `proxy`; a missing
operator or state variable is `blocked`; and a cause is `not identified` when
multiple hidden states share the same visible outputs.

Release 0.8 freezes the ideal fixed-proton, non-relativistic state

\[
|\Psi(t)\rangle=\frac{|1s\rangle+e^{-i\omega t}|2p_z\rangle}{\sqrt2},
\qquad
\omega=\frac{E_{2}-E_{1}}{\hbar}.
\tag{BJ.2}
\]

It compiles this one state into \(\rho=|\psi|^2\), the field-free probability
current \(\mathbf j\), electron charge density \(-e\rho\), charge current
\(-e\mathbf j\), the expectation dipole, a coherent far-zone radiation field,
and the earlier operator-filtered harmonic catalog. These are multiple
observation channels of one model; they are not additional ontological layers.

### BJ.2. Coherent electromagnetic projection

Let

\[
d=e|\langle1s|z|2p_z\rangle|,
\qquad
|\langle1s|z|2p_z\rangle|
=\frac{256}{243\sqrt2}a_0.
\tag{BJ.3}
\]

For the packet in Equation BJ.2,

\[
\mathbf p(t)=-d\cos(\omega t)\,\hat{\mathbf z}.
\tag{BJ.4}
\]

In the electric-dipole radiation zone, at retarded time,

\[
\mathbf E_{\rm rad}(\mathbf n,R,t)
=\frac{\mathbf n\times(\mathbf n\times\ddot{\mathbf p})}
{4\pi\epsilon_0c^2R},
\qquad
\mathbf B_{\rm rad}=\frac{\mathbf n\times\mathbf E_{\rm rad}}{c}.
\tag{BJ.5}
\]

The resulting power density is radial and nonnegative, with the dipole
\(\sin^2\theta\) pattern. Integration over solid angle gives

\[
P_{\rm coh}(t)=\frac{|\ddot{\mathbf p}(t)|^2}
{6\pi\epsilon_0c^3},
\qquad
\overline P_{\rm coh}=\frac{d^2\omega^4}
{12\pi\epsilon_0c^3}.
\tag{BJ.6}
\]

The finite engine stores 48 synchronized \(65\times65\) density/current and
charge/current slices plus a 721-direction radiation audit. At one metre the
declared packet gives peak equatorial \(|\mathbf E|=1.5179677846\times10^{-4}\)
V/m and cycle-average coherent power
\(2.5620237532\times10^{-10}\) W. These are deterministic model outputs, not
measurements of an individual atom.

### BJ.3. Exact counterexample to a complete classical projection

The ideal electric-dipole spontaneous rate is

\[
A=\frac{\omega^3d^2}{3\pi\epsilon_0\hbar c^3}.
\tag{BJ.7}
\]

Equation BJ.2 has excited-state population \(1/2\). Its initial quantum E1
energy-loss rate is therefore

\[
P_{\rm quantum}=\frac12\hbar\omega A
=\frac{d^2\omega^4}{6\pi\epsilon_0c^3}.
\tag{BJ.8}
\]

Equations BJ.6--BJ.8 imply the exact ratio

\[
\boxed{\frac{\overline P_{\rm coh}}{P_{\rm quantum}}=\frac12}.
\tag{BJ.9}
\]

The production calculation returns respectively
\(2.5620237532\times10^{-10}\) W and
\(5.1240475064\times10^{-10}\) W; the independent implementation also returns
exactly 0.5 at displayed precision. A pure excited state supplies an even
sharper counterexample: its expectation dipole is stationary and radiates no
classical coherent field, while Equation BJ.7 gives a nonzero spontaneous
decay rate.

Thus the visible classical wave from an expectation value is not the complete
photon observable. Vacuum fluctuations, incoherent emission, and photon
statistics require a quantized electromagnetic field or another explicitly
declared stochastic/quantum operator. The result agrees with the primary
mixed quantum--classical analysis of Chen et al., *J. Chem. Phys.* 150, 044102
(2019), https://doi.org/10.1063/1.5057365, which shows that ordinary Ehrenfest
fields miss essential spontaneous-emission physics.

### BJ.4. Verification, evidence type, and boundary

All eight frozen production checks pass: analytic continuity residual zero;
finite-grid neutrality error \(1.06955\times10^{-3}\); transversality relative
error \(1.33921\times10^{-16}\); plane-wave \(|B|=|E|/c\) error
\(4.44089\times10^{-16}\); angular power-integral error
\(4.53077\times10^{-12}\); cycle-average error
\(2.01789\times10^{-16}\); a zero dipole field for the parity-silent
\(1s+2s\) control; and Equation BJ.9. A separate implementation imports none
of the production compiler modules and passes twelve scalar and saved-array
checks.

The evidence classes are:

1. Equations BJ.1--BJ.9 are written conditional mathematics.
2. Stored fields, quadratures, tests, and the observer map are finite
   deterministic computation.
3. No new spectrum, photon count, or atomic movie was measured.
4. A deeper four- or five-dimensional generator remains a hypothesis because
   it still lacks an action, coupling, projection operator, and distinguishing
   prediction.
5. The classical/quantum difference does not identify one hidden cause and
   has no theological label.

The next bounded build is a two-level quantized-radiation channel with
density-matrix decay and photon-number statistics. It must freeze probability
conservation, total emitted energy, coherent/incoherent decomposition, and the
pure-excited-state counterexample before another visual layer is added.

Revision 1.14's complete deterministic closure remains idempotent. Revision
1.15's integer-tick non-crossing limitation, angular-impulse formula, and
finite-prefix generator non-identifiability remain unchanged. The recurrence
learner's \(2K\)-sample continuation guarantee still assumes a noiseless
homogeneous constant-coefficient linear recurrence of order at most \(K\) from
the initial state. Nothing here proves seven physical dimensions, \(P=NP\), a
universal grid/tick, or a theological identification.

## Appendix BR. Primary-equation CW/LD truth-tower benchmark

### BR.1. Bounded question and frozen assumptions

Appendix BQ showed exactly why a mass sequence is not yet a reconstructed
ATLAS spectrum. The next bounded question is narrower:

> What numerical spin-2 clockwork/linear-dilaton tower follows from one
> published ATLAS validation benchmark, and is that truth-level tower narrow
> compared with its masses and adjacent mode spacing?

Protocol `EPL-CWLD-TRUTH-TOWER-2026-09-20-v1` fixes

\[
k=500\ \mathrm{GeV},\qquad M_5=6000\ \mathrm{GeV},\qquad
\overline M_P=2.4\times10^{18}\ \mathrm{GeV},\qquad m_n\leq5000\ \mathrm{GeV}.
\tag{BR.1}
\]

ATLAS explicitly generated samples at \(k=300,500,1000,2000\) GeV with
\(M_5=6\) TeV to validate its analytic CW/LD templates. The 500 GeV point is
therefore reproducible rather than chosen after inspecting a computed tower.
It is not fitted here, not declared allowed, and not treated as a preferred
physical point. The 5 TeV cap matches the upper diphoton range and lies inside
the dielectron range.

The primary theory source is G. F. Giudice et al., *JHEP* **06** (2018) 009,
arXiv:1711.08437v2, https://arxiv.org/html/1711.08437v2. The primary
experimental source is ATLAS Collaboration, *JHEP* **10** (2023) 079,
arXiv:2305.10894v2, https://arxiv.org/html/2305.10894v2.

### BR.2. Radius, masses, and couplings

The published Planck relation is

\[
\overline M_P^2=\frac{M_5^3}{k}\left(e^{2\pi kR}-1\right).
\tag{BR.2}
\]

Solving rather than approximating it gives

\[
R=\frac{\log\!\left(1+k\overline M_P^2/M_5^3\right)}{2\pi k}
=0.0206137688164\ \mathrm{GeV}^{-1},
\tag{BR.3}
\]

so \(kR=10.3068844082\) and \(R^{-1}=48.5112649176\) GeV. The
spin-2 spectrum and coupling scale are

\[
m_n^2=k^2+\frac{n^2}{R^2},\qquad
\Lambda_n^2=M_5^3\pi R\left(1-\frac{k^2}{m_n^2}\right)^{-1}.
\tag{BR.4}
\]

There are 102 positive modes under the frozen cap. Selected rows are

| \(n\) | \(m_n\) (GeV) | \(m_{n+1}-m_n\) (GeV) | \(\Lambda_n\) (GeV) |
|---:|---:|---:|---:|
| 1 | 502.347831 | 6.978559 | 1,224,735.890 |
| 10 | 696.659373 | 34.609717 | 169,847.203 |
| 50 | 2476.561540 | 47.531304 | 120,758.312 |
| 102 | 4973.346835 | 48.267848 | 118,873.894 |

The first-mode spacing is much smaller than the asymptotic \(1/R\) because the
tower begins just above the mass gap \(k\). The spacing approaches \(1/R\) at
high mode number. This nonuniformity is a derived property of BR.4, not a
golden-ratio or universal-harmony input.

### BR.3. Conditional widths and channel ratios

The published massless-Standard-Model expression is

\[
\Gamma_{n\to\mathrm{SM}}
=\frac{283}{960\pi}\frac{m_n^3}{\Lambda_n^2}.
\tag{BR.5}
\]

The primary partial-width equations also give

\[
\Gamma_{\gamma\gamma}=\Gamma_0,\qquad
\Gamma_{e^+e^-}=\frac12\Gamma_0,\qquad
\Gamma_0=\frac{m_n^3}{80\pi\Lambda_n^2}.
\tag{BR.6}
\]

Thus, within this approximation only,

\[
\mathcal B_{\gamma\gamma}=\frac{12}{283}=0.0424028\ldots,
\qquad
\mathcal B_{e^+e^-}=\frac{6}{283}=0.0212014\ldots.
\tag{BR.7}
\]

Across the compiled modes,

\[
\max_n\frac{\Gamma_{n\to\mathrm{SM}}}{m_n}
=1.64244126\times10^{-4},\qquad
\max_n\frac{\Gamma_{n\to\mathrm{SM}}}{m_{n+1}-m_n}
=0.0169231288.
\tag{BR.8}
\]

The paper's stated resolved-tower condition is

\[
m_n\lesssim6.8\left(\frac{k}{M_5}\right)^{1/7}M_5.
\tag{BR.9}
\]

It evaluates to 28.6083 TeV, so every compiled mode lies below it. This says
that natural widths are narrow compared with neighboring truth modes in the
stated effective theory. It does not say that ATLAS reconstructs 102 separate
peaks; detector resolution and analysis classification are independent
observer maps.

The high-mode approximation

\[
\frac{\Gamma_{n\to\mathrm{cascade}}}{\Gamma_{n\to\mathrm{SM}}}
\simeq0.041\sqrt{m_n/k}
\tag{BR.10}
\]

runs from 0.0411 to 0.1293 over the table, but the engine deliberately stores
it only as a diagnostic. Equation BR.10 is not promoted to an exact low-mode
total width. Exact thresholds, low-mode cascades, finite Standard-Model masses,
the scalar tower, and stabilization-dependent interactions remain outside the
compiled object.

### BR.4. What changed in the recast inventory

For the frozen benchmark and mass cap, the numerical mass spectrum and
coupling scale are now instantiated. The massless-SM partial-width and channel
ratio tables are also instantiated with their approximation labels. This is a
real advance over Appendix BQ's then-current inventory, which remains preserved
as the historical release-0.7 contract.

The exact detector-level recast is still blocked. The resonant production
formula contains parton luminosities and therefore still needs a frozen PDF
set, perturbative order, and scale prescription. The engine also lacks the
complete acceptance/efficiency map, detector mass-transfer matrix, binned
observations and background vector, nuisance likelihood and correlations, and
an executable equivalent of the trained classifier. Appendix BQ's exact
response counterexample is unchanged: equal truth weights can produce
different reconstructed spectra under different admissible response maps.

The official HEPData record again presented persistent human verification to
the available browser. That is recorded only as an access limitation. It is
not evidence that the machine-readable record is absent, incomplete, or
inconsistent with the publication.

### BR.5. Verification, evidence classes, and next test

Eleven new production tests verify the frozen manifest, exact Planck-relation
round trip, contiguous modes, cap boundary, positive couplings, narrow
massless-SM widths, adjacent spacing, massless branching relations, the
published resolved-tower condition, serialization, and the continuing recast
block. Together with all prior Element Projection Lab checks, 73 tests pass.

An independent standard-library implementation imports no production tower
module. It replays the JSON and CSV tables, performing 2,454 checks across all
102 modes. Its largest absolute difference is
\(7.21775\times10^{-9}\) on quantities as large as approximately
\(1.22\times10^6\), arising from independent `log` versus `log1p` evaluation.

The evidence classes remain distinct:

1. Equations BR.2--BR.10 are written conditional mathematics taken from the
   declared CW/LD model.
2. The 102-row tower, diagnostics, tests, and independent replay are finite
   computations.
3. No new collision measurement is produced; the ATLAS null result remains
   external measured evidence.
4. The CW/LD geometry is a physical hypothesis whose consequences are being
   compiled, not an observed fifth dimension.
5. No mismatch or residual is assigned to a unique hidden variable or
   theological cause.

The next bounded test is to instantiate the published per-mode production
formula with a frozen PDF set, perturbative order, scale choices, and uncertainty
prescription, then independently reproduce the resulting cross sections. That
would yield truth-level channel weights, still not reconstructed ATLAS counts.
If the official response or likelihood cannot be obtained, the honest measured
test is a separately calibrated public-spectrum analysis under a new identity,
not an alleged exact recast.

Revision 1.14's complete deterministic closure remains idempotent. Revision
1.15's integer-tick non-crossing limitation, angular-impulse formula, and
finite-prefix generator non-identifiability remain unchanged. The recurrence
learner's \(2K\)-sample continuation guarantee still assumes a noiseless
homogeneous constant-coefficient linear recurrence of order at most \(K\) from
the initial state. Nothing here proves seven physical dimensions, \(P=NP\), a
universal grid/tick, or a theological identification.

## Appendix BS. Measured HEPData constraint on the compiled benchmark

### BS.1. Source and bounded question

The user supplied the exact JSON content of HEPData Table 1 twice; both copies
were identical. The table is identified by DOI
`10.17182/hepdata.140955.v1/t1`, Figure 9a, and contains the observed 95%
confidence-level exclusion contour in the `k-M5` plane for the dielectron
channel with mass thresholds, using 139 inverse femtobarns at 13 TeV.

The bounded question is:

> Does revision 1.69's `k=500 GeV`, `M5=6000 GeV` truth-tower benchmark
> survive the observed Table 1 contour?

This question does not reconstruct ATLAS's classifier, pseudo-experiments,
nuisance likelihood, or detector response. It classifies one fixed point using
the collaboration's published measured exclusion boundary.

### BS.2. Local contour calculation

The two adjacent Table 1 vertices bracketing `k=500 GeV` are

| `k` (GeV) | observed `M5` limit (GeV) |
|---:|---:|
| 489.1 | 6973.0 |
| 502.3 | 7082.4 |

The frozen evaluator uses no extrapolation. Linear interpolation on this local
published contour segment gives

\[
M_{5,\mathrm{limit}}(500)
=6973.0+\frac{500-489.1}{502.3-489.1}(7082.4-6973.0)
=7063.3378788\ \mathrm{GeV}.
\tag{BS.1}
\]

Revision 1.69 used `M5=6000 GeV`, which lies

\[
7063.3378788-6000=1063.3378788\ \mathrm{GeV}
\tag{BS.2}
\]

below the observed boundary. ATLAS states that the signal cross-section scales
approximately as `M5^-3` and that reducing `M5` makes the signal more prominent.
The lower-`M5` side is therefore the excluded side. The point is excluded at
95% CL by this channel and threshold scenario.

As a model-dependent diagnostic, the approximate signal strength relative to
the boundary is

\[
\left(\frac{7063.3378788}{6000}\right)^3=1.6314591.
\tag{BS.3}
\]

Equation BS.3 inherits the approximate scaling. The exclusion classification
itself comes from the observed contour, not from treating the scaling as exact.

### BS.3. Correction and scope

There is no contradiction between the point being used for simulation
validation and being excluded by collision data. Template validation asks
whether the analytic signal template reproduces generated events at a chosen
point. Experimental viability asks whether the signal is compatible with the
observed data. These are distinct claims.

The corrected status is therefore

\[
\boxed{\texttt{benchmark excluded by observed ee mass-threshold contour}.}
\tag{BS.4}
\]

This is a genuine negative measured-input result for the exact point used by
the compiler. It does not invalidate the tower arithmetic, because that
arithmetic states what the assumed model predicts. It invalidates treating
this particular parameter choice as viable under the declared ATLAS result.
Nor does it exclude every CW/LD point, every compactification, or every
extra-dimensional model.

### BS.4. Submission inventory and recast boundary

The user also supplied the complete HEPData record index. It enumerates 24
tables:

1. observed, median expected, and expected plus/minus one and two standard
   deviation contours;
2. dielectron and diphoton channels;
3. analyses with and without mass thresholds.

The index exposes CSV, JSON, ROOT, YAML, YODA, and YODA-HDF5 download links for
each table and for the whole submission. It advertises exclusion contours and
image resources, but no observed invariant-mass spectrum, background-bin
vector, detector response matrix, acceptance/efficiency grid, covariance,
workspace, likelihood, or trained classifier artifact. Thus the archive can
support a complete published-limit audit. On the supplied inventory it cannot
by itself support an exact detector-level reconstruction of Equation BQ.1.

The direct archive endpoint remained behind automated access verification in
this environment. That is an access limitation; the index and Table 1 values
already supplied are used as authoritative measured inputs. The remaining 23
table values must be ingested from the downloaded archive rather than guessed.

### BS.5. Verification and next test

Ten production tests verify the table DOI and qualifiers, axis orientation,
bracketing interval, interpolation, exclusion, approximate scaling, scope,
serialization, and extrapolation rejection. Together with the earlier suite,
all 83 Element Projection Lab tests pass.

A separate implementation imports no production benchmark module and uses
50-digit Decimal arithmetic. All eight independent checks pass, reproducing
the limit, 1063.3378788 GeV margin, 1.6314591 scaling diagnostic, and exclusion
decision.

The evidence classes remain distinct:

1. The Table 1 contour is published external measured evidence.
2. Interpolation, classification, tests, and the independent replay are finite
   computations.
3. The `M5^-3` diagnostic is conditional model mathematics.
4. No new collision measurement or independent likelihood is produced here.
5. The negative result supplies no positive dimensional or theological
   identification.

The next bounded test is to ingest all 24 tables and evaluate the same fixed
benchmark against every observed channel/threshold contour while retaining the
expected bands as diagnostics. If every observed scenario excludes it, that is
a stronger cross-channel constraint on this point, still not a universal
model exclusion. No unlisted detector or likelihood object will be imputed.

Revision 1.14's complete deterministic closure remains idempotent. Revision
1.15's integer-tick non-crossing limitation, angular-impulse formula, and
finite-prefix generator non-identifiability remain unchanged. The recurrence
learner's \(2K\)-sample continuation guarantee still assumes a noiseless
homogeneous constant-coefficient linear recurrence of order at most \(K\) from
the initial state. Nothing here proves seven physical dimensions, \(P=NP\), a
universal grid/tick, or a theological identification.

## Appendix BP. Retrospective ATLAS cross-channel dimension-evidence audit

### BP.1. Bounded question and source

Revision 1.66 required any physical-dimension claim to survive a fixed model,
a conventional baseline, multiplicity and systematics, reserved validation,
and replication. This appendix asks one narrower question:

> Does the exact public result reported by the ATLAS periodic dielectron and
> diphoton search pass a necessary cross-channel screen for positive evidence
> of an extra physical dimension?

ATLAS analysed \(139.0\,\mathrm{fb}^{-1}\) of \(13\,\mathrm{TeV}\) proton--proton
collisions recorded in 2015--2018. The model-independent analysis published
four signed significance values per channel. Positive values indicate that
the data reconstruction loss exceeds the background median; negative values
point in the opposite direction. The exact published matrix is

| threshold | dielectron | diphoton |
|---|---:|---:|
| \(R<10\%\) | \(+0.4\) | \(-1.8\) |
| \(R<50\%\) | \(+1.5\) | \(-0.2\) |
| no threshold | (+0.7) | (-0.7) |
| scale threshold | (+1.5) | (-0.6) |

The \(R<50\%\) row is fixed as the primary periodicity-focused screen. The
paper's separate model-dependent clockwork/linear-dilaton search reports no
significant deviation from the background-only hypothesis.

Primary source: ATLAS Collaboration, *JHEP* **10** (2023) 079,
https://arxiv.org/html/2305.10894v2, with associated HEPData DOI
`10.17182/hepdata.140955.v1`.

### BP.2. Frozen necessary screen

For a reported signed significance \(z\), the one-sided positive tail is

\[
p_{\mathrm{local}}(z)=\frac12\operatorname{erfc}
\left(\frac{z}{\sqrt2}\right).
\tag{BP.1}
\]

The four threshold variants use the same collision sample and are not treated
as independent. A Bonferroni adjusted value

\[
p_{\mathrm{adj}}=\min\{1,4p_{\mathrm{local}}\}
\tag{BP.2}
\]

controls this four-variant family without an independence assumption. The
frozen necessary screen requires all three conditions:

1. both primary \(R<50\%\) signed significances are at least \(+3\);
2. in each channel the most positive of the four values has
   \(p_{\mathrm{adj}}\le 0.00135\);
3. the locked model-dependent analysis reports a significant positive result,
   not background compatibility and exclusion.

Passing would still not prove spatial dimension. It would trigger, not replace,
tests of masses, spins, widths, couplings, branching fractions, detector
response, conventional alternatives, and independent experimental
replication.

### BP.3. Result and counterexamples

The primary values (+1.5) and (-0.2) fail the first condition. The largest
positive value in each channel is likewise (+1.5) and (-0.2), giving

\[
\begin{aligned}
p_{ee,\mathrm{local}}&=0.0668072013,&
p_{ee,\mathrm{adj}}&=0.2672288051,\\
p_{\gamma\gamma,\mathrm{local}}&=0.5792597094,&
p_{\gamma\gamma,\mathrm{adj}}&=1.
\end{aligned}
\tag{BP.3}
\]

The model-specific condition also fails because ATLAS reports no significant
deviation. Therefore

\[
\boxed{\text{the published ATLAS result does not pass this necessary
cross-channel evidence screen}.}
\tag{BP.4}
\]

Three constructed controls prevent the rule from becoming rhetorical. A
(+4.0,+4.2) primary pair with a positive model-specific result passes the
software screen. A (+3.1,+3.1) pair passes the uncorrected primary rule but
has \(p_{\mathrm{adj}}=0.00387041285\) in each channel and therefore fails the
declared familywise threshold. A (+6.0,0.0) pair fails because one strong
channel is not cross-channel support and is not independent replication.
These are finite synthetic controls, not observations.

### BP.4. Exact procedural correction and limits

This audit is retrospective. The ATLAS publication and its null conclusion
were public before protocol `EPL-ATLAS-PUBLICATION-GATE-2026-09-20-v1` was
frozen. Freezing now improves reproducibility but cannot retroactively make the
test blind or preregistered.

This release also does not reproduce the ATLAS event- or binned-spectrum
likelihood. It evaluates exact printed summary values and the published
model-dependent conclusion. The complete event data, nuisance likelihood,
covariance, detector response, and trained networks are not implemented here.
Accordingly, Equation BP.4 is a faithful classification of the publication
summary, not an independent ATLAS analysis.

All 53 Element Projection Lab tests pass. The independent audit imports no
production module, verifies the protocol digest and exact eight-value matrix,
and passes 12/12 checks of the tail calculation, gate result, retrospective
flag, and likelihood boundary.

Evidence types remain separate:

1. Equations BP.1--BP.2 are written standard probability mathematics.
2. The eight significance values and model-dependent conclusion are external
   measured-data summaries produced by ATLAS.
3. Their deterministic replay and constructed controls are finite computation.
4. A physical extra dimension remains a model-dependent hypothesis, not the
   inferred cause of an unconstrained residual.
5. No theological interpretation is tested or identified.

The next valid step is to obtain the official machine-readable spectrum and
statistical model for a retrospective calibration study, then freeze one
complete tower model before genuinely independent future data are opened. A
failure may reflect a wrong dimensional hypothesis, omitted conventional
physics, miscalibration, statistical fluctuation, or another model error. It
cannot automatically be counted as evidence for missing hidden variables.

Revision 1.14's complete deterministic closure remains idempotent. Revision
1.15's integer-tick non-crossing limitation, angular-impulse formula, and
finite-prefix generator non-identifiability remain unchanged. The recurrence
learner's \(2K\)-sample continuation guarantee still assumes a noiseless
homogeneous constant-coefficient linear recurrence of order at most \(K\) from
the initial state. Nothing here proves seven physical dimensions, \(P=NP\), a
universal grid/tick, or a theological identification.

## Appendix BO: Dimension evidence requires a distinguishing physical constraint (revision 1.66)

### BO.1. The bounded question

The project has repeatedly rendered three-dimensional observations of states
whose mathematical descriptions use four or five coordinates. That is useful
for visualizing a proposed latent dynamics. It is not automatically physical
evidence that space has another spatial dimension. This appendix asks the
narrow question:

> Can the current linear 3D projection of the declared five-coordinate
> norm-preserving motion distinguish a physical fifth spatial dimension from
> ordinary temporal oscillators in 3+1-dimensional physics?

The answer is no. The obstruction is exact and holds for the full continuous
record, so increasing frame rate or sampling duration cannot remove it.

### BO.2. Projection-equivalence theorem

Let \(A\) be a real \(5\times5\) skew-symmetric generator. The real spectral
theorem supplies an orthogonal basis in which

\[
 A=J(\omega_1)\oplus J(\omega_2)\oplus 0,
 \qquad
 J(\omega)=\begin{pmatrix}0&-\omega\\\omega&0\end{pmatrix},
 \tag{BO.1}
\]

for nonnegative \(\omega_1,\omega_2\). For an initial state \(x_0\in
\mathbb R^5\) and linear observer \(B\in\mathbb R^{3\times5}\), define
\(y(t)=B\exp(At)x_0\). Expanding the two rotation blocks gives five constant
three-vectors \(c,a_1,b_1,a_2,b_2\) such that

\[
 y(t)=c+a_1\cos(\omega_1t)+b_1\sin(\omega_1t)
       +a_2\cos(\omega_2t)+b_2\sin(\omega_2t).
 \tag{BO.2}
\]

Equation (BO.2) is also the output of two ordinary temporal harmonic
oscillators coupled to a three-vector detector. Thus the two model
descriptions are observationally equivalent through this observer for every
real \(t\). No estimator, neural network, Fourier transform, Hankel matrix, or
human observer receiving only \(y(t)\) can identify which representation is
the physical ontology.

A large minimal realization or Hankel rank may identify the number of
dynamical state variables within a declared linear class. It does not count
spatial dimensions: ordinary 3+1D media and fields can have arbitrarily many
internal modes. This distinction corrects the inference, without denying the
mathematical validity of the projection.

The result is conditional. Nonlinear couplings, back-reaction, interactions,
nonlinear observers, or different topologies define different model classes.
They require explicit equations and new distinguishing predictions; they are
not rescued merely by adding unconstrained harmonics to Equation (BO.2).

### BO.3. What a constrained extra-dimensional signature can look like

For one field on one flat compact circle of radius \(R\), Fourier separation
along the compact coordinate gives the model-specific tower

\[
 m_n^2=m_0^2+\frac{n^2}{R^2},\qquad n=0,1,2,\ldots.
 \tag{BO.3}
\]

Given labelled modes \(n=0\) and \(n=1\), any positive pair \(m_0<m_1\)
calibrates

\[
 R^{-1}=\sqrt{m_1^2-m_0^2}.
 \tag{BO.4}
\]

Those two peaks are therefore not a test. Mode \(n=2\) is the first mass that
Equation (BO.3) constrains after calibration, and further modes add correlated
checks. A finite mass pattern is still not unique to geometry. A credible
claim also needs derived widths, spins, coupling ratios, branching fractions,
and correlations across detector channels, with look-elsewhere effects and
conventional resonance alternatives included.

Equation (BO.3) is not presented as a universal prediction of all
extra-dimensional theories. Warping, boundary conditions, branes, clockwork
geometries, field content, and interactions change the spectrum. A failed
circle tower rejects the declared circle model, not every possible hidden
dimension.

### BO.4. Executable checks and counterexample

Element Projection Lab 0.5.0 compiles Equation (BO.2) algebraically for 512
random frequency pairs, initial states, and observer matrices. Across 512,512
three-vector samples, the explicit five-coordinate evolution and ordinary
two-oscillator output have maximum component disagreement
\(5.329070518200751\times10^{-15}\). The largest squared-norm drift is
\(1.4210854715202004\times10^{-14}\).

The finite tower test uses six synthetic masses satisfying Equation (BO.3).
After calibrating only \(n=0,1\), the four held-out masses agree exactly in the
executed binary arithmetic. A counterexample changes those four masses by
\(+0.07,-0.05,+0.11,-0.09\) TeV and yields a held-out RMSE of
\(0.08306623862918076\) TeV. Both lists share the same two perfect calibrants;
only the reserved later modes distinguish them.

An independent scalar implementation imports no production dimension module.
It checks 299,919 output components in 257 new trials, with maximum disagreement
\(3.552713678800501\times10^{-15}\), and independently reproduces both tower
results. These finite computations check the algebra; they are not measured
evidence for an extra dimension.

### BO.5. Physical-dimension evidence gate

The project may label a future result physical evidence for another spatial
dimension only if all of the following are present:

1. a fixed extra-dimensional model identity;
2. an explicit dimensional coupling mapped to measured units;
3. a matched conventional baseline;
4. a distinguishing cross-observable constraint;
5. parameters frozen before reserved validation is opened;
6. a measured excess that survives declared systematic uncertainties and
   multiplicity correction; and
7. independent experimental replication.

Passing this gate would support the specified model over the specified
alternatives; it would not constitute deductive proof against every logically
possible conventional explanation. Failing it blocks the stronger wording.
The current projected-harmonic animation has a measurable output map and a
conventional baseline, but lacks a fixed physical dimensional coupling, a
distinguishing constraint, reserved measured validation, a measured excess,
and experimental replication. It therefore fails the gate.

### BO.6. Current measured status

The following primary experimental results are relevant measured evidence,
but none is positive evidence for the present theory:

- ATLAS searched 139 fb\(^{-1}\) of 13 TeV dielectron and diphoton data for
  periodic signals, including a clockwork-gravity interpretation, and in the
  absence of a signal set 95% confidence exclusions (ATLAS Collaboration,
  *JHEP* 10 (2023) 079, https://arxiv.org/abs/2305.10894).
- CMS searched 138 fb\(^{-1}\) of 13 TeV high-mass diphoton data and found no
  statistically significant excess. Its model-specific 95% limits include
  ADD scales of 7.1--11.1 TeV depending on convention and Randall--Sundrum
  graviton masses in the reported coupling-dependent ranges (CMS
  Collaboration, *JHEP* 08 (2024) 215,
  https://cms-results.web.cern.ch/cms-results/public-results/publications/EXO-22-024/).
- A torsion-balance test over 52 micrometres to 3.0 millimetres found
  Newtonian gravity to fit the data and limited gravitational-strength Yukawa
  interactions to ranges below 38.6 micrometres at 95% confidence (Lee et al.,
  *Physical Review Letters* 124, 101101 (2020),
  https://arxiv.org/abs/2002.11761).

These null results constrain declared parameter regions. They do not exclude
all extra-dimensional models, and unexplained residuals in unrelated atomic,
optical, water, or acoustic calculations cannot be substituted for them.

### BO.7. Next experiment and retained boundaries

The next admissible step is one frozen public-data tower search. Before
opening the reserved region, it must specify a published spectrum, dimensional
model, mode relation, widths, spin/coupling/channel ratios, background family,
fit window, nuisance parameters, and multiplicity correction. Model parameters
must be fitted only in a declared training region and then scored without
repair on a reserved mass or channel region. A lone peak, improved animation,
or freely added oscillator does not pass.

Revision 1.14's complete deterministic closure for a fixed transition family
remains idempotent. Revision 1.15's integer-tick non-crossing limitation,
angular-impulse formula, and finite-prefix generator non-identifiability remain
unchanged. The recurrence learner's \(2K\)-sample continuation guarantee still
assumes a noiseless homogeneous constant-coefficient linear recurrence of
order at most \(K\) from the initial state. Nothing here establishes seven
physical dimensions, a fixed universal grid/tick, \(P=NP\), or a theological
identification.

## Appendix BK. All-element atomic projection atlas

### BK.1. The word “shape” needs an observer

An isolated atom is not assigned one rigid classical boundary. This revision
therefore defines two different observer channels rather than silently
equating them.

For any one-electron density \(\rho(\mathbf r)\), the unpolarized rotational
average is

\[
\overline{\rho}(\mathbf r)
=\int_{SO(3)}\rho(R^{-1}\mathbf r)\,d\mu(R)
=\overline{\rho}(|\mathbf r|),
\tag{BK.1}
\]

where \(d\mu\) is normalized Haar measure. Equation BK.1 is spherical by
construction and is the whole-atom display used by the atlas. It does not say
that every oriented open-shell atomic state has an exactly spherical density.

The second channel selects a declared spatial frame and displays one canonical
real differentiating-subshell basis. Its lobes are an orbital/state
visualization, not a hard surface or a localized electron trajectory. Rotating
the chosen basis rotates the picture without changing the free atom's
field-free physics.

### BK.2. Exact angular result

For a complete subshell, the spherical-harmonic addition theorem gives

\[
\sum_{m=-l}^{l}|Y_{lm}(\Omega)|^2=\frac{2l+1}{4\pi}.
\tag{BK.2}
\]

Thus a uniformly occupied closed subshell has no angular density structure.
For one oriented basis

\[
\psi_{nlm}(\mathbf r)=R_{nl}(r)Y_{lm}(\Omega),
\tag{BK.3}
\]

the product \(|Y_{lm}|^2\) has only even spherical-harmonic ranks allowed by
\[
L=0,2,\ldots,2l.
\tag{BK.4}
\]

The atlas reports these ranks for \(s,p,d,f\) signatures as respectively
\(\{0\}\), \(\{0,2\}\), \(\{0,2,4\}\), and
\(\{0,2,4,6\}\). Equations BK.1--BK.4 are written mathematics. They do not
depend on the radial approximation below and they do not establish a physical
fourth or fifth spatial dimension.

### BK.3. All-118 finite atlas

Element Projection Lab 0.1.0 stores one neutral gas-phase configuration record
for each atomic number \(1\le Z\le118\). It checks electron number, subshell
capacity, shell population, periodic-table position, differentiating
subshell, radial and angular node counts, a Hund-rule unpaired-electron proxy,
and the angular-rank catalog. Configurations for \(109\le Z\le118\) are marked
predicted. The earlier records are compiled reference inputs rather than 118
new measurements.

The radial comparison is explicitly a low-cost proxy. Each occupied subshell
uses a normalized hydrogenic radial function with the nuclear charge replaced
by the Slater effective charge

\[
Z_{\mathrm{eff}}=Z-S.
\tag{BK.5}
\]

The total displayed radial probability is

\[
P_Z(r)=\frac1Z\sum_{nl}q_{nl}\,r^2
\left|R_{nl}^{(Z_{\mathrm{eff},nl})}(r)\right|^2,
\qquad
\int_0^\infty P_Z(r)\,dr=1.
\tag{BK.6}
\]

This supplies comparable \(r_{50}\), \(r_{90}\), and radial-peak coordinates
in Bohr radii for every element. It is not Hartree--Fock, configuration
interaction, density-functional, Dirac--Fock, or measured electron density.
Relativistic contraction, spin--orbit coupling, electron correlation, nuclear
size, external fields, ionization, bonding, and condensed-matter band
structure are outside this release.

The atlas partitions the 118 display cells into 14 \(s\)-block, 36
\(p\)-block, 40 \(d\)-block, and 28 \(f\)-block records. The interactive
observer can switch between the spherical average and the canonical oriented
subshell signature, preserving their different provenance.

### BK.4. Verification and counterexample

All eleven production tests pass. They cover all 118 atomic numbers,
configuration electron counts, subshell capacities, reference exceptions,
shell totals, hydrogenic radial normalization, saved-data round trip,
configuration provenance, allowed angular ranks, and Equation BK.2 for
\(l=0,1,2,3\). The largest saved per-electron radial normalization error is
\(1.35\times10^{-7}\).

A separate program imports none of the production module and passes thirteen
checks. It independently reparses all 118 configurations, verifies unique
periodic-grid cells, reconstructs the \(14/36/40/28\) block partition, checks
the prediction boundary, and reproduces Equation BK.2 with maximum sampled
error \(5.55\times10^{-16}\).

The required counterexample is immediate: a canonical \(p_z\) display and its
rotated \(p_x\) display have different lobes but describe bases related by a
rotation. Therefore the electron configuration alone does not identify a
unique oriented 3D picture. Conversely, rotational averaging erases that
orientation. Treating either picture as the atom's one literal surface would
confuse a chosen observer with the state.

Primary sources and provenance: IUPAC, Periodic Table of Elements,
https://iupac.org/what-we-do/periodic-table-of-elements/; W. C. Martin et al.,
NIST Ground Levels and Ionization Energies for the Neutral Atoms,
https://www.nist.gov/publications/ground-levels-and-ionization-energies-neutral-atoms;
J. C. Slater, *Physical Review* **36**, 57 (1930),
https://doi.org/10.1103/PhysRev.36.57; H. A. Fertig and W. Kohn,
*Physical Review A* **62**, 052511 (2000),
https://doi.org/10.1103/PhysRevA.62.052511.

No new atomic density was measured. No fourth- or fifth-dimensional rotor was
inferred. Isolated-atom geometry does not by itself predict molecular shape,
phase of matter, or metallic conductivity; those require interaction and
many-body operators. A residual does not identify God or another unique
cause.

The next bounded test is to freeze a diverse subset of closed- and open-shell
atoms, replace the Slater radial proxy with independent ab initio
Hartree--Fock or Dirac--Fock reference densities, and score \(r_{50}\),
\(r_{90}\), radial peak positions, and angular multipoles on data not used to
choose the proxy.

Revision 1.14's complete deterministic closure remains idempotent. Revision
1.15's integer-tick non-crossing limitation, angular-impulse formula, and
finite-prefix generator non-identifiability remain unchanged. The recurrence
learner's \(2K\)-sample continuation guarantee still assumes a noiseless
homogeneous constant-coefficient linear recurrence of order at most \(K\) from
the initial state. This appendix does not prove seven physical dimensions,
\(P=NP\), a universal fixed grid/tick, or a theological identification.

## Appendix BL. Bulk gold and silver color observer

### BL.1. The corrected object of study

Color is not assigned to a single isolated Au or Ag atom. An isolated atom has
discrete scattering and transition amplitudes; a macroscopic metal surface has
a collective, frequency-dependent response built from an occupied crystal and
its excitations. The observer chain is therefore

\[
\text{atomic basis}\longrightarrow\text{periodic solid}\longrightarrow
\varepsilon(\omega)\longrightarrow R(\lambda)\longrightarrow
\text{illuminant}\longrightarrow\text{finite color observer}.
\tag{BL.1}
\]

Element Projection Lab 0.2.0 implements only the last three arrows with a
measured dielectric input. It does not silently treat the revision-1.62
Slater-screened atom picture as a solid-state calculation.

### BL.2. Measured-input Fresnel map

Johnson and Christy measured the room-temperature complex optical constants
of evaporated copper, silver, and gold films from reflection and transmission
over 0.5--6.5 eV. For

\[
\widetilde n(\lambda)=n(\lambda)+i\kappa(\lambda),\qquad
\varepsilon(\lambda)=\widetilde n(\lambda)^2,
\tag{BL.2}
\]

the normal-incidence field coefficient from air is

\[
r(\lambda)=\frac{1-\widetilde n(\lambda)}{1+\widetilde n(\lambda)},
\tag{BL.3}
\]

and the reflected power fraction is

\[
R(\lambda)=|r(\lambda)|^2
=\frac{(n-1)^2+\kappa^2}{(n+1)^2+\kappa^2}.
\tag{BL.4}
\]

Equation BL.4 is written electromagnetic boundary-condition mathematics. The
values of \(n\) and \(\kappa\) are published measurements, not outputs of this
theory. The implementation linearly interpolates the compiled 367.9--821.1 nm
samples only within the frozen 380--780 nm observer range.

For illuminant spectrum \(S(\lambda)\) and color matching functions
\(\overline x,\overline y,\overline z\), the finite observer is

\[
\begin{pmatrix}X\\Y\\Z\end{pmatrix}
=K\int S(\lambda)R(\lambda)
\begin{pmatrix}\overline x(\lambda)\\\overline y(\lambda)\\\overline z(\lambda)\end{pmatrix}
\,d\lambda,
\tag{BL.5}
\]

where \(K\) gives a perfect reflector unit luminance. The renderer uses a
declared analytic approximation to the CIE 1931 two-degree functions and
Bradford adapts the source white to the D65 display white. Display RGB and hex
values are therefore observer/renderer summaries, not intrinsic material
constants.

### BL.3. Finite result and counterexample

The frozen reflectances are

| vacuum wavelength | Au | Ag |
|---:|---:|---:|
| 450.9 nm | 0.4082203341 | 0.9803469959 |
| 520.9 nm | 0.6434199990 | 0.9835410941 |
| 659.5 nm | 0.9625853747 | 0.9905659438 |

For Au the mean changes from 0.4134016869 in 400--500 nm to
0.9507743769 in 600--700 nm. The corresponding Ag means are 0.9766352201
and 0.9897903942. Under the declared equal-energy observer the displayed
approximations are `#FFDFA2` and `#FEFDFC`, respectively. These numbers
quantify the familiar statement: gold suppresses blue/cyan reflection while
silver returns the visible spectrum much more uniformly.

The counterexample to a coarse-geometric explanation is immediate. Au and Ag
are both group-11 metals with a filled d shell plus one s electron in their
neutral reference configurations, and both ordinarily crystallize in the fcc
family, yet their visible spectra differ strongly. Configuration labels and a
coarse lattice class therefore do not identify the optical response.

Established band calculations supply a standard mechanistic bridge.
Christensen and Seraphin's relativistic and nonrelativistic gold calculations
found shifts and splittings comparable to approximately 1 eV band gaps and
connected the calculated interband structure to reflectance. Rangel et al.
later found many-body corrections opening the Au 5d--6sp interband gap by
approximately 0.4--0.8 eV. The present executable cites these results; it does
not reproduce either solid-state calculation.

### BL.4. Verification, evidence type, and boundary

All eleven revision-1.62 atlas tests and eight new optical tests pass. The new
tests cover table integrity, the complex-amplitude/closed-form equivalence in
Equation BL.4, dielectric square-root recovery, physical reflectance bounds,
gold blue suppression, silver visible neutrality, their 450.9 nm separation,
and the perfect-reflector white control. A second implementation imports no
production optics module and replays 802 one-nanometre reflectance values with
maximum absolute discrepancy \(5.55\times10^{-16}\), plus eleven point and
ordering checks.

The evidence classes remain separate:

1. Equations BL.2--BL.5 are written conditional mathematics.
2. Johnson--Christy \(n,\kappa\) values are published measured inputs.
3. Interpolation, Fresnel evaluation, band averages, and display color are
   finite computation.
4. Relativistic-band attribution is established external theory and
   computation, not a result newly derived here.
5. A four- or five-dimensional harmonic cause remains a hypothesis without a
   Hamiltonian, coupling, optical operator, and held-out spectral advantage.
6. A residual cannot identify a unique physical or theological cause.

Surface roughness, morphology, contamination or tarnish, grain structure,
temperature, incidence angle, finite thickness, and nanoparticle resonances
can alter observed color. The present result is restricted to an ideal
normal-incidence bulk-like surface using the cited film data.

The next bounded test is to freeze a relativistic Au/Ag tight-binding or
ab-initio-derived band model, compute optical conductivity and
\(\varepsilon(\omega)\) without fitting the validation wavelengths, and score
its held-out reflectance against the same measurements. A higher-dimensional
term earns support only if it improves that holdout beyond the ordinary
relativistic baseline under an equal or stricter parameter budget.

Primary sources: P. B. Johnson and R. W. Christy, *Physical Review B* **6**,
4370 (1972), https://doi.org/10.1103/PhysRevB.6.4370; N. E. Christensen and
B. O. Seraphin, *Physical Review B* **4**, 3321 (1971),
https://doi.org/10.1103/PhysRevB.4.3321; T. Rangel et al., *Physical Review B*
**86**, 125125 (2012), https://doi.org/10.1103/PhysRevB.86.125125.

Revision 1.14's complete deterministic closure remains idempotent. Revision
1.15's integer-tick non-crossing limitation, angular-impulse formula, and
finite-prefix generator non-identifiability remain unchanged. The recurrence
learner's \(2K\)-sample continuation guarantee still assumes a noiseless
homogeneous constant-coefficient linear recurrence of order at most \(K\) from
the initial state. Nothing here proves seven physical dimensions, \(P=NP\), a
universal grid/tick, or a theological identification.

## Appendix BM. Frozen interband-response holdout

### BM.1. The bounded question

Appendix BL showed the observed color contrast by passing measured optical
constants through the exact Fresnel observer. It did not ask which compact
response class can predict values hidden from fitting. This appendix asks one
narrow question:

> Does one explicit interband-response term improve untouched visible
> reflectance over a free-electron response for gold and silver?

This is not yet an ab-initio band calculation. In particular, deleting a
Lorentz oscillator does not turn a relativistic Hamiltonian into a
nonrelativistic one. The comparison is between two phenomenological causal
response classes only.

### BM.2. Frozen equations and split

Writing photon energy as \(E=hc/\lambda\), the three-parameter baseline is

\[
\epsilon_D(E)=\epsilon_\infty-
\frac{A_D}{E(E+i\gamma_D)}.
\tag{BM.1}
\]

The six-parameter candidate adds one damped bound/interband oscillator,

\[
\epsilon_{DL}(E)=\epsilon_D(E)+
\frac{A_L}{E_L^2-E^2-iE\gamma_L}.
\tag{BM.2}
\]

Positive strengths and damping make the implemented response passive over the
tested range. Complex refractive index and normal reflectance follow from

\[
\widetilde n(E)=\sqrt{\epsilon(E)},\qquad
R(E)=\left|\frac{1-\widetilde n(E)}{1+\widetilde n(E)}\right|^2.
\tag{BM.3}
\]

For each metal the optimizer receives the eleven Johnson--Christy rows at
zero-based indices

\[
0,1,3,4,6,7,9,10,12,13,15.
\tag{BM.4}
\]

It minimizes unweighted \(n,\kappa\) residuals. Indices

\[
2,5,8,11,14
\tag{BM.5}
\]

are never passed to the optimizer. They correspond to 397.4, 450.9, 520.9,
616.8, and 756.0 nm. This is a deterministic train/holdout split within one
published measurement table, not a later claim of prospective preregistration.

### BM.3. Held-out result

The untouched normal-reflectance errors are

| material | Drude MAE | Drude--Lorentz MAE | reduction |
|---|---:|---:|---:|
| Au | 0.1546905703 | 0.0197184999 | 87.2529399% |
| Ag | 0.0068397504 | 0.0027273560 | 60.1249190% |

For gold the absolute separation is decisive within these classes. The
free-electron curve cannot reproduce the measured low blue reflectance and
high red reflectance simultaneously. The added oscillator predicts all five
untouched points much more closely. For silver the free-electron baseline is
already close in the visible; the candidate still improves the held-out MAE,
but only by 0.0041123944 in absolute reflectance.

This pattern matches the standard condensed-matter interpretation that gold's
visible response contains a strong interband contribution, whereas silver's
principal interband onset lies farther toward the ultraviolet. The Lorentz--
Drude construction follows the established effective optical modeling family
of Rakic et al. It does not reproduce their multi-oscillator parameterization
and does not derive a microscopic band structure.

### BM.4. Counterexamples and model strain

The held-out improvement does not uniquely identify the microscopic law:

1. The candidate has six fitted parameters while the baseline has three.
   Holding out wavelengths limits direct interpolation but does not remove all
   small-sample selection advantage.
2. The fitted Au Drude damping reaches its declared lower bound of 0.001 eV.
   The fitted Ag background permittivity reaches its lower bound of zero.
   Both flags are retained rather than presented as exact physical constants.
3. Multiple oscillator sets can approximate a finite spectrum. The fitted
   coefficients are effective coordinates, not unique particles or rotations.
4. All observations come from one thin-film measurement series. Sample
   morphology, preparation, contamination, temperature, and experimental
   systematics are shared between training and holdout points.
5. An ordinary three-dimensional electromagnetic response model supplies the
   gain. Therefore the gain is not evidence for a fourth/fifth dimension.

These counterexamples also answer the proposal that failure can only mean a
missing hidden variable. Failure can instead arise from a wrong response
class, insufficient parameterization, biased sample, inappropriate loss, or
non-identifiability. Missing variables remain one possibility, not the only
one.

### BM.5. Verification and evidence type

Element Projection Lab 0.3.0 adds nine production tests for the exact split,
holdout isolation, passivity, physical reflectance bounds, both material
improvements, the much larger Au correction, and the retained boundary
diagnostics. Together with the eleven atlas and eight optical-observer tests,
all 28 package tests pass.

A second implementation imports no production band-response code. It
reconstructs complex epsilon, refractive index, reflectance, and every reported
metric using scalar complex arithmetic. It replays 64 point/model predictions,
performs 218 checks, and agrees with the aggregate metrics to at most
\(2.1511\times10^{-16}\).

The evidence classes remain distinct:

1. Equations BM.1--BM.3 are written conditional mathematics.
2. The optical constants are published measured inputs, not new measurements
   by this project.
3. Parameter fitting, held-out scoring, and the independent replay are finite
   computations.
4. The external relativistic and many-body band results cited in Appendix BL
   remain established outside work, not computations performed here.
5. A physical higher-dimensional coupling remains an unconstrained hypothesis
   and receives no support from an ordinary model's improvement.

The next test freezes the 0.3.0 coefficients and scores them without refitting
against an independent modern Au/Ag optical dataset. In a separate experiment,
matched relativistic and nonrelativistic solid-state Hamiltonians must be
implemented before attributing spectral gain to relativity. Only after those
ordinary baselines are fixed can an explicit higher-dimensional coupling earn
support through fresh out-of-sample improvement under an equal or stricter
parameter budget.

Primary sources: A. D. Rakic et al., *Applied Optics* **37**, 5271--5283
(1998), https://doi.org/10.1364/AO.37.005271; P. B. Johnson and R. W.
Christy, *Physical Review B* **6**, 4370 (1972),
https://doi.org/10.1103/PhysRevB.6.4370; N. E. Christensen and B. O.
Seraphin, *Physical Review B* **4**, 3321 (1971),
https://doi.org/10.1103/PhysRevB.4.3321; T. Rangel et al., *Physical Review B*
**86**, 125125 (2012), https://doi.org/10.1103/PhysRevB.86.125125.

Revision 1.14's complete deterministic closure remains idempotent. Revision
1.15's integer-tick non-crossing limitation, angular-impulse formula, and
finite-prefix generator non-identifiability remain unchanged. The recurrence
learner's \(2K\)-sample continuation guarantee still assumes a noiseless
homogeneous constant-coefficient linear recurrence of order at most \(K\) from
the initial state. Nothing here proves seven physical dimensions, \(P=NP\), a
universal grid/tick, or a theological identification.

## Appendix BN. Frozen gold/silver external-validation contract

### BN.1. What is frozen

The revision-1.64 response experiment is now immutable under the identity

\[
\texttt{EPL-AU-AG-COLOR-2026-09-20-v1}.
\tag{BN.1}
\]

The freeze manifest locks:

1. the exact Au and Ag Drude and Drude--Lorentz coefficient vectors, encoded
   as hexadecimal IEEE-754 values;
2. the eleven training indices and five untouched indices;
3. Equations BM.1--BM.3 and the 367.9--821.1 nm domain;
4. the fit target and future scoring rules;
5. SHA-256 hashes of the measurement table, fitting implementation, fitted
   result JSON, and point-prediction CSV;
6. the parameter-boundary warnings, limitations, and negative claims.

Its manifest digest is

\[
\texttt{b46a29610b9c9040f0fdf791722caf48cf1a31c3f99b8f66a24c1610847e0262}.
\tag{BN.2}
\]

### BN.2. Evaluation-only rule

The frozen evaluator contains no optimizer or fitting path. Given a material,
a declared model, and wavelengths inside the fixed domain, it decodes the
stored hexadecimal coefficients and evaluates the dielectric and Fresnel maps.
For external measured \(n,\kappa\), it reports normal-reflectance MAE, RMSE,
and maximum absolute error.

Before prediction it checks the manifest digest and every locked artifact
hash. A coefficient edit, source-data edit, regenerated result with different
bytes, changed prediction table, or changed protocol therefore produces a
failure instead of silently defining a new experiment.

Under freeze identity BN.1:

\[
\boxed{\text{refit}=\text{bound change}=\text{model selection}=\text{false}.}
\tag{BN.3}
\]

A later model may be created, but it must receive a new identity and fresh
validation data. It cannot replace this benchmark and retain the original
claim.

### BN.3. Verification and boundary

Eight new production tests verify the manifest digest, artifact hashes,
bit-exact parameter round trips, replay of all saved predictions, the
evaluation-only holdout score, rejected unknown/out-of-domain requests, and
absence of optimizer dependencies. Together with the earlier 28 tests, all 36
Element Projection Lab tests pass.

An independent implementation imports no frozen evaluator. It verifies the
manifest and four artifact hashes, decodes the coefficients directly, and
replays all 64 saved predictions. Its maximum absolute discrepancy is
\(4.440892098500626\times10^{-16}\).

Freezing changes experimental discipline, not physical evidence. The
coefficients still originate from one Johnson--Christy measurement series; the
current holdout still shares that series' preparation and systematic effects;
and the effective oscillator remains non-unique. The freeze establishes what
will be tested before seeing the next data. It does not establish a new
particle, a relativistic Hamiltonian, a fourth/fifth dimension, or a
theological cause.

The next admissible test is to apply the unchanged evaluator to a separately
published Au/Ag optical dataset with no refitting. Both success and failure
must be retained. A poor result cannot be repaired using that validation set;
any repair begins a new model identity and requires another fresh dataset.

Revision 1.14's complete deterministic closure remains idempotent. Revision
1.15's integer-tick non-crossing limitation, angular-impulse formula, and
finite-prefix generator non-identifiability remain unchanged. The recurrence
learner's \(2K\)-sample continuation guarantee still assumes a noiseless
homogeneous constant-coefficient linear recurrence of order at most \(K\) from
the initial state. Nothing here proves seven physical dimensions, \(P=NP\), a
universal grid/tick, or a theological identification.

## Appendix BQ. Exact-recast completeness before dimensional interpretation

### BQ.1. Bounded question

Appendix BP classified the exact significance values printed in the ATLAS
periodic-spectrum publication. It did not reproduce the detector-level signal
or likelihood. The next bounded question is therefore procedural and
mathematical:

> Do the primary-source ingredients currently instantiated in this project
> uniquely determine an exact reconstructed ATLAS spectrum for one fixed
> clockwork/linear-dilaton tower?

The ATLAS model is genuinely relevant to the dimensional proposal: it is a
five-dimensional clockwork/linear-dilaton construction with a narrowly spaced
tower of spin-2 Kaluza--Klein gravitons parameterized by \(k\) and \(M_5\).
That relevance does not remove the need to calculate the experiment's forward
observer.

The primary experimental source is ATLAS Collaboration, *JHEP* **10** (2023)
079, arXiv:2305.10894v2, last revised 30 November 2023,
https://arxiv.org/html/2305.10894v2. The primary tower reference is G. F.
Giudice et al., *JHEP* **06** (2018) 009, arXiv:1711.08437v2,
https://arxiv.org/abs/1711.08437.

### BQ.2. The forward equation and exact ambiguity

ATLAS expresses the reconstructed contribution of the graviton modes as

\[
N_i^{\mathrm{reco}}=\mathcal L\sum_G
\sigma_G\,\mathcal B(G\to XX)\,
P(M_i\mid M_G)\,(A\epsilon)_G.
\tag{BQ.1}
\]

Here \(\mathcal L\) is luminosity, \(\sigma_G\) the mode production cross
section, \(\mathcal B\) the channel branching fraction, \((A\epsilon)_G\) the
acceptance times efficiency, and \(P(M_i\mid M_G)\) the probability that a mode
of truth mass \(M_G\) is reconstructed in bin \(i\). Thus a list of masses does
not determine \(N_i^{\mathrm{reco}}\).

The executable counterexample makes that underdetermination exact. Hold fixed

\[
(\sigma\mathcal B)_G=(10,6),\qquad (A\epsilon)_G=(0.8,0.5),
\qquad \mathcal L=1.
\tag{BQ.2}
\]

Both response maps below are nonnegative and each truth-mode column sums to
one:

\[
P_A=\begin{pmatrix}1&0\\0&1\\0&0\end{pmatrix},\qquad
P_B=\begin{pmatrix}1/2&1/4\\1/2&1/4\\0&1/2\end{pmatrix}.
\tag{BQ.3}
\]

Equation BQ.1 then gives

\[
N_A=(8,3,0),\qquad N_B=(4.75,4.75,1.5).
\tag{BQ.4}
\]

Both totals equal 11, but their bin shapes differ. Consequently even a fixed
truth tower and fixed efficiency vector do not identify the reconstructed
spectrum without the detector response. Unspecified efficiencies, branching
fractions, or production weights create additional ambiguities.

### BQ.3. Frozen readiness gate

Protocol `EPL-ATLAS-RECAST-COMPLETENESS-2026-09-20-v1` enumerates thirteen
required components: model identity and parameters; mass spectrum; spin;
widths; production rates; channel branching fractions; acceptance and
efficiency; detector mass transfer; observed binned spectra; background bin
expectations; nuisance model and correlations; trained classifier mapping;
and the selections, binning, and luminosity.

Three are ready at the required level. Ten are not yet numerically instantiated
in this engine. In particular, the publication's approximate
\(\sigma\propto M_5^{-3}\) scaling is not a per-mode production table; the
narrow-width approximation is not a frozen width calculation; and a plotted
or truth-level mass sequence is not the response-folded spectrum used by the
analysis. The rule is therefore

\[
\boxed{\texttt{recast\_blocked\_without\_imputation}.}
\tag{BQ.5}
\]

The official HEPData record linked from the ATLAS page was not materialized in
this run because its web challenge blocked the available client. Equation
BQ.5 does not turn that access result into a claim that numerical tables were
not published. After the record is obtained, its actual contents must be
inventoried. An exact ATLAS replay additionally requires the nuisance model and
an executable equivalent of the trained neural-network test statistic; a
separately calibrated replacement would be a new analysis identity.

### BQ.4. Verification, evidence classes, and consequence

Nine production tests verify the frozen contract digest, present gate failure,
a fully ready synthetic pass, exact folding, response non-uniqueness,
column-stochastic validation, domain rejection, and the HEPData access
boundary. Together with the prior tests, all 62 Element Projection Lab tests
pass. An independent standard-library implementation imports no production
module and passes 15/15 checks, reproducing both spectra and every blocking
component identifier.

The evidence classes remain separate:

1. Equation BQ.1 and the construction BQ.2--BQ.4 are written conditional
   mathematics.
2. The completeness inventory and two folds are finite computations.
3. No new collision measurement is produced; Appendix BP's ATLAS summaries
   remain externally measured inputs.
4. A complete CW/LD tower remains a physical hypothesis to be compiled and
   tested, not a discovered cause.
5. No residual is assigned to hidden variables, another dimension, or a
   theological interpretation.

This corrects a recurring inference: model failure can reflect omitted
variables, but it can also reflect a wrong law, wrong response, wrong
likelihood, calibration error, or fluctuation. "Add more strings" is therefore
not the sole permitted repair. Every added variable must enter Equation BQ.1
or another declared observable map and earn predictive credit on data not used
to choose it.

The next bounded test is to materialize and checksum the official HEPData
tables through a non-challenged interface, compile the primary CW/LD equations
into one numerical tower, and rerun this inventory. If the full ATLAS response
or classifier remains unavailable, the honest next object is a separately
calibrated public-spectrum analysis with a new identity--not a claimed exact
recast. Any model replacement must be frozen before new validation data are
opened.

Revision 1.14's complete deterministic closure remains idempotent. Revision
1.15's integer-tick non-crossing limitation, angular-impulse formula, and
finite-prefix generator non-identifiability remain unchanged. The recurrence
learner's \(2K\)-sample continuation guarantee still assumes a noiseless
homogeneous constant-coefficient linear recurrence of order at most \(K\) from
the initial state. Nothing here proves seven physical dimensions, \(P=NP\), a
universal grid/tick, or a theological identification.

## Appendix BT. Complete HEPData contour audit of the unchanged benchmark

### BT.1. Bounded question and provenance

The original YAML archive requested in Appendix BS is now available. This
appendix completes that promised test: keep `k=500 GeV`, `M5=6000 GeV` fixed,
read every curve from the original bytes, and classify the point against each
observed channel/threshold contour. It does not select a new parameter point.
The study is retrospective: the public results and their null interpretation
were known beforehand. It is not a blind preregistration.

The uploaded file is gzip-compressed tar despite its final `.tar` extension.
Its SHA-256 is
`cdfa32efdd4415054aa55554b6e318132d2cde115ef7b19059094efe2f86407e`.
The original upload identity is `libfile_43530ad787688191b67648fe1be65259`.
Element Projection Lab 0.10.0 preserves the identical bytes in `sources/` and
records hashes for each archive member. The record DOI is
[10.17182/hepdata.140955.v1](https://doi.org/10.17182/hepdata.140955.v1).
The primary interpretation is [ATLAS, JHEP 10 (2023) 079, arXiv v2 dated
30 November 2023](https://arxiv.org/html/2305.10894v2), especially Figures 9--10.

### BT.2. Input inventory and geometry

The archive contains one directory plus 25 regular YAML files, totaling
103,527 uncompressed file bytes: submission metadata and 24 contour tables
with 1,978 coordinate pairs. Tables 1, 7, 13, and 19 are observed limits. The
remaining twenty tables are expected medians and expected plus/minus one and
two sigma band curves. They are not additional observations or independent
experiments. Source labels are preserved even where a plus-sigma band has a
smaller M5 coordinate; expected bands are not error bars on the observed curve.

The independent column is `M5` and the dependent column is `k`, both in GeV.
The program preserves vertex order and treats each contour as an open
piecewise-linear curve. For two successive vertices `(M0,k0)` and `(M1,k1)`
bracketing a target k with `k0 != k1`, the intersection is

\[
M_{\rm limit}(k)=M_0+\frac{k-k_0}{k_1-k_0}(M_1-M_0).
\tag{BT.1}
\]

The engine adds no closing edge and performs no sorting, smoothing, convex
hull, or extrapolation. Shared vertices are deduplicated. Multiple distinct
intersections and nonzero horizontal overlaps block a one-boundary verdict.
Eleven horizontal segments occur elsewhere in the supplied curves, but all
24 slices at `k=500 GeV` are unique and have no horizontal overlap.

### BT.3. Checked observed result

The original point lies in the plotted perturbative side `k < M5`. Under the
publication's CW/LD interpretation, smaller M5 at fixed k is the excluded side
of the observed boundary. The unchanged point has the following status:

| HEPData table | Channel | Mass thresholds | Interpolated M5 boundary (TeV) | Boundary minus benchmark (TeV) | Individual observed result |
|---|---|---|---:|---:|---|
| 1 | dielectron | yes | 7.063338 | 1.063338 | excluded |
| 7 | diphoton | yes | 9.815289 | 3.815289 | excluded |
| 13 | dielectron | no | 8.735511 | 2.735511 | excluded |
| 19 | diphoton | no | 10.896422 | 4.896422 | excluded |

Each result inherits the individual published 95% CL procedure. ATLAS analyses
the channels separately because event selections can overlap; threshold
variants also reuse data. This appendix does not multiply confidence levels,
construct a joint likelihood, report a combined p-value, or count the results
as independent experimental replication. The result extends the known Table 1
exclusion to the other three scenarios.

The exact rational evaluations of Equation BT.1 from rounded decimal inputs
are respectively `4661803/660`, `30672778/3125`, `16815858/1925`, and
`4238708/389` GeV. Exact rational arithmetic is numerical bookkeeping; it does
not imply that ATLAS experimentally determined a boundary to those many digits.

### BT.4. Verification and remaining reconstruction gap

All 120 Element Projection Lab unit tests pass, including 37 new archive,
schema, interpolation, ambiguity, provenance and domain controls and the
83 earlier tests. The targeted audit is runnable with `bash run_hepdata.sh`.

The production floating-point importer is checked against a separately
implemented parser that reads decimal YAML scalars directly as exact Fractions
and imports no production code. All 24 intersections, original brackets,
table identities, row counts, classifications, and observed margins agree
through 349 comparisons. The maximum boundary discrepancy is
`1.8189894035458565e-12 GeV`. The independent auditor also passes nine synthetic
geometry controls. This verifies program arithmetic, not the ATLAS likelihood.

Metadata contains references to eight image filenames and one external ATLAS
auxiliary page, none embedded as extra files. There are no event spectra,
background-bin vectors, acceptance/efficiency grids, response matrices,
covariance, classifier weights, or likelihood workspaces in this archive.
This conclusion is limited to the uploaded archive, not a claim of global
non-publication. The prior archive-access blocker is resolved; the exact
detector-level reconstruction gap remains. Historical release reports retain
their then-current access status for provenance.

### BT.5. Evidence classes, correction, and next test

1. The ATLAS observed contours are externally measured, model-dependent
   exclusion results; the project made no new collision measurement.
2. Intersections, archive checks, tests, and independent replay are finite
   computations. Equation BT.1 is an interpolation identity, not a new law.
3. CW/LD remains an assumed physical model. This borrowed benchmark is not
   derived from the project's fixed grid/tick or string-rotation hypotheses.
4. The point is excluded in each evaluated observed scenario. A surviving
   region would mean not excluded, not confirmed or uniquely dimensional.
5. No residual or exclusion is assigned to God or a unique missing variable.

The next bounded test is an explicitly exploratory map of the published
parameter domain with source-coverage, contour-fold and non-perturbative-region
guards. The excluded benchmark stays excluded; this release does not tune it
into a surviving point. Any new predictive model requires its own identity
and genuinely fresh validation after its parameters and observable map are
fixed. Theory 1.70 is preserved as a complete historical snapshot.

Revision 1.14's complete deterministic closure for a fixed transition family
remains idempotent. Revision 1.15's distinction between integer-tick capture
and continuous non-crossing remains intact, as does the angular-impulse formula
`Delta L = r cross J` with no additional time factor. Finite prefixes still
do not identify arbitrary generators. The recurrence learner's 2K-sample
continuation guarantee assumes a noiseless homogeneous constant-coefficient
linear recurrence of order at most K from the initial state. Nothing here
proves seven physical dimensions, P=NP, or a universal physical grid/tick.

## Appendix BU. Guarded exploration of published contour sides

### BU.1. A map is not a new likelihood

Revision 1.72 and Element Projection Lab 0.11.0 implement the bounded follow-up
to Appendix BT. The original source archive and benchmark are unchanged. Four
observed curves are mapped separately, retaining their original vertex order
and open endpoints. The twenty expected curves are not promoted to observed
evidence or combined with these four.

The declared display window is `1000 <= M5 <= 12000 GeV`,
`100 <= k <= 5500 GeV`. Increments of 100 and 50 GeV produce 12,099 nodes per
curve, or 48,396 contour-relative labels. These are computational sampling
choices, not a discovered physical grid or time tick. The window is not a
certified two-dimensional likelihood scan domain; the source archive supplies
curves, not a likelihood grid. Source vertex extrema similarly do not certify
likelihood coverage.

The guard order is explicit: outside chosen window; nonperturbative `k>M5`;
missing or ambiguous contour slice; then below, above, or exactly on a unique
interpolated boundary. At fixed k, the smaller-M5 side is interpreted as the
excluded side only within the source CW/LD model and observed-curve convention.
Above the contour means not excluded by this contour-relative lookup, never
confirmed, independently viable, or uniquely dimensional. Equality is not
silently assigned to either side. `k=M5` is the displayed model boundary, not
a certificate of accurate perturbation theory.

### BU.2. Counterexamples to overconfident mapping

The observed curves contain nine nonzero horizontal segments at five distinct
table/k levels, plus one repeated adjacent vertex of zero length. A repeated
vertex is one intersection, not an ambiguous interval. The regular 50 GeV k
grid hits none of the five genuinely ambiguous levels: passing a regular grid
alone would therefore miss these source features.

At `k=2781.8 GeV, M5=6000 GeV`, Table 1 contains a horizontal overlap and the
engine returns unresolved, not a silently selected branch. The same point is
below Tables 7 and 19 but above Table 13. This is a retrospective diagnostic
point, not a proposed replacement benchmark. At `k=5500 GeV, M5=11000 GeV`,
all four lack tabulated k coverage; missing coverage is not survival. Synthetic
fold, shared-vertex and open-versus-closed-polyline controls test additional
failure modes without claiming these are new measured features.

### BU.3. Verification and evidence class

An independent parser and geometry implementation uses exact Fractions from
source decimal strings and imports no production functions. It agrees on
all 48,396 grid classifications and 2,916 classifications for 729 additional
probe records (712 distinct coordinate pairs, each evaluated in four tables).
The largest discrepancy in serialized boundary coordinates is 3e-12 GeV.
The 1e-8 GeV comparison tolerance applies only to displayed binary-float
coordinates; no tolerance is used in classification comparisons and neither
number represents experimental precision. The full independent run passes
54,586 checks, including 20 synthetic controls.

An initial audit run had a swapped hardcoded channel/threshold metadata
expectation for Tables 7 and 13. It is preserved with its explanation; deriving
those labels from source metadata resolves the audit error without changing
source points, physical parameters, or classification outcomes.

The offline explorer selects precomputed nodes, while the CLI evaluates
off-grid points using the original checked interpolator. The unchanged
benchmark is permanently marked. No joint CL, independent-experiment count,
parameter-survival probability, or optimal replacement point is computed.
The old source parser, physical models, frozen optical parameters and historic
results are unchanged; the complete previous paper is preserved as Theory 1.71.

This contribution is finite arithmetic on externally measured model-dependent
limits, synthetic verification, and display engineering. It is not new measured
evidence, a derivation of CW/LD from the project's fixed-grid hypothesis, or a
unique identification of extra dimensions or theological causes.

### BU.4. Sources and next bounded test

Primary sources remain [HEPData v1](https://doi.org/10.17182/hepdata.140955.v1)
and [ATLAS arXiv:2305.10894v2, 30 November 2023](https://arxiv.org/html/2305.10894v2),
Sections 1, 11–12 and Figures 9–10. The figure frames are displays, not a new
declaration of likelihood coverage. No newer collision measurement is claimed.

Next: implement an explicitly labelled detector-response interface with an
injected-signal closure test. An assumed toy response must remain distinct from
ATLAS calibration; the real-data recast gate stays closed until production,
response, efficiency and likelihood inputs are instantiated. Any new physical
prediction must be frozen before fresh validation.

Earlier corrections remain unchanged: complete deterministic closure for a
fixed transition family is idempotent; integer-tick capture does not establish
continuous non-crossing; angular impulse is `Delta L = r cross J` with no extra
time factor; finite observed prefixes do not identify arbitrary generators.
The recurrence learner's 2K continuation guarantee requires a noiseless
homogeneous constant-coefficient linear recurrence of order at most K from the
initial state. No seven physical dimensions, P=NP or universal grid is proved.

## Appendix BV. Explicit detector losses and source–efficiency non-identifiability

### BV.1. Observation map and declared synthetic experiment

This appendix executes Appendix BU's proposed detector-response closure test.
It adds a new software interface, not a measured ATLAS response. For truth
mode j and observed bin i, let R_ij be the probability that an event from j is
reconstructed in i, including efficiency and the finite observed window. Thus
R_ij is nonnegative and each column sums to at most one. Define the missed
probability q_j = 1 - sum_i R_ij. For nonnegative truth expectations t_j,

\[
s_i=\sum_j R_{ij}t_j,\qquad
N_{\mathrm{missed}}=\sum_j q_jt_j,\qquad
\sum_i s_i+N_{\mathrm{missed}}=\sum_j t_j.
\]

The last equality follows by summing the completed columns. It is a written
accounting identity; numerical conservation is checked to roundoff. Missed
events are not redistributed into the observed spectrum, nor treated as an
observed bin with a fabricated zero. The earlier conditional migration matrix
with separately supplied acceptance/efficiency remains unchanged and valid.

The locally frozen protocol is `EPL-SYNTHETIC-RESPONSE-CLOSURE-v1`, SHA-256
`41310f7c9d9390b65ea9363e6ec8da6190b00f73cb1150233dc461ad966e63cf`.
Its 102 line masses are the unchanged Appendix BR benchmark values. The
synthetic response integrates a Gaussian of standard deviation 0.02 m_j into
92 bins from 400 to 5000 GeV, with pre-window efficiency 0.8. Equal truth weights
sum to 500 expected events at unit strength. Background is fixed at 5/bin.
The line weights are not calculated hadronic production rates, and natural
widths are not used in this narrow-line toy. No detector parameter is measured
or fitted from experimental data here.

At unit strength the forward sum gives 396.970157301805 reconstructed and
103.029842698195 missed events. The original k=500 GeV, M5=6000 GeV benchmark
remains excluded under each previously audited observed contour. Using it as
a software fixture does not make it experimentally viable.

### BV.2. Fixed-background fitting and exact ambiguity

Let observed counts be independent Poisson variables with means
lambda_i(mu) = b_i + mu s_i, for known positive b_i, a fixed nonzero
nonnegative template s, and nonnegative strength mu. Up to count-dependent
constants, the log likelihood and score are

\[
\ell(\mu)=\sum_i[n_i\log(b_i+\mu s_i)-(b_i+\mu s_i)],\qquad
U(\mu)=\sum_i s_i\left(\frac{n_i}{b_i+\mu s_i}-1\right).
\]

The derivative U'(mu) = -sum_i n_i s_i^2/(b_i+mu s_i)^2 is nonpositive.
Therefore U(0)<=0 puts the maximum at zero; otherwise the engine brackets the
interior score root and solves it by bisection. This is a fixed-shape,
fixed-background nonnegative MLE, not a detector-nuisance fit or a discovery
test. Mean-data (Asimov) checks use nonnegative real counts; sampling checks
use integer Poisson counts.

For any positive c such that R'=cR remains an admissible response,

\[
\boxed{b+(\mu/c)R't=b+\mu Rt.}
\]

The two parameter descriptions thus define the identical entire observed
Poisson distribution. If source yield and overall efficiency are both free,
these observations identify only their product. This is structural
non-identifiability even with unlimited counts from the same observation map,
not simply low power in a small experiment. Independent calibration or another
identifying constraint is needed.

For c=0.8, replacing efficiency 0.8 with 0.64 raises every nonnegative fitted
strength by a factor 1.25 while preserving fitted bin means and likelihood.
A perfect spectrum fit therefore does not prove that the assumed detector or
underlying source strength is correct. Likewise, adding an unconstrained
hidden variable can preserve a fit without selecting its physical cause.

### BV.3. Checked finite results and negative control

The protocol was written before these synthetic draws; it is not a blind
preregistration of a physical discovery. Strengths 0, 1 and 2 each receive
2,000 histories using the specified PCG64 seed 202609210173. Each history is
fitted using the correct response, the efficiency-scaled response, and a
deliberately incorrect response whose observed columns were renormalized to
one. The last control deletes inefficiency/window losses and is never the
release default.

| Injected strength | Mean fit, correct response | Mean fit, efficiency scaled by 0.8 | Mean fit, columns incorrectly normalized |
| --- | ---: | ---: | ---: |
| 0 | 0.020766 | 0.025957 | 0.016563 |
| 1 | 1.003297 | 1.254121 | 0.797039 |
| 2 | 2.003607 | 2.504508 | 1.591424 |

Mean-data injection at strength 1 yields exactly 1 and 1.25 for the first two
fits, with zero deviance. The renormalized response yields 0.794444422585 and
deviance 0.210515352445. These are software controls, not measured spectra or
calibrated goodness-of-fit probabilities.

For strength 1, the correct-response sample standard deviation is 0.072991
and Monte Carlo standard error is 0.001632; its mean 1.003297 is retained
unchanged and does not establish exact unbiasedness. Under zero strength,
997/2000 correct-response fits are zero. Positive constrained estimates in
the other histories are not detections: no significance or coverage rule was
specified. Across all 6,000 histories, the correct and efficiency-scaled fitted
bin means agree within 1.43e-14 and deviances within 8.53e-14.

All 192 unit tests pass (162 previous and 30 new). An independent implementation
imports no production module and uses scalar math.erf integration, math.fsum
summation and safeguarded Newton fitting. It passes 1,231,728 checks, replaying
9,384 response entries, 552,000 counts and 18,000 fits. Maximum matrix difference
is 1.39e-16 and maximum fitted-strength difference is 7.96e-13. Count replay uses
the same specified PCG64 algorithm; it checks reproducibility, not independent
randomness. All raw synthetic arrays, summaries, code hashes and audit results
are preserved. The full 1.72 paper and earlier result files are retained.

### BV.4. Sources, remaining limits and next test

This response convention follows [Koch, revised 21 March 2019](https://arxiv.org/abs/1903.06568v2).
[Zhu et al., revised 27 January 2026](https://arxiv.org/abs/2509.00892v2)
address finite response-estimation uncertainty, distinct from model
misspecification; that method is relevant but not implemented here. The
[ATLAS publication](https://arxiv.org/html/2305.10894v2) remains a source of
the previously audited negative measured result, not of this toy calibration.

The current real-data recast remains blocked. The old v0.7 readiness counts
are retained explicitly as a historical snapshot, not represented as a fresh
inventory. Synthetic response closure does not supply calibrated efficiencies,
hadronic production, backgrounds, nuisance correlations or classifier
likelihood. No measured excess, confidence level, physical extra dimension,
universal grid/tick or theological identification follows from this test.

Next: add a separate synthetic efficiency-calibration sample and derive how
its finite uncertainty constrains the yield–efficiency ambiguity. Freeze that
new study before fresh validation; do not tune it on this release's histories.
All earlier mathematical corrections and recurrence model-class assumptions
remain unchanged.

## Appendix BW. Primitive crystal cells, projected views and sunlight color

### BW.1. Smallest repeating pattern versus smallest stable object

The user-directed crystal branch represents a selected periodic phase by
three primitive lattice vectors and its atomic basis. With vectors as rows
of A, fractional site f and integer translation n, positions are

\[
\mathbf r=(\mathbf n+\mathbf f)A.
\]

This specifies a repeated crystal, not a finite cluster's stability. In a
monatomic FCC crystal the primitive basis contains one atom; the conventional
cubic cell contains four. The eight corner images of the primitive cell have
one-eighth sharing weights and therefore one atom total. The repeated
environment, rather than the corner picture alone, defines the bulk material.
See the [IUCr primitive-cell definition](https://dictionary.iucr.org/Primitive_cell).

All 118 identities are retained. The pinned official ASE reference compilation
provides enough information for 67 audited primitive cells: 37 with one basis
atom, 29 with two and one with three. Another 51 entries remain unresolved:
10 explicitly missing bases, 6 ambiguous phase/basis entries, 5 isolated
molecular/atomic references, 29 absent references and 1 unsupported structure.
Missingness is not interpreted as inability to crystallize.

The reference values are rounded and lack datum-specific temperature,
pressure and uncertainty. Carbon selects diamond, not graphite. Noble-gas
crystals and solid mercury refer to cooled phases; helium requires pressure
as well. Hydrogen and oxygen molecular references are not silently converted
to monatomic cells. Complete conditions and source caveats accompany each
record. AFLOW prototypes corroborate the vector/basis constructions, while
the numerical constants remain the explicitly separate ASE compilation.

### BW.2. Optical observation requires an electronic response

The new optical layer uses the measured Johnson–Christy complex indices
N(lambda)=n(lambda)+i k(lambda) for eleven elements: Au, Ag, Cu, Ti, V, Cr,
Mn, Fe, Co, Ni and Pd. It computes normal-incidence power reflectance from air:

\[
R(\lambda)=\frac{(n-1)^2+k^2}{(n+1)^2+k^2},\qquad
(X,Y,Z)=\frac{\int S R(\bar x,\bar y,\bar z)\,d\lambda}
{\int S\bar y\,d\lambda}.
\]

S is either a 5772 K Planck approximation to sunlight or official CIE D65
standard daylight. Official tabulated CIE 1931 two-degree functions are
integrated over 380–780 nm at 1 nm spacing. The source white is calculated
separately; Bradford adaptation to D65 and sRGB conversion produce a display
swatch. Raw XYZ and unclipped linear RGB remain available. Gold exceeds the
display gamut slightly, so the hex code is not an exact intrinsic color.

| Material | Primitive type / atoms | R(450 nm) | R(650 nm) | Adapted solar display |
| --- | --- | ---: | ---: | --- |
| Gold | FCC / 1 | 0.408194 | 0.956522 | #FFDEA3 |
| Silver | FCC / 1 | 0.980236 | 0.989834 | #FEFDFC |
| Copper | FCC / 1 | 0.538258 | 0.935218 | #F8CFBF |

Their normalized primitive Gram matrices agree exactly in this construction,
but their reflectances differ. This rejects a color rule based only on FCC
topology; their angstrom lattice constants are not identical. The optical
indices are measured inputs, not predictions obtained from those constants
or from higher-dimensional rotation. Reconstructing familiar appearances
does not supply held-out empirical validation of a new electronic theory.

The film measurements are applied as effective scalar responses to an ideal
clean, smooth, optically thick planar surface. The transition-metal samples
were polycrystalline. They do not determine every single-crystal dielectric
tensor or finite-cluster color. Oxidation, roughness, coatings, finite size,
multiple scattering, fluorescence, atmospheric solar modification and
individual observer differences are outside the calculation.

### BW.3. Two geometric checks

First, for FCC the largest spacing associated with a nonzero reciprocal-lattice
vector is d111=a/sqrt(3). First-order free-space Bragg diffraction requires
lambda=2d sin(theta), and hence lambda<=2d. The maxima are 0.471118 nm for Au,
0.472273 nm for Ag and 0.416847 nm for Cu. Visible wavelengths here start at
380 nm. Ordinary diffraction from these atomic planes therefore does not
explain the calculated visible colors. This does not remove the role of
periodicity in electronic bands or exclude structural color from much larger
photonic patterns.

Second, the optional 4D shadow has an explicit, limited definition:

\[
(x,y,z,0)\longmapsto
(x\cos\theta,y,z,x\sin\theta)\longmapsto
(x\cos\theta,y,z).
\]

The first map preserves the four-dimensional norm, and the projection can
discard information. At 90 degrees, distinct x coordinates share one 3D
shadow. This is a mathematical display convention, not an inferred hidden
physical motion. It neither changes nor predicts the optical constants.

### BW.4. Validation, provenance and next test

All 218 unit tests pass, including 26 new tests. A finite primitive-basis
check tests every candidate translation taking one basis site to another;
all 67 supplied cells lack a redundant translation, while a deliberately
nonprimitive BCC description is detected. Boundary sharing and 27-cell
repetition are checked independently of the visual impression.

An independent implementation reconstructs geometry from the raw ASE snapshot
and optics from the original numerical tables, importing no production
module. It checks 118 identities, 67 cells, 31 nontrivial translation
candidates, 8,822 interpolated n/k values, 4,411 reflectances and 22 colors.
Maximum reflectance disagreement is 6.66e-16 and maximum color-component
disagreement is 5.22e-15. Fifteen optical-source hashes and four official CIE
checksums pass. This is implementation verification, not independent
experimental replication. The UI passes 633 Canvas/interaction checks;
full-browser CSS validation remains unperformed. The static poster and
Canvas previews were visually inspected.

The complete previous paper, old source/results and frozen optical analysis
are preserved. The new observer tabulations are a separately versioned layer.
Sources include [ASE data](https://docs.ase-lib.org/ase/data.html), pinned
commit c77931fd38c5fdaa560bc204488fad5e4c50f4c8 (18 September 2026),
[AFLOW FCC](https://aflow.org/p/A_cF4_225_a-001/),
[Johnson–Christy 1972](https://doi.org/10.1103/PhysRevB.6.4370),
[Johnson–Christy 1974](https://doi.org/10.1103/PhysRevB.9.5056),
[CIE observer](https://cie.co.at/datatable/cie-1931-colour-matching-functions-2-degree-observer)
and [CIE D65](https://cie.co.at/datatable/cie-standard-illuminant-d65).
Raw source data, metadata, checksums and reuse notices are bundled.

Next: specify an electronic Hamiltonian and response calculation for one
chosen phase and surface, freeze its predicted dielectric function, then test
an independent reflectance measurement without fitting target colors. Until
then the engine reconstructs bulk colors from measured response; it does not
derive them from the proposed dimensional geometry. The detector-calibration
task remains open. Earlier closure, angular-impulse, non-crossing and recurrence
model-class corrections remain unchanged, as do the ATLAS exclusions. No
new physical dimension, universal tick, P=NP or theological cause is inferred.


## Appendix BX. Complete element records and source-qualified crystal/optical coverage

Date: 21 September 2026. This release expands the current crystal table using
source-backed data and keeps every earlier geometry and optical input intact.

## Coverage

| Layer | Previous | Current |
|---|---:|---:|
| Selectable elements | 118 | 118 |
| Typed general-property fields per element | — | 15 |
| Populated general-property values | — | 1,544 / 1,770 |
| Complete selected reference crystal cells | 67 | 90 |
| Additional complete phase options | 0 | 2 |
| Elements with visible optical constants | 11 | 30 |

The 226 null property fields include 34 elements for which CIAAW assigns no
standard atomic weight; these are marked **not defined**, not zero mass or an
ordinary missing measurement. Other blanks mean this source snapshot does not
supply a value. All 118 identities and property records are present, but this
is not a claim that all properties of all elements have been measured.

The 15 fields are standard atomic weight, reported atomic mass, electron
configuration, electronegativity, van der Waals radius, first ionization
energy, electron affinity, oxidation states, reference state, melting point,
boiling point, density, category, discovery, and CPK illustration color.
The last is explicitly not optical color and never enters the optical model.

## Reference sources and their limits

[CIAAW Standard Atomic Weights 2024](https://ciaaw.org/atomic-weights.htm)
supplies dimensionless recommended values; isotope intervals and uncertainties
are preserved verbatim. [PubChem's periodic table](https://pubchem.ncbi.nlm.nih.gov/periodic-table/)
supplies the other reference fields. Both snapshots were retrieved on
21 September 2026 and are bundled with hashes. The compact PubChem export does
not consistently specify pressure, temperature, phase or measurement
uncertainty. Consequently source-reported thermal and density values are not
automatically certified ambient-pressure measurements. Explicit predictions,
helium/carbon/arsenic thermal cautions, and Lr/Ds/Rg/Og source conflicts are
retained. The old engine electron configurations are not overwritten.

New selected complete atomic structures are **He, B, N, O, F, P, S, Cl, Mn,
Ga, Se, Br, Te, I, La, Pr, Nd, Ra, Pa, U, Np, Pu and Am**. Raw CIFs,
experimental citations, metric parameters, full symmetry-expanded coordinates,
conditions, and source caveats are archived in `data/completion_*`.
These include alpha-boron's 12-site primitive motif, nitrogen's 8 sites,
sulfur's 32 sites, alpha-manganese's 29 sites, and alpha-plutonium's 16 sites.
The selected alpha-boron phase is an explicit allotrope, not an assertion
that it is the unique stable room-temperature form.

Graphite is an experimental-reference alternative to the existing diamond
carbon cell. Hydrogen has a separate **DFT-predicted C2/c phase at 300 GPa**,
from [Pickard and Needs (2007)](https://doi.org/10.1038/nphys625); this does not
increase experimental element coverage. The main phase-I hydrogen record
contains a sourced molecular-center lattice. Its orientational disorder does
not justify inserting fixed atomic orientations. The two molecular centers
are not represented as two hydrogen atoms.

The remaining 28 selected atomic-coordinate gaps are H, Pm, At, Rn, Fr, Cm,
Bk, Cf, Es and Fm through Og. For Pm/Rn/Cm/Bk/Cf/Es, the primary numeric-source
ingestion gap is distinguished from an unmeasured physical property. No
fictitious atomic basis is assigned to short-lived superheavy elements.

Two incomplete COD Pr/Nd transcriptions omitted the 2a sites, giving half
the correct site density. This release uses complete 2a+2c dhcp bases from
the independently corroborated records at the same metrics; the rejected
transcriptions and an explicit origin-shift comparison remain archived.
An obsolete chlorine prototype is likewise not silently used in place of the
22 K experimental molecular structure.

## Geometry and display

Conventional single-element cells are reduced only by translations that
actually permute every atomic site. The reduction verifies the lattice index,
site multiplicities, periodic round trips and unchanged number density. A
left-handed-input metadata error found during the independent audit was
corrected: the reported transformation includes the initial axis permutation.
No atomic displacement, guessed basis or forced symmetry is introduced.

The default crystal-motif view shows the full primitive basis for complex
and molecular crystals, and local neighbors for simple structures. Local,
primitive and 27-cell views remain available. The local cutoff is 1.10 times
the shortest site distance from basis site 0; lines are distance guides, not
chemical bonds. Inequivalent sites may have different environments. The full
lattice matrix and fractional coordinates are inspectable. A primitive cell
is a repeating unit, not the smallest stable isolated crystallite. Fractional
sites at opposite boundaries reconnect through periodic copies.

The 4D mode remains an explicitly defined embedding and mathematical
projection. These source coordinates do not establish a physical extra
dimension or a unique hidden trajectory.

## Optical expansion

The 19 additions are **Al, Bi, Cs, Ge, K, Li, Mg, Mo, Na, Nb, Pb, Pt, Rb, Rh,
Si, Ta, W, Zn and Zr**. Raw tabulations are pinned to refractiveindex.info
database commit `c5c2f188e848453def5970e347399d653df2ffc2` (4 September 2026).
The original 11 optical records are unchanged. Primary references include
[Aspnes and Studna (1983)](https://doi.org/10.1103/PhysRevB.27.985),
[Werner et al. (2009)](https://doi.org/10.1063/1.3243762), and
[Palm et al. (2018)](https://doi.org/10.1021/acsphotonics.8b01243).

Each accepted source spans 380–780 nm, avoiding extrapolation. The source
samples and inference methods differ: ellipsometry, reflectance or REELS
derived constants, prepared films, and a compiled experimental Rh tabulation
are labeled separately. Si/Ge samples are lightly doped. Scalar inputs for
anisotropic materials are not orientation-resolved dielectric tensors.
Identical duplicate ultraviolet rows in Nb/W were removed only from the
normalized interpolation arrays; original files and all differing values
remain unchanged. Conflicting duplicate wavelengths would fail ingestion.

The existing Fresnel/CIE pipeline reconstructs ideal smooth, opaque,
semi-infinite surface reflection under a 5772 K Planck approximation or D65.
It does not predict optical constants from a crystal shape. It omits oxide,
roughness, coatings, finite-cluster response and sample-specific uncertainty.
Changing the displayed phase does not create phase-resolved optical data.

## Verification and next work

All **259 unit tests passed**, including the 228 previous tests. Independent
raw-CIF reconstructions checked 10 light-region phase records (including the
two alternatives), nine middle-region cells and six heavy-region cells.
The middle audit additionally confirms 1,793 periodic site comparisons in the
integrated data. Independent scalar optics reproduced **7,619 reflectances,
15,238 interpolated n/k values and 38 Sun/D65 colors**. All hex colors matched;
maximum reflectance discrepancy was 6.67e-16. All 11 original optical records
and 67 original crystal geometries remained unchanged.

The actual view script passed **14,105 interaction and data checks**. Native
Canvas motif/comparison previews and the scientific poster were inspected.
Full browser CSS/layout QA was unavailable and is not claimed. The complete
JSON, raw inputs, scripts, independent audits, source hashes, old results and
previous paper remain in the runnable package.

Next: retrieve the remaining primary numeric crystal records for Pm, Rn and
transplutonium metals; then add phase-matched optical constants and uncertainty
before comparing reconstructed colors against independent measured spectra.
This release compiles external measurements and performs checked computation;
it adds no new physical measurement and does not close earlier detector,
fixed-grid, identifiability or dimensional-evidence gaps.


## Appendix BY. Crystal interference, finite size and exact inverse ambiguity

21 September 2026. The new observer calculates constructive and destructive
interference from the frozen crystal coordinates. Earlier atomic properties,
crystal phases, optical constants and color reconstructions are unchanged.

## Forward model

For row lattice matrix A and fractional sites b_j, reciprocal row vectors are
B=A^(-T), measured in cycles/Å. An integer reflection h has q=hB and spacing
d=1/|q|. Identical unit-strength point scatterers give the normalized amplitude

    F(h) = (1/N) sum_j exp(2πi h·b_j).

The geometric weight is |F|². Powder shells sum that weight over equal-|q|
directions, including their multiplicity. Bragg accessibility requires
λ|q|≤2, and 2θ=2asin(λ|q|/2). The forward beam is excluded. These are
established [IUCr structure-factor](https://dictionary.iucr.org/Structure_factor)
and [reciprocal-lattice](https://dictionary.iucr.org/Reciprocal_lattice)
relations applied to the stored reference cells.

The finite enumeration is complete within its declared sphere because
h_i=q·a_i implies |h_i|≤Q||a_i||. Boundary comparisons use a relative 1e-12
tolerance. Integer h=(0,0,0), rather than an arbitrary small-q threshold,
identifies the forward beam. The independent audit found and corrected two
extreme-scale cutoff errors before release: dropping small nonzero q and
using an absolute tolerance that became larger than a tiny requested sphere.
An explicit enumeration limit rejects oversized requests without truncation.

The release computes **56,656 reciprocal points in 92 phase records**, of
which **4,766** are numerically suppressed at the stated 1e-12 intensity
tolerance. There are **5,962 powder shells**, with **5,229 nonzero shells**.
Numerical suppression for rounded source coordinates is not a general exact
symbolic proof. Exact analytic BCC/FCC/diamond extinction rules are tested
separately. The 92 records comprise 90 selected cells, graphite as an
additional experimental-reference phase, and one explicitly predicted
high-pressure hydrogen phase. Missing atomic coordinates remain missing.

The interactive observer selects a crystal, radiation wavelength and
reflection. It shows the powder interference weights and the complex
contribution from each basis site. For example, diamond-silicon basis vectors
cancel at primitive reflection (1,1,0). Miller indices refer to the **stored
primitive basis**, not automatically to conventional cubic indexing.

## Published reference check

[NIST SP260-245, February 2024](https://nvlpubs.nist.gov/nistpubs/SpecialPublications/NIST.SP.260-245.pdf)
certifies SRM640g silicon's powder lattice parameter at 22.5°C as
5.431109 ± 0.000080 Å (expanded uncertainty, k=2). Its eleven tabulated angles
are **computed, non-certified information values**, using λ=1.5405929 Å.
The separate SRM benchmark reproduces all eleven within their 0.001°
rounding: maximum difference **0.000435606927°**. This checks units, reciprocal
geometry and Bragg conversion. It is not eleven independent measured-peak
confirmations and does not validate relative intensities.

The raw publication, its 22 powder lattice refinements, source transcription,
conditions and hashes are bundled. The old atlas Si cell uses rounded
a=5.43 Å; using it for the same comparison shifts a peak by as much as
**0.0596987303°**. That record is preserved. A generic rounded lattice must
not be silently treated as the calibration specimen, and the difference
does not establish new physics. The certified lattice interval is separately
mapped through the monotone angular formula; table rounding is not an
experimental uncertainty estimate.

## Finite crystal interference

For N_i repetitions along each row a_i, the finite amplitude factors into the
basis amplitude and three finite geometric sums. The implementation uses
stable sinc ratios and checks them against explicit atomic sums. With a
one-dimensional 1 Å chain, the first zero beside the q=1 Å^-1 peak occurs at
an offset 1/N Å^-1: longer chains sharpen the interference peak.

Changing primitive basis usually changes a finite parallelepiped's boundary
atoms. Finite clusters are equivalent only when their actual Cartesian point
sets agree. Global translations are preserved without independently wrapping
boundary atoms. The primitive/conventional equivalence test applies to the
infinite periodic Bragg pattern, not arbitrary equal-repeat finite clusters.

## Exact ambiguity that the engine must preserve

Consider two equal-scatterer one-dimensional arrangements:

    A = {0, 1, 4, 10, 12, 17}
    B = {0, 1, 8, 11, 13, 17}.

They are not translations or reflections of one another, but their ordered
pair-difference multisets are exactly equal. Expanding

    |sum_j exp(2πi q x_j)|² = sum_jk exp(2πi q(x_j-x_k))

therefore proves their intensities agree for **every real q**. The integer
multiset check and the algebra are distinct from the finite floating-point
replay; the latter agrees to roughly 1e-14. More intensity samples alone
cannot distinguish this pair within the declared model. This is an explicit
homometry control, consistent with [Grimm and Baake (2008)](https://arxiv.org/abs/0808.0094).
Embedding these points on a line in 3D preserves the same ambiguity.

## Scope and verification

The model is elastic, coherent, kinematic scattering from identical points.
It omits element-dependent atomic form factors, thermal displacement,
absorption, multiple scattering, texture, background and instrument response.
Consequently the displayed weights are **geometry-only interference weights**,
not calibrated detector counts or material-color predictions. A real
diffraction measurement can require physics that the point model omits.

All available phases have no nonzero atomic Bragg shell accessible in the
stated 380–780 nm visible band. This follows from the independently checked
380 nm bound and monotonic wavelength accessibility. It does not imply no
visible reflection, absorption or scattering; electronic optical response
remains the separate existing color calculation.

All **272 unit tests pass**, including the previous 259. A separately written
scalar audit recomputes every reciprocal point using cross-product reciprocal
vectors and larger enumeration boxes. Maximum normalized intensity difference
is **8.80e-15**. It also checks **3,993 analytic extinction cases**, **132
explicit finite-cluster cases**, origin/rotation/scale invariance,
unimodular cell changes, primitive/conventional equivalence, and the exact
homometric counterexample. Audit hashes bind the source atlas, implementation
and calculated output. The view passes **48,786 DOM/SVG interaction checks**,
retaining all 92 phases and 5,962 shells and matching every basis phase sum
against the engine. Full browser CSS/layout QA was unavailable after the
browser download failed; the deterministic fixture uses approximate text
metrics. Its results and limitations are recorded separately in
`outputs/diffraction_view_smoke.json`.

These results comprise written conditional mathematics, finite computation,
and use of external certified measurements. They establish neither new
measurements by this project nor a unique hidden geometry, extra physical
dimension or theological cause. Existing closure, recurrence-class,
continuous-collision, angular-impulse and identifiability corrections remain.

Next bounded test: add element-dependent scattering factors and an explicit
instrument response, then reserve an independent measured powder intensity
profile for validation. For the demonstrated homometric pair, test a specified
phase-sensitive reference measurement; unreferenced intensity alone cannot
resolve the proven ambiguity.

## Run

`bash run_crystal_diffraction.sh` rebuilds numerical outputs, audits and
figures. Open `outputs/crystal_wave_observer.html` for the standalone observer.
The original periodic-table launcher and files remain available unchanged.
See `DIFFRACTION_PROTOCOL_v1.json` for the declared numerical choices and
`DIFFRACTION_MATHEMATICAL_AUDIT.md` for the detailed proofs and assumptions.


---

## Appendix BZ. Reestablishing the foundation: memory, dynamics and a discrete Coulomb benchmark

Revision 1.77, 21 September 2026. Fixed Grid String Lab 0.2.0 adds a bounded foundation test to its unchanged earlier integer-string and direct-tick modules. It starts from the saved revision-1.76 paper and current claim ledgers, including Rope Tomography Lab 0.14.0 and RRLS audited extension 2. Those engines, Hydrogen Matrix Lab 0.8.0 and Element Projection Lab 0.15.0 are unchanged. No new physical measurements or measured validation are used.

### BZ.1 Corrected claims and counterexamples **[M/C]**

Section 4.2 now defines minimum fixed-width memory using the actual maximum of future-class counts across observation fibers. The two exact finite examples in that section are implemented by partition refinement and independently verified by exhaustive paired future trajectories. They refute the previous substitution of an arbitrary upper bound for an exact minimum. Section 4.1 now distinguishes a weighted one-step conditional from Markov closure; a uniform-phase four-cycle with observation sequence 0011 supplies an exact rational counterexample. Section 16.1 now compares joint ensemble observation records rather than substituting individual-state equalities for distributional matching. These changes preserve the deterministic descent and idempotent complete-closure results.

### BZ.2 One defined bound-state model **[M]**

In atomic units, with an infinitely massive point proton and S-wave radial reduction, declare

\[
(H_au)_j=-\frac{u_{j+1}-2u_j+u_{j-1}}{2a^2}-\frac{u_j}{ja},
\quad j\ge1,\quad u_0=0,\quad a>0.
\]

This is a bounded self-adjoint operator on \(\ell^2(\mathbb N)\). Its negative bound-state energies are exactly

\[
E_n(a)=-\frac{1}{n^2\left(1+\sqrt{1+(a/n)^2}\right)},\quad n=1,2,\ldots.
\]

The accompanying `FOUNDATION_DERIVATION.md` proves this with a generating function: square summability forces the interior singularity to be removable, giving \(\sinh\kappa_n=a/n\). It includes normalized ground-state coefficients, excited-state coefficients, finite-box variational bounds and the limit

\[
E_n(a)=-\frac1{2n^2}+\frac{a^2}{8n^4}+O(a^4).
\]

Finite-difference S-state hydrogen already has primary research precedent in [Wall, published 15 August 1986](https://www.pnas.org/doi/10.1073/pnas.83.16.5753). The generating function relates to [NIST DLMF 18.23.4](https://dlmf.nist.gov/18.23.E4). These references establish prior context, not physical confirmation of this implementation or endorsement of Wall's proposed relativistic interpretation. The displayed Hamiltonian and local derivation fix the conventions used here.

The potential, Hilbert amplitudes, Schrödinger evolution and Born probabilities are imported assumptions. A radial mesh is not a translationally invariant three-dimensional physical lattice. This calculation does not derive electromagnetic coupling, photon dynamics or hydrogen from higher-dimensional strings.

### BZ.3 Frozen finite computation **[C]**

`FOUNDATION_PROTOCOL_0.2.json` was frozen before the numerical run; its SHA-256 is `af00364dd565503bb6c3fb256eb7c36e16e00f7bb04aac77a5b624bf6eda619a`. The spectrum reference string contains one redundant closing parenthesis as a cosmetic text typo; the equations above, executable formula and derivation give the unambiguous intended expression. The frozen bytes are retained.

The test retains the four lowest levels at spacings \(a=0.4,0.2,0.1,0.05\), with outer Dirichlet walls at \(R=80,160\), giving eight matrices and 32 eigenvalues. The primary spacing is \(a=0.2\), with an equal coherent superposition of its lowest four states and \(\tau=0.1\) atomic time units. No parameter is fitted.

| Checked quantity | Result |
|---|---:|
| Maximum production/reference energy difference at R=160 | \(4.75\times10^{-14}\) Hartree |
| Maximum eigenvector residual | \(1.35\times10^{-13}\) |
| Maximum production/independent Sturm eigenvalue difference | \(4.59\times10^{-14}\) Hartree |
| Unit tests, including retained older tests | 23 passed |
| Independent audit assertions | 102 passed, 0 failed |

The smaller box retains fourth-level errors around \(1.22\)–\(1.29\times10^{-9}\) Hartree, demonstrating why box and spacing errors are separate. These finite differences are diagnostics, not certified uniform domain-error bounds.

The dimensionless diagnostic \(D=(E_3-E_1)/(E_2-E_1)\) tends to \(32/27\). At primary spacing the finite-matrix value is 1.186807500408563. Relative half-line departures from the ideal continuum value are approximately 5251.58, 1368.83, 345.95 and 86.73 ppm for the four decreasing spacings. Their convergence checks numerical behavior. It does not establish that discretization error is a new physical effect.

All retained states are S states; their energy gaps alone are not predictions of observed single-photon electric-dipole emission, whose S-to-S transitions are forbidden. The existing primary [1S–3S two-photon measurement by Fleurbaey et al., 2018](https://arxiv.org/abs/1801.08816) illustrates why an actual spectroscopy comparison needs explicit interaction and precision modeling. No value from that measurement is used for fitting or validation here; already inspected references are not a future blinded holdout.

### BZ.4 What fixed ticks do and do not show **[M/C]**

For the same finite Hermitian matrix \(H\), define \(U_\tau=e^{-i\tau H}\). Then

\[
U_\tau^m=e^{-im\tau H}
\]

exactly. Thus the tick model is observationally identical at integer ticks to sampled continuous-time evolution of that matrix. This is not exact continuum-space hydrogen. Numerical agreement cannot identify a fundamental clock.

The 129 sampled packet states have maximum norm drift \(8.99\times10^{-15}\) and maximum discrete/sample amplitude difference \(1.30\times10^{-15}\). Adding \(2\pi/\tau\) to one energy preserves all tick phases, while changing sub-tick evolution: the tested half-tick packet fidelity is 0.25. The aliased generator need not remain a nearest-neighbor Coulomb Hamiltonian; the ambiguity concerns unrestricted generators. Independently evaluated dense exponentials and matrix powers reproduce tick equivalence. A matrix exponential is generally dense, so no strictly one-site-per-tick propagation law follows from the local Hamiltonian stencil.

### BZ.5 Foundation status and the next gate **[H/E boundary]**

The supported result is a corrected claim framework plus a defined, independently checked mathematical benchmark. It supplies no new measured evidence for extra spatial dimensions, a physical grid or a fundamental tick. Prior failed benchmarks, projection ambiguities, optical-data provenance, and exclusion-contour results remain in force. RRLS's recurrence continuation guarantee still requires a noiseless homogeneous constant-coefficient recurrence of order at most K from the initial state; integer-tick capture still does not enforce continuous non-crossing, angular impulse has no extra time multiplier, and complete deterministic closure for fixed laws remains idempotent. No unconstrained residual is identified with God, and no seven-physical-dimension or P=NP claim is adopted.

Revision 1.78 executes the angular-momentum-1 and dipole baseline proposed here; Appendix CA records its outcome. It tests a measurement coupling within the imported quantum model and does not derive that coupling from strings. A physical promotion test still must predict a distinguishable observable after standard reduced-mass, relativistic, radiative and nuclear corrections, with a frozen uncertainty/decision rule before unused measured validation. If an extension only changes latent coordinates while leaving every observation probability unchanged, it must be recorded as equivalent rather than as a discovery.

---

## Appendix CA. Hydrogen light coupling: the frozen 1S→2P baseline

Revision 1.78, 21 September 2026. Fixed Grid String Lab 0.3.0 starts from the saved 0.2.0 engine and full revision-1.77 paper. It adds one bounded light–matter benchmark and leaves Rope Tomography Lab, RRLS, Hydrogen Matrix Lab and Element Projection Lab unchanged. The complete 0.2.0 files and paper are preserved under `history/pre_0.3.0`. No measured wavelength is fitted or scored.

### CA.1 Frozen model and exact targets **[M]**

In nonrelativistic atomic units, separate \(\psi_{lm}=u_l(r)Y_{lm}/r\) and declare

\[
h_l=-\frac12\frac{d^2}{dr^2}+\frac{l(l+1)}{2r^2}-\frac1r.
\]

The finite Dirichlet matrix uses \(r_j=ja\), diagonal

\[
\frac1{a^2}-\frac1{ja}+\frac{l(l+1)}{2(ja)^2},
\]

and nearest-neighbor entries \(-1/(2a^2)\). The initial state is the lowest \(l=0\) eigenstate; the final state is the lowest \(l=1\) eigenstate. This continuum labeling corresponds to 1S and 2P.

The exact infinite-mass continuum values used as analytical references are

\[
\Delta E=\frac38,\qquad
I_r=\langle u_{2P}|r|u_{1S}\rangle=\frac{256}{81\sqrt6},
\]

\[
\langle2P,0|z|1S\rangle=\frac{256}{243\sqrt2},
\qquad
f=2\Delta E|\langle z\rangle|^2=\frac{8192}{19683}.
\]

The last expression is the z-direction oscillator strength and, for an initial S state in this degenerate nonrelativistic model, equals the usual multiplet-summed isotropic line oscillator strength. `DIPOLE_DERIVATION.md` gives normalized wavefunctions and the integral.

For z polarization, the exact spherical-harmonic matrix element is nonzero only for \(m'=m\) and \(l'=l\pm1\). The allowed 1S→2P \(m'=0\) angular factor is \(1/\sqrt3\). The declared 1S→2S, 1S→3D and 1S→2P \(m'=1\) controls vanish. These are consequences of the supplied E1 operator, not a derivation of its origin.

### CA.2 Frozen computation **[C]**

`DIPOLE_PROTOCOL_0.3.json` was hashed before production execution. Its SHA-256 is `0eafcae663b9a53c93734acb6b336a3ef2a57b67957386a1337e24da92f10277`. It fixes \(a=0.4,0.2,0.1,0.05\,a_0\), outer walls \(R=80,160\,a_0\), zero fitted parameters and the acceptance thresholds.

| \(a/a_0\) at \(R=160a_0\) | Gap (Hartree) | \(I_r/a_0\) | Oscillator strength | Relative \(f\) error |
|---:|---:|---:|---:|---:|
| 0.40 | 0.356029069945 | 1.359023447440 | 0.438377343138 | 5.3294% |
| 0.20 | 0.369992762914 | 1.307695101009 | 0.421808147112 | 1.3483% |
| 0.10 | 0.373730129725 | 1.294639278893 | 0.417603770303 | 0.3381% |
| 0.05 | 0.374681377112 | 1.291360476703 | 0.416548743989 | 0.08458% |
| Analytic continuum | 0.375 | 1.290266201960 | 0.416196717980 | — |

All three errors decrease strictly. Their final empirical convergence orders are 1.995, 1.999 and 1.999 respectively, consistent with the declared second-order stencil in this tested regime. This finite observation is not a uniform convergence proof. At \(a=0.05a_0\), moving the outer wall from 80 to 160 \(a_0\) changes these three quantities by at most \(7.60\times10^{-14}\) in their respective atomic-unit quantities. The maximum eigenvector residual over the 16 computed states is \(8.26\times10^{-14}\).

All 32 package tests pass. The independent audit imports no production functions; it uses sparse shift-invert eigenpairs, direct Gauss–Legendre/Fourier angular quadrature and adaptive continuum radial quadrature. It passes 58/58 checks. Both calculations share the declared equations and NumPy/SciPy runtime, so this is implementation verification rather than independent physical replication.

### CA.3 Relation to observed hydrogen **[E boundary]**

The calculation establishes that the engine can represent a standard E1 transition once ordinary quantum dynamics and the dipole operator are supplied. NIST's critically evaluated spectrum lists strong resolved H I reference components near 1215.67 Å; the cited table attributes those entries to calculations. The values are context only: the release neither tunes to them nor claims their precision, and it consumes no measured validation.

A precision comparison would require at least the proton reduced mass, relativistic and radiative terms, Lamb shift, recoil, finite nuclear size, natural width, external-field and detector/line-shape models. None is silently assigned to a hidden dimension. Consequently, the result is not new measured evidence, not a string derivation of electromagnetism, and not evidence for four, five or seven physical dimensions.

The next physical-development gate is one explicit project-specific interaction term that yields a numerical contrast from the precision-complete standard model. Its parameters, uncertainty budget, observable, competitors and rejection rule must be frozen before opening genuinely unused measured validation. A latent rotation or dimensional coordinate that leaves every accessible probability unchanged cannot be distinguished by that experiment and must remain an interpretation rather than an identified cause.

---

## Appendix CB. Multichannel acoustic support: force balance, not mass loss

Revision 1.79, 21 September 2026. Fixed Grid String Lab 0.4.0 starts from the
saved 0.3.0 release and the full revision-1.78 paper. It answers the bounded
question raised by acoustic levitation: can coordinated vibrations supply the
external force needed to hold matter up? The calculation uses established
linear acoustics and the small-sphere radiation potential. It does not assume
that matter is literally made of these waves, and it does not infer hidden
dimensions from successful levitation.

### CB.1 Frozen field and particle **[M]**

The protocol fixes two opposed 8 by 8 arrays, 0.12 m apart, with 0.01 m element
pitch. All 128 ideal isotropic monopoles run at 40 kHz in illustrative dry air,
with density \(\rho_0=1.204\ \mathrm{kg\,m^{-3}}\) and sound speed
\(c_0=343\ \mathrm{m\,s^{-1}}\). Phases focus both arrays at
\(\mathbf x_0=(0,0,0.06\ \mathrm m)\), with an additional \(\pi\) phase on
the upper array. For \(R_j=|\mathbf x-\mathbf s_j|\),

\[
p(\mathbf x)=C\sum_{j=1}^{128}\frac{e^{i(kR_j+\phi_j)}}{R_j},
\qquad k=\frac{2\pi f}{c_0},
\]

and the linearized particle-velocity phasor is

\[
\mathbf v(\mathbf x)=-\frac{i}{\rho_0\omega}\nabla p(\mathbf x).
\]

Each array alone is normalized to 3000 Pa peak at the geometric focus. The
opposed phases produce a pressure node there but do not eliminate the velocity
field or its gradients.

The declared object is a solid-polystyrene sphere with radius 0.5 mm, density
1050 kg/m³ and longitudinal sound speed 2350 m/s. Its computed mass is
\(5.497787143782139\times10^{-7}\ \mathrm{kg}\), and its gravitational force
parameter is \(mg=5.391487429357111\times10^{-6}\ \mathrm N\). These material
numbers are illustrative inputs rather than measurements of a particular bead.

For particle volume \(V\), the Gor'kov potential used in the release is

\[
U_G=V\left[
\frac{f_1|p|^2}{4\rho_0c_0^2}
-\frac{3f_2\rho_0|\mathbf v|^2}{8}
\right],
\]

\[
f_1=1-\frac{\rho_0c_0^2}{\rho_pc_p^2},\qquad
f_2=\frac{2(\rho_p-\rho_0)}{2\rho_p+\rho_0}.
\]

Gravity adds \(mgz\), so the modeled force is

\[
\mathbf F=-\nabla(U_G+mgz).
\]

This formula is conditional on linear, lossless acoustics and the small-sphere
approximation. It is not a fundamental equation of the projective framework.

### CB.2 Equilibrium, threshold and controls **[C]**

`ACOUSTIC_PROTOCOL_0.4.json` was frozen before the release run. Its SHA-256 is
`e4b79836e34c251e3b0fa3626424ab9400ea8564181515794d3a75e3011c6148`.
The production calculation finds a stable equilibrium at

\[
z_\mathrm{eq}=0.05991342552219733\ \mathrm m,
\]

or 86.5744778 micrometres below the pressure node. The force residual is
\(5.66\times10^{-17}\ \mathrm N\). The Hessian eigenvalues of the total
potential are

\[
(0.00415165392590,\ 0.00415165392603,\ 0.0620028890531)\ \mathrm{N\,m^{-1}},
\]

so the tested equilibrium is locally stable in all three spatial directions.
The corresponding undamped small-oscillation frequencies are 13.8305, 13.8305
and 53.4481 Hz.

Because pressure is linear in source amplitude and the radiation potential is
quadratic, the force scales as pressure squared. The engine's half-pressure
control gives a force ratio of exactly 0.25 in the tested arithmetic. Scaling
the maximum upward reference force gives the frozen support threshold

\[
P_\mathrm{th}=1010.8584779716566\ \mathrm{Pa\ peak}
\]

per array at the coherent focus. The declared scan finds no stable equilibrium
at 900 or 1000 Pa and one at 1100, 1500, 2000, 3000 and 4500 Pa. As pressure
increases, the calculated sag below the node decreases from 765.315 micrometres
at 1100 Pa to 38.411 micrometres at 4500 Pa.

All six frozen production gates and all 53 package tests pass. The independent
audit imports no production calculation; it uses scalar complex loops, custom
bisection and independently written finite differences. It passes 24/24 checks,
reproducing the equilibrium within \(7.49\times10^{-16}\ \mathrm m\) and the
threshold within \(7.92\times10^{-11}\ \mathrm{Pa}\). These are finite software
checks under shared physical assumptions, not independent experimental
replication.

### CB.3 What a scale would measure **[M/C]**

At equilibrium the acoustic force on the sphere is \(+mg\), so a hypothetical
support directly under the sphere can read zero in the ideal static model. That
is an apparent-weight effect. The sphere's mass parameter is unchanged.

Momentum conservation supplies the decisive countercheck: the arrays and their
mounts receive the equal and opposite acoustic reaction. For the closed
apparatus-plus-sphere system,

\[
\mathbf F_{\mathrm{acoustic\ on\ sphere}}
+\mathbf F_{\mathrm{reaction\ on\ source}}=0.
\]

The production and independent codes both verify that the closed-system excess
over the full object weight is zero within the frozen tolerance. Therefore
acoustic levitation redistributes force through an externally powered field; it
does not remove gravitational mass, screen gravity or make the complete setup
weigh less.

### CB.4 Evidence boundary and next test **[E/H boundary]**

Acoustic levitation is measured physics: published phased-array experiments
have trapped particles and counteracted gravity. Those external experiments
establish the general phenomenon, not this release's numerical threshold or
equilibrium. The present run uses no measured pressure map, force curve,
levitation position or electrical-power data.

The ideal-monopole model omits finite-piston directivity, element-to-element
amplitude and phase errors, air absorption, reflections, nonlinear propagation,
acoustic streaming, viscous and thermal boundary layers, electromechanical
coupling, particle-scattering backreaction and calibrated material uncertainty.
Those omissions can make a real device disagree even when the implementation is
correct. A mismatch would not, by itself, identify missing strings or another
dimension; the omitted conventional mechanisms are explicit competing causes.

The next admissible test is to freeze one build geometry, calibration method,
uncertainty budget and rejection rule before collecting validation data. Measure
the complex pressure field and bead position independently, compare the frozen
threshold and equilibrium prediction, and place the complete apparatus on a
load cell while levitating the bead. The expected conventional outcome is that
the bead's local support force vanishes while the total apparatus load remains
unchanged. Only a repeatable residual that survives calibrated conventional
models and independent replication could motivate a new-physics hypothesis.

All prior corrections remain in force: complete deterministic closure for one
fixed transition family is idempotent; integer-tick point capture does not
enforce continuous non-crossing; angular impulse has no extra time factor;
finite prefixes do not identify arbitrary generators; and the 2K continuation
guarantee requires a noiseless homogeneous constant-coefficient recurrence of
order at most K from the initial state. No residual is identified with God, no
seven physical dimensions or P=NP are assumed, and an observationally
equivalent hidden-dimensional account remains an interpretation rather than an
identified cause.

## Appendix CC. Material and phase acoustics: density, contrast, interfaces

Revision 1.80, 21 September 2026. Fixed Grid String Lab 0.5.0 answers one
bounded continuation of Appendix CB: which declared equal-radius solids and
liquid droplets are easier to support in the same frozen acoustic field, and
why are gases a different model class?

### CC.1 Corrected static load **[M]**

Appendix CB used `mg` as the entire downward load. An inclusion immersed in a
host fluid also experiences hydrostatic buoyancy. With particle density
\(\rho_p\), host density \(\rho_0\), volume \(V\), and gravitational
acceleration \(g\), the radiation field must balance

\[
W_\mathrm{eff}=(\rho_p-\rho_0)Vg.
\]

The new engine retains gravitational weight, buoyant force and their difference
as separate quantities. The historical 0.4 result remains in the record. For
the declared polystyrene sphere in air, the corrected threshold is

\[
P_\mathrm{PS}=1010.2787528754241\ \mathrm{Pa\ peak/array},
\]

versus 1010.8584779716566 Pa before the correction, a 0.0573498% reduction.
The small magnitude follows from \(\rho_\mathrm{air}\ll\rho_\mathrm{PS}\); it
does not make buoyancy optional near density matching or for gases.

### CC.2 The material connection **[M/H boundary]**

For a small compressible sphere, retain the Gor'kov factors

\[
f_1=1-\frac{\rho_0c_0^2}{\rho_pc_p^2},\qquad
f_2=\frac{2(\rho_p-\rho_0)}{2\rho_p+\rho_0},
\]

and define the one-dimensional standing-wave contrast

\[
\Phi=\frac{f_1}{3}+\frac{f_2}{2}.
\]

The production engine still evaluates the pressure and velocity terms
separately in the three-dimensional array field; \(\Phi\) is a diagnostic, not
a replacement field model. For condensed matter in air, density and bulk
modulus are much larger than the host values, giving \(f_1\approx f_2\approx1\)
and \(\Phi\approx5/6\). Because acoustic force and effective weight both scale
with \(V\), the support threshold in this small-sphere model is approximately

\[
P_\mathrm{th}\propto
\sqrt{\frac{\rho_p-\rho_0}{\Phi}}.
\]

Thus the tested mathematical regularity is not atomic number, molecular mass or
a hidden-dimensional winding. It is an approximate square-root density law
inside a declared non-resonant model class. Composition affects the result
through emergent density, compressibility, elasticity, interfaces and geometry.

### CC.3 Frozen equal-radius results **[C]**

All objects have radius 0.5 mm in the exact ideal 128-source, 40 kHz field of
Appendix CB. The size parameter is \(ka=0.3664\), below one but not
asymptotically small.

| Material | Phase | \(\Phi\) | Threshold (Pa peak/array) |
|---|---:|---:|---:|
| Lithium | solid | 0.831642 | 720.396 |
| Ethanol | liquid | 0.832146 | 875.918 |
| Water ice | solid | 0.832346 | 944.111 |
| Water | liquid | 0.832408 | 985.047 |
| Polystyrene | solid | 0.832466 | 1010.279 |
| Glycerol | liquid | 0.832607 | 1106.726 |
| Aluminum | solid | 0.832999 | 1620.145 |
| Iron | solid | 0.833218 | 2766.822 |
| Copper | solid | 0.833232 | 2951.469 |
| Silver | solid | 0.833247 | 3193.543 |
| Gold | solid | 0.833286 | 4331.773 |
| Osmium | solid | 0.833293 | 4686.466 |

The dense-limit estimate differs from the exact field result by less than
0.046% across this table. The same chemical formula need not give the same
answer: liquid water and ice differ because phase changes bulk properties and
boundary behavior.

The full-element snapshot contains 118 records. The strict density screen keeps
85 entries labeled solid or liquid with a reported density. Under fixed
\(\Phi=5/6\), Li, K, Na, Rb and Ca form the low-threshold end; Os, Ir, Pt, Re
and Np form the high end. This screen omits element-specific sound speed,
allotrope, surface state and shape, so it is a prioritization device rather than
a universal periodic-table prediction.

### CC.4 Phase classes and gas counterexample **[M/H]**

A gas supports longitudinal pressure waves but no static shear and is highly
compressible. More importantly, an unconfined gas parcel in air lacks the
persistent boundary assumed by the sphere model and mixes with its host. The
engine therefore reports ideal-gas density and the buoyancy direction of a
hypothetically sealed sample, but it refuses to rank a free parcel beside a
solid or droplet.

A liquid droplet has a surface-tension boundary, can deform, and can develop
internal circulation and acoustic streaming. A solid retains shape and can
support shear and elastic mode conversion. A gas bubble or membrane introduces
an interface whose radial resonance can dominate the non-resonant contrast
calculation. Consequently gas, liquid and solid responses are not obtained by
changing density in one universal point-particle equation.

### CC.5 Verification and evidence boundary **[C/E boundary]**

The frozen production protocol SHA-256 is
`c7aa4d927f29bca3765b69734304aae68cece1eb1077f3d7ec39eeed33c710a1`.
All five production gates pass. All 70 package tests pass, including the 53
preserved tests from release 0.4. An independently written scalar complex-field
implementation passes 86/86 checks; its largest threshold difference is
\(2.82\times10^{-11}\) Pa. The implementations share the same physical
assumptions, so this is finite software verification, not experimental
replication.

No new measurement was made. The material inputs are rounded representative
values; source conditions are incomplete for the element density compilation.
The model omits finite-piston behavior, calibrated drive, absorption,
nonlinearity, streaming, boundary layers, deformation, elastic shear,
scattering backreaction and resonance. The result does not support mass loss,
a fundamental grid/tick or an extra physical dimension.

The next test is a frozen three-material experiment with equal-radius
polystyrene, water and ethanol samples. Independently measure dimensions,
temperature and the complex pressure field, then test without retuning the
predeclared ordering

\[
P_\mathrm{ethanol}<P_\mathrm{water}<P_\mathrm{polystyrene}.
\]

All earlier mathematical corrections remain binding: complete deterministic
closure for one fixed transition family is idempotent; integer-tick point
capture does not enforce continuous non-crossing; angular impulse has no extra
time factor; finite prefixes do not identify arbitrary generators; and the
2K-sample continuation guarantee assumes a noiseless homogeneous
constant-coefficient recurrence of order at most K from the initial state.

## Appendix CD. Mathematical exceptionalism, scoped topology and evidence

Revision 1.81, 21 September 2026. Fixed Grid String Lab 0.6.0 asks one bounded
question: do the supplied exceptional results in mathematics provide evidence
for a physical fourth dimension?

### CD.1 Smooth structures: the quantifier correction **[M]**

The supplied catalog asserted both that every dimension except four has one
smooth structure and that dimension seven has 28 different spheres. Those
sentences are inconsistent if the first quantifies over all manifolds.

The valid exceptional result is domain-specific. Euclidean four-space admits
uncountably many pairwise nondiffeomorphic smooth structures, while Euclidean
`R^n` has the standard smooth structure up to diffeomorphism for `n != 4`.
This does not imply that every other `n`-manifold has one smoothing.

Kervaire and Milnor's result concerns smooth homotopy 7-spheres. In the
oriented category the relevant group has order 28, including the standard
sphere. These manifolds are homeomorphic but not all diffeomorphic; “28
spheres” must not be read as 28 topologically distinct underlying spaces.

The Poincare entry also requires typed categories. The topological generalized
Poincare conjecture is resolved. The naive smooth statement is false in
dimensions with exotic spheres. The remaining four-dimensional smooth
Poincare question asks specifically whether every smooth homotopy 4-sphere is
diffeomorphic to the standard `S^4`.

### CD.2 Physical-promotion theorem **[M]**

Let `D` denote an extra-dimensional physical hypothesis, `D0` a declared
lower-dimensional rival, `I` a fixed intervention and `y` an observation. A
dimension test needs bridge models

\[
B_D:(D,I)\mapsto p(y\mid D,I),\qquad
B_0:(D_0,I)\mapsto p(y\mid D_0,I).
\]

Without the bridges and a measured `y`, the likelihood ratio

\[
\Lambda(y)=\frac{p(y\mid D,I)}{p(y\mid D_0,I)}
\]

is undefined. Therefore a valid theorem about a mathematical object cannot by
itself be empirical evidence that nature instantiates a dimension associated
with that object. The new executable gate requires a measured observation, a
fixed physical bridge, a dimension-specific prediction and a test of declared
alternatives. This is a claim-type rule, not a theorem that exceptional
mathematics is physically irrelevant.

### CD.3 Catalog corrections **[M/C]**

The audit represents 14 catalog entries with explicit domains and evidence
classes. Six require correction, five survive with narrowed scope, one is
conditional, one is an active model-class controversy and one is an
unsupported physical inference.

The prime-count entry contains a numerical error. Littlewood proved sign
changes of `pi(x)-li(x)` but did not locate the first one. Chao and Plymen prove
positive values for some integers in

\[
[\exp(727.951858),\exp(727.952178)]
=[1.3978965,1.3983439]\times10^{316}\ \text{approximately}.
\]

The whole interval is above, not below, `10^316`.

Banach–Tarski uses non-Lebesgue-measurable pieces, so ordinary volume is not
defined on each piece. It neither breaks physical conservation nor describes a
finite cutting process for matter. Chaitin's halting probability is also not
one canonical number: each universal prefix-free machine `U` defines its own
`Omega_U`. Goodstein independence is specifically relative to first-order
Peano arithmetic, not all mathematical reasoning. These statements remain
deep mathematics after their scope is corrected.

### CD.4 Complex quantum theory: measured but model-relative **[E/H]**

The 2022 superconducting experiment reports game score `8.09(1)`, violating
the `7.66` bound for the declared real-valued standard formalism by 43 standard
deviations. This is measured evidence against that formalism under its
composition and source assumptions.

It is not evidence that `i` is a physical substance or that spacetime has an
extra dimension. A peer-reviewed paper published 18 June 2026 constructs a
real-variable formulation with a different composition rule that reproduces
standard finite-dimensional quantum predictions. Additional 2025–2026 work
quantifies or disputes the needed source-independence assumptions. The broad
claim that complex numbers are uniquely necessary in nature is therefore
model-class dependent; the experiment does not identify a unique ontology.

### CD.5 Finite result and boundary **[C/E boundary]**

Zero of the 14 entries passes the physical-dimension gate. Exactly one contains
measured evidence, and that entry lacks an extra-dimensional bridge, a
dimension-specific prediction and exclusion of all declared alternatives. All
six production gates and all 85 package tests pass. A separate implementation
passes 21/21 checks and independently reproduces the scope counterexample and
the prime-interval conversion. This verifies the finite audit logic and
arithmetic; it does not machine-check the cited theorems or replicate the
physical experiment.

The next admissible test is not another search for remarkable constants.
Freeze competing 3D/internal-state and 4D-geometric chirality mechanisms that
predict distinct calibrated outcome distributions under one intervention. Set
nuisance parameters, uncertainty and rejection rules before opening unused
data. If the predicted observable distributions coincide, the measurement
cannot identify dimension.

All earlier corrections remain binding: complete deterministic closure for one
fixed transition family is idempotent; integer-tick point capture does not
enforce continuous non-crossing; angular impulse has no extra time factor;
finite prefixes do not identify arbitrary generators; and the 2K-sample
continuation guarantee assumes a noiseless homogeneous constant-coefficient
recurrence of order at most K from the initial state. Mathematical dimensions,
theological interpretation and measured physical dimensions remain separate.

## Appendix CE. Ordered slices, projection aliases and a fifth generator index

Revision 1.82, 21 September 2026. Fixed Grid String Lab 0.7.0 asks what information about a finite four-dimensional set is retained by complete three-dimensional slices, finitely many shadows, and one parameter that changes the four-dimensional set.

### CE.1 Three distinct operations **[M]**

Let `M subset Z^4` have coordinates `(x,y,z,w)`. A labelled slice is `M_c={(x,y,z):(x,y,z,c) in M}`. A coordinate-axis X-ray instead records line sums after one coordinate is discarded. A five-coordinate family adds an index `lambda` and forms `W={(p,lambda):p in M_lambda}`. Slice, projection and family index are not interchangeable. The names `w` and `lambda` do not make either coordinate a physical spatial dimension.

### CE.2 Complete labelled slices reconstruct exactly **[M/C]**

By direct set identity, `M = union_c (M_c x {c})`. The labels and registration are part of the data. The frozen discrete four-ball contains 137 voxels. Its five `w` slices contain `7,33,57,33,7` voxels and reconstruct all 137 with symmetric difference zero. One boundary-like slice, an unlabelled stack or a merged projection does not contain the same information.

### CE.3 An exact alias survives all four axis projections **[M/C]**

Define the even- and odd-parity subsets of `{0,1}^4`. They are disjoint and contain eight points each. For any coordinate axis and fixed retained triple, exactly one omitted bit gives even parity and one gives odd parity. Every corresponding line sum is one for both sets. All four coordinate-axis projections agree although the sets differ.

Their labelled `w` slices do differ and reconstruct the two sets separately. This is a written counterexample with exhaustive finite replay. It does not claim ambiguity for every transform: discrete-tomography uniqueness depends on the object class and measurement directions.

### CE.4 Scope correction for a scalar fifth parameter **[M]**

The statement “one scalar can only trace a curve” is false without a regularity restriction. One integer can enumerate any finite catalog; the engine explicitly decodes `0..48` onto all 49 members of a two-parameter box family. Peano's continuous construction maps an interval onto a square.

For a `C1` map `g:R->R^2`, the derivative is a 2 by 1 matrix with rank at most one. On compact intervals the map is locally Lipschitz, its image has two-dimensional measure zero and it cannot contain a two-dimensional open set. The frozen example `g(lambda)=(1+lambda,1+lambda^2)` has sampled rank one at all 101 diagnostic points. The general statement is written mathematics, not an inference from those samples.

The declared path `a=b` visits seven of 49 distinct four-dimensional boxes and misses 42. That result concerns the frozen regular path, while the finite catalog decoder prevents a broader coding claim.

### CE.5 Evidence status and next test **[C/H/E boundary]**

All nine production gates, all 103 package tests and 61/61 independently implemented checks pass. No physical measurement is used. The visualization is a synthetic coordinate model, not a measurement of hidden geometry or proof that a fifth spatial dimension generates four-dimensional form.

The next admissible physical test must freeze two bridge models—ordinary 3D plus internal state and an explicit extra-dimensional dynamics—that predict different calibrated distributions for one intervention. Parameters, uncertainties, competitors and rejection rule must be fixed before unused measured validation. If the accessible distributions agree, the dimension remains unidentified.

All earlier corrections remain binding: complete deterministic closure for one fixed transition family is idempotent; integer-tick point capture does not enforce continuous non-crossing; angular impulse has no extra time factor; finite prefixes do not identify arbitrary generators; and the 2K-sample continuation guarantee assumes a noiseless homogeneous constant-coefficient recurrence of order at most K from the initial state. No seven physical dimensions, P=NP or theological identification is inferred.


---

## Appendix CF. Wave–Observer Formulation: representation, dynamics and observation

**Formulation 0.1 · Theory revision 1.83 · 21 September 2026**  
**Originating research direction:** Anthony Michael Collado  
**Status:** A proposed organization of explicit mathematical models and checked
finite calculations. It is not an experimentally established universal law.

The useful central proposal is this: represent a declared system by modes,
specify how those modes interact and evolve, and calculate what a declared
observer receives. Each physical application must supply its own equations,
units, couplings and evidence. A common representation is possible without
assuming that sound, light and quantum probability are the same physical field.

The formulation connects the existing string, hydrogen, light, crystal and
acoustic modules. It adds no measured physical evidence and derives neither
the Standard Model nor gravity from the project's higher-dimensional hypothesis.

### WF.1. The compact wave equation

For a closed finite quantum model, declare a Hermitian Hamiltonian
\(H_\theta=H_\theta^\dagger\), a normalized amplitude vector \(a\), and an
orthonormal basis \(\phi_\mu\). Then the core is

\[
\boxed{
i\hbar\frac{d a(t)}{dt}=H_\theta a(t),\qquad
\psi(x,t)=\sum_\mu a_\mu(t)\phi_\mu(x),\qquad
p(y\mid b,t)=a(t)^\dagger E_y^{(b)}a(t).
}
\tag{WF.1}
\]

Here \(\theta\) specifies the model, \(b\) the measurement setting, and
\(E_y^{(b)}\succeq0\), \(\sum_y E_y^{(b)}=I\), the measurement effects.
Born probabilities are an imported quantum postulate, not derived from a
projection metaphor. \(x\) is a coordinate of the declared model; it need
not be a point in ordinary physical space. A many-particle configuration
coordinate and an internal spin index are not additional spatial dimensions.

For constant \(H_\theta\),

\[
a_{n+1}=U_\tau a_n,\qquad
U_\tau=\exp(-i\tau H_\theta/\hbar),\qquad t_n=n\tau.
\tag{WF.2}
\]

In an energy eigenbasis this becomes

\[
\psi(x,t)=\sum_\mu c_\mu\phi_\mu(x)e^{-iE_\mu t/\hbar}.
\tag{WF.3}
\]

This is an existing quantum evolution law used as an engine interface. Writing
an unspecified \(H_\theta\) does not predict a material or solve a theory of
everything. For time-dependent generators use the time-ordered propagator;
Equation WF.2 cannot generally use one instantaneous matrix exponential.

The model must list every Hamiltonian term and its domain. For example, the
existing hydrogen baseline supplies a kinetic stencil, Coulomb potential,
boundaries and a dipole operator. A future interaction term must have fixed
units and consequences; it cannot be an unrestricted residual-fitting field.

### WF.2. What “all functions in waves” can mean exactly

**Finite representation theorem [M].** Let \(x\in\mathbb C^N\) be any finite
array, and let

\[
F_{kj}=N^{-1/2}e^{-2\pi i kj/N},\quad
\widehat x=Fx,\quad x=F^\dagger\widehat x.
\tag{WF.4}
\]

Every such array has an exact wave-basis representation because
\(F^\dagger F=I\). To prove orthogonality, sum the geometric series
\(N^{-1}\sum_j e^{2\pi i(k-\ell)j/N}\): it equals one when
\(k=\ell\pmod N\), and zero otherwise. Tensor products give the same result
for finite multidimensional arrays.

For any linear operator \(A\), its wave-coordinate form is

\[
\widehat A=FAF^\dagger,\qquad
F(Ax)=\widehat A\widehat x.
\tag{WF.5}
\]

For any specified, evaluable function \(f:\mathbb C^N\to\mathbb C^N\),

\[
\widehat f(a)=F f(F^\dagger a).
\tag{WF.6}
\]

This is a conjugate representation of the same computation. It does not make
\(f\) linear, simpler, computable when it was not, or cheaper by theorem.
A nonlinear function is not generally a collection of independent rotations.

For pointwise squaring, direct substitution in WF.4 gives

\[
\widehat{(x^2)}_k=\frac1{\sqrt N}\sum_{p=0}^{N-1}
\widehat x_p\widehat x_{(k-p)\bmod N}.
\tag{WF.7}
\]

Thus an interaction mixes modes. On 32 sites, squaring
\(\cos(2\pi 5j/32)\) creates a constant mode and modes \(\pm10\), since
\(\cos^2 u=(1+\cos2u)/2\). Retaining only \(|k|\le5\) leaves the constant
and loses relative Euclidean norm \(1/\sqrt3\approx57.735\%\). The engine
must track generated modes, truncation and aliasing rather than declaring every
new harmonic to be a new particle or spatial axis.

For continuous functions, representation needs a declared function space and
convergence notion. Square-integrable periodic functions have Fourier
representations in the mean-square sense; this is not a promise of pointwise,
uniform or finite-term reconstruction for every conceivable function.
Standard convergence qualifications are summarized by
[NIST DLMF §1.8](https://dlmf.nist.gov/1.8).

Finite sample recovery also does not identify a unique continuous function:
on \(x_j=2\pi j/N\), adding \(\epsilon\sin(Nx)\) leaves every sample
unchanged while changing values between samples. This is a written
counterexample, not a limitation that more decimal places alone can remove.

### WF.3. Exact bridge from a classical string to complex modes

Consider a finite conservative mechanical system

\[
M\ddot q+Kq=0,\qquad p=M\dot q,
\tag{WF.8}
\]

where \(M\) and \(K\) are real symmetric positive definite matrices. Set

\[
Q=M^{1/2}q,\quad P=M^{-1/2}p,\quad
\Omega=(M^{-1/2}KM^{-1/2})^{1/2},
\]

\[
a=\frac{\Omega^{1/2}Q+i\Omega^{-1/2}P}{\sqrt2}.
\tag{WF.9}
\]

Then exactly

\[
\boxed{i\dot a=\Omega a},\qquad
\mathcal E=\tfrac12P^TP+\tfrac12Q^T\Omega^2Q=a^\dagger\Omega a.
\tag{WF.10}
\]

**Proof [M].** WF.8 implies \(\dot Q=P\) and
\(\dot P=-\Omega^2Q\). Differentiate WF.9 and collect terms to obtain
\(\dot a=-i\Omega a\). Expanding \(a^\dagger\Omega a\) cancels the
imaginary cross terms and yields WF.10. Inverting real and imaginary parts
recovers both \(q\) and \(p\). Since \(\Omega\) is Hermitian, both the
mode norm and mechanical energy are conserved. □

This makes the connection to rotating strings exact for this class. The
complex coefficients have units of square root of action; they are not
normalized quantum probabilities. Multiplying a classical equation by
\(\hbar\) would not quantize the system or derive Born's rule.

**Necessary scope boundary.** A zero-frequency coordinate obeys
\(\ddot q=0\) and may drift, \(q=q_0+v_0t\). WF.9 is then singular.
Unstable modes can grow exponentially. These cases need explicit drift or
growth channels; a finite sum of bounded real-frequency oscillations cannot
represent their unbounded evolution for all time.

For damping, nonlinear interactions and drive, use the explicit extension

\[
M\ddot q+\Gamma\dot q+Kq+\nabla V_{\rm nl}(q)=J(t).
\tag{WF.11}
\]

When \(\Gamma\succeq0\), the classical energy obeys
\(\dot{\mathcal E}=-\dot q^T\Gamma\dot q+\dot q^TJ\), including
\(V_{\rm nl}\) in the energy. Declaring these terms does not determine their
material-specific values. This nonlinear/damped extension is specified here;
the released mechanical benchmark tests the conservative linear case only.

### WF.4. Why phase and the observer must stay in the state model

For a position-density observer,

\[
|\psi(x,t)|^2=\sum_{\mu,\nu}c_\mu c_\nu^*
\phi_\mu(x)\phi_\nu(x)^*
e^{-i(E_\mu-E_\nu)t/\hbar}.
\tag{WF.12}
\]

The visible interference involves **energy differences** and an observation
operator. Shifting all energies by one constant changes only global phase.
An energy eigenstate has static position density even though its amplitude
phase evolves. Static structure therefore does not require an eye averaging
unresolved classical particle orbits.

A finite counterexample makes lost phase explicit. In units \(\hbar=1\),
let \(H=\begin{pmatrix}0&1\\1&0\end{pmatrix}\) and

\[
\psi_+=\frac{(1,i)^T}{\sqrt2},\qquad
\psi_-=\frac{(1,-i)^T}{\sqrt2}.
\]

Both initially give site probabilities \((1/2,1/2)\). But
\(U(t)=\cos t\,I-i\sin t\,H\), so at \(t=\pi/4\):

\[
p_+=(1,0),\qquad p_-=(0,1).
\tag{WF.13}
\]

The visible density alone is not a sufficient autonomous state here. The
maximally mixed state \(I/2\), which has the same initial site probabilities,
instead remains \(I/2\). This is a direct bridge to the paper's earlier
observation-fiber and hidden-memory results. It demonstrates phase sensitivity,
not a physical extra dimension.

For a multi-channel observer the natural interface is

\[
p(y\mid b,t)=\operatorname{Tr}(E_y^{(b)}\rho(t)).
\tag{WF.14}
\]

Sequential measurements also need a state-update instrument. For outcome
\(y\), an explicitly chosen operation
\(\mathcal J_y(\rho)=\sum_rK_{yr}\rho K_{yr}^\dagger\) gives
\(p_y=\operatorname{Tr}\mathcal J_y(\rho)\) and conditional state
\(\rho_y=\mathcal J_y(\rho)/p_y\) when \(p_y>0\). Measurement effects alone
do not specify disturbance. These are established quantum rules, not claims
that human consciousness selects an outcome.

### WF.5. The broader evolution equation includes the environment

For a specified finite Markovian open quantum model, use

\[
\boxed{
\dot\rho=-\frac{i}{\hbar}[H_\theta,\rho]
 +\sum_r\left(L_r\rho L_r^\dagger
 -\frac12\{L_r^\dagger L_r,\rho\}\right),\qquad
p(y\mid b,t)=\operatorname{Tr}(E_y^{(b)}\rho).
}
\tag{WF.15}
\]

The braces denote an anticommutator. \(H\) has energy units and \(L_r\)
units s⁻¹ᐟ². With fixed operators in finite dimension this generates a
completely positive, trace-preserving semigroup. This is the established
Lindblad form ([Lindblad, June 1976](https://doi.org/10.1007/BF01608499)); it
is not newly discovered by this project. The observer and quantum-operation
interpretations are detailed in
[Preskill, Chapter 3](https://www.preskill.caltech.edu/ph219/chap3_15.pdf).

The assumptions matter. Not every subsystem has autonomous Markovian dynamics;
initial system–environment correlations and retained environmental memory can
require a larger state or history. The framework cannot use WF.15 as an
unrestricted law of all reduced systems.

**Checked example [M/C].** In dimensionless units, set
\(H=\mathrm{diag}(0,1.7)\),
\(L=\sqrt{0.4}|0\rangle\langle1|\), and
\(\rho(0)=|+\rangle\langle+|\). Then

\[
\rho_{11}(t)=\tfrac12e^{-0.4t},\quad
\rho_{00}(t)=1-\rho_{11}(t),\quad
\rho_{01}(t)=\tfrac12e^{-0.2t}e^{1.7it}.
\tag{WF.16}
\]

The state is generally mixed during decay, so a single normalized subsystem
wavefunction is insufficient. Moreover, an initially excited state has zero
expectation of the off-diagonal dipole operator but positive jump rate
\(\operatorname{Tr}(L^\dagger L\rho)=0.4\). A vanishing coherent average
therefore does not imply no photon emission. This preserves Appendix BJ's
boundary between classical expectation-field radiation and the quantum
emission channel. The rate 0.4 is supplied for a synthetic test, not predicted
for any atom or measured by this project.

### WF.6. What the symbols mean in the engine

| Object | Definition and role | What it does not establish |
|---|---|---|
| \(\Lambda,\Delta x\) | Declared computational sites and spacing | A measured spacetime lattice |
| \(\tau\) | Evaluation or update interval | A fundamental clock |
| \(\phi_\mu\) | A complete basis of the retained representation | Hidden physical spatial axes |
| \(a_\mu\) | Mode amplitude and phase | A classical particle trajectory by itself |
| \(H_\theta\) | Specified quantum generator, including interactions | A prediction when its entries remain unspecified |
| \(M,K,\Gamma,V_{\rm nl},J\) | Mechanical inertia, stiffness, loss, interaction and drive | Universal material constants |
| \(\rho\) | Quantum state including mixtures | A uniquely identified microscopic ontology |
| \(L_r\) | Declared environmental transition/decoherence coupling | A missing-force explanation inferred from a residual |
| \(E_y^{(b)},\mathcal J_y\) | Outcome probability and state-update model | A complete observer merely by drawing a camera |
| \(\Pi\) | Projection or coarse observation of a richer state | Invertibility or a unique hidden cause |

For real \(H\), writing \(a=u+iv\) gives
\(\hbar\dot u=Hv\), \(\hbar\dot v=-Hu\). The two real quadratures
describe phase. Doubling coordinates in this representation does not prove
that physical space doubled its dimension.

### WF.7. Connecting the existing physical modules

| Existing module | Wave formulation | Remaining physical work |
|---|---|---|
| Rope/string | Classical displacement, tension, inertia, boundary drive; WF.8–11 after a declared discretization | Calibrated forces, contact and constitutive laws; sampled collision is not continuous non-crossing |
| Hydrogen | Coulomb \(H\), complex amplitudes, density/current and dipole matrix elements | Precision corrections and independent measured validation; standard quantum ingredients are imported |
| Light | Electromagnetic vector fields, polarization, gauge constraints and detector coupling | A scalar display wave is not full Maxwell/QED dynamics; quantum emission needs an appropriate channel |
| Crystal interference | Coherent sums with lattice phases and a scattering observer | Intensity inverse problems can be nonunique; diffraction does not by itself predict electronic color |
| Gold/silver color | Frequency-dependent complex optical response, illumination and spectral observer | Imported/fitted optical response is not a new prediction from FCC geometry |
| Acoustics | Pressure/velocity modes and radiation-force coupling | A supported object's weight is balanced by a force; its mass is not removed |
| Molecules and phases | Interactions, collective normal modes and occupations | Many-body binding, exchange, anharmonicity and phase thermodynamics are not supplied by a shape label |

Waves can therefore be a shared mathematical language. Their physical state
spaces and measurement rules remain typed. Many-body quantum states require
appropriate tensor/configuration or Fock spaces, exchange symmetry and spin;
a single scalar wave over ordinary 3D space is not automatically sufficient.

Thermal equilibrium concerns populations and energy exchange. Heating and
cooling are not, in general, opposite signs of a wave's phase rotation.
Similarly, an integer octave reduction preserves frequency ratios but adds no
new atom–sound interaction. The prior musical-note plotting branches and
reconstructed spectral equalities remain representation checks.

### WF.8. Higher dimensions: a declared extension, not a fitted rescue term

The project may propose fields \(\Phi(x,\chi,t)\) with extra coordinates
\(\chi\), a fixed higher-dimensional generator, and an observation map into
3D records. To make that proposal physical, specify the metric/topology,
boundary conditions, units, coupling and preparation. Then freeze a numerical
prediction that differs from the conventional competitor through the same
observer. A displayed extra coordinate could instead be an internal variable.

The existing exact aliases remain: a linear five-coordinate rotation can
produce the same observed signal as two ordinary temporal oscillators; different
4D sets can share specified projections. Additional modes alone do not remove
those ambiguities. No dimension count, fine-structure constant, golden-ratio
law, universal tick, or \(P=NP\) result is derived here.

### WF.9. Executed checks and interpretation

The frozen protocol is `FGSL-WAVE-OBSERVER-0.8`, SHA-256
`1f47d7383c5e131a5087c1f1ee9c60e0e6eb374c6fddb0c8fe31ecb30fe984ff`.

| Test | Executed result |
|---|---:|
| DFT round-trip, N=8,16,32 | Maximum relative error 3.78 × 10⁻¹⁵ |
| Linear operator in wave coordinates | Maximum relative error 4.34 × 10⁻¹⁵ |
| Nonlinear squaring versus circular convolution | Maximum relative error 5.68 × 10⁻¹⁵ |
| Deliberately discarded new harmonics | Relative error 0.577350269190 |
| Eight coupled oscillators versus real block evolution | Maximum component difference 3.00 × 10⁻¹⁵ |
| Separate time-domain oscillator ODE audit | Maximum component difference 2.55 × 10⁻¹³ |
| Equal initial densities, opposite phases | Final probabilities (1,0) and (0,1), within 6.67 × 10⁻¹⁶ |
| Markovian decay versus exact solution | Maximum component difference 1.58 × 10⁻¹⁶ |
| Separate density-matrix ODE audit | Maximum component difference 1.55 × 10⁻¹³ |
| Production gates | 9/9 passed |
| Complete package unit suite | 112/112 passed |
| Independent implementation audit | 29/29 passed |

The independent audit imports no production functions; it uses FFT routines,
direct time-domain ODEs and analytic solutions. Both implementations share the
declared mathematics and NumPy/SciPy environment. This is implementation
verification, not experimental replication. The first development execution
encountered a NumPy-boolean JSON serialization error; converting output flags
to Python booleans fixed reporting without changing the frozen mathematics,
parameters or tolerances.

Written proofs, finite computation and synthetic controls support the scoped
claims above. There is no new measured evidence in this release. The existing
v1.14 idempotent-closure result and v1.15 collision, impulse and finite-prefix
corrections remain. The recurrence learner's \(2K\)-sample guarantee still
requires noiseless homogeneous constant-coefficient recurrence order at most
\(K\), valid from the initial state. Theological interpretation remains a
separate category; an unconstrained residual is not identified with God.

### WF.10. What makes this a testable candidate theory

The proposed organizing principle is **state → mode dynamics → interaction →
observer → checked prediction**. Its equations become a physical model only
after the generator and observer are specified enough to rule out outcomes.

The next bounded target is one frozen hydrogen emission case. Declare the
preparation, Hamiltonian, dipole/emission coupling, profile and detector; compute
line positions and strengths with an explicit error budget; then compare with
independently sourced measured data without retuning. Include standard physical
corrections at the precision being claimed. Any added higher-dimensional term
must make a predeclared difference and face fresh validation after modification.

That gives the reality-render engine a coherent mathematical architecture now,
while leaving the physical law and its evidence open to correction.

---

## Appendix CG — Physical coverage and the electromagnetic wave foundation

Theory of Reality 1.84 · Fixed Grid String Lab 0.9.0 · 21 September 2026

### CG.1 — What physical completeness would require

The goal is to recover established physical predictions in their experimentally
tested domains, explain how effective descriptions follow from the proposed
foundation, and make discriminating new predictions. This release does not
claim that goal has been reached.

Mathematics alone does not choose the physical state space, interactions,
constants, initial conditions or observation rule. Different consistent
mathematical models predict different worlds. Experimental results constrain
those choices. Also, different formulations can describe the same physics:
recovering their common predictions and proving the relevant equivalences is
more meaningful than collecting every equation ever written.

A wave representation is an architecture, not a unique law of nature. The
preceding formulation, i hbar da/dt = H a, becomes predictive only after H,
the state preparation and the measurement map are specified. This release
fills one bounded part of that specification: nonrelativistic charged-wave
coupling to a prescribed U(1) connection. It adds a written Maxwell-action
derivation as a separate, explicitly postulated field sector. The runnable
module does not yet evolve that field sector.

The ledger below is a baseline of requirements, not an exhaustive catalog of
all mathematical physics or a fresh audit of every historical engine. Earlier
benchmark results are referenced as earlier records; only CG.2–CG.5 numerical
claims are newly executed here.

Evidence must retain separate labels:

- **Postulate or imported law:** a physical choice supplied to the model.
- **Conditional derivation:** a written consequence of stated assumptions.
- **Numerical verification:** the implementation agrees with a mathematical
  reference within a declared tolerance.
- **Measured validation:** predictions confront independent measured data,
  including uncertainty and detector response.
- **Hypothesis or interpretation:** an additional proposed explanation not
  established by the preceding labels.

Passing an implementation test cannot promote an imported law into a derivation
from strings or promote a synthetic calculation into measured evidence.

### CG.2 — Coverage ledger and acceptance obligations

| ID | Physical sector and required relation | Current defensible status | Missing derivation or acceptance gate |
|---|---|---|---|
| PC-01 | State space, units, preparation, observation and likelihood | Foundation contract and wave/observer map exist; finite quantum rules are imported | Derive or declare the ontology, admissible state preparations and observer assumptions; preserve observational equivalence |
| PC-02 | Continuous and discrete dynamics; Fourier equivalence | Finite DFT and positive-stiffness classical mode transformation checked in CF | Control truncation, unstable/zero modes, nonlinear mode generation and continuum errors for each actual application |
| PC-03 | Classical mechanics and conservation | Scoped oscillator, collision and acoustics models use specified laws | Recover Hamilton/Newton equations and Noether identities from a declared action; account for boundaries, external driving and open systems |
| PC-04 | Special relativity, causality and fundamental grid/tick | No derivation identifying a physical grid or clock | Recover Lorentz-compatible observables within stated precision; a local matrix exponential is not a strictly local finite-speed update |
| PC-05 | Charged matter coupling: (p-qA)^2/(2m)+qPhi | Conditional continuum construction and finite link model in this release; gauge and continuity checks pass | Derive the charged degrees of freedom, masses and charges from the proposed deeper model, or retain them as empirical inputs |
| PC-06 | Maxwell equations and field energy | Written derivation below from a postulated classical field action | Implement a self-consistent field–matter model, maintain Gauss constraints and account for total energy/charge; then confront calibrated data |
| PC-07 | Quantum probabilities and measurement | Born/POVM rules and a scoped channel are explicit imported assumptions | Recover preparation and measurement statistics, including correlations; explanatory reformulation is not a derivation of the Born rule |
| PC-08 | Spin, identical particles and quantum statistics | No universal derivation from the projection model | Implement spin representations, fermion antisymmetry, boson symmetry and their measurable consequences |
| PC-09 | Relativistic matter and QED | Not supplied by the scalar finite-ring model | Recover Dirac/field-theory limits, pair processes, radiation and precision corrections with a controlled approximation |
| PC-10 | Strong interaction and hadrons | No checked derivation from the proposed foundation | Recover SU(3) gauge dynamics and hadronic observables; a U(1) ring is not a QCD calculation |
| PC-11 | Weak interaction, electroweak symmetry and Higgs sector | No checked derivation from the proposed foundation | Recover chiral interactions, symmetry breaking, masses, mixing and relevant decay/scattering observations |
| PC-12 | Gravity and spacetime | Extra-coordinate proposals remain hypotheses | Recover Einstein dynamics and the appropriate Newtonian limit, equivalence-principle tests and gravitational observations |
| PC-13 | Atomic and molecular structure | Imported Coulomb Hamiltonian and hydrogen dipole benchmark verified in CA; reference element/crystal data compiled | Many-electron interactions, spin, nuclear/recoil/relativistic corrections, chemical bonds and independent precision spectra |
| PC-14 | Solids, bands, optical response and transport | Sourced crystal geometry, diffraction and measured-input optical reconstruction exist | Predict electronic response and transport from a declared many-body model; equal lattice topology does not imply equal color |
| PC-15 | Statistical mechanics, temperature and phase changes | Harmonic and material examples are scoped; no universal phase derivation | Derive thermodynamic limits, entropy/free energies, heat capacities and phase boundaries; heat/cold are not assigned opposite universal rotations |
| PC-16 | Fluids, elasticity, sound and radiation pressure | Conditional acoustic support and material comparison checked in CB/CC | Recover constitutive laws, dissipation, nonlinear flow and calibrated force/pressure data; acoustic support does not remove mass |
| PC-17 | Open systems, decoherence and irreversibility | Finite Markovian Lindblad example checked in CF | Justify environmental approximations and rates; retain memory when a Markovian approximation fails |
| PC-18 | Constants and dimensionless ratios | Constants and interaction scales remain inputs | Predict and independently validate a dimensionless relation before calling it derived; musical octave mappings do not determine the fine-structure constant |
| PC-19 | Cosmology and high-energy tests | Published ATLAS clockwork limits already exclude the recorded benchmark in its tested scenarios | Recover applicable cosmological/particle observations with calibrated likelihoods; exclusions and unavailable data remain explicit |
| PC-20 | Deeper unification and model identification | No unique higher-dimensional cause identified | Freeze alternatives that predict distinguishable observed distributions, reserve fresh data, and quantify all stated limitations |

The Standard Model's electroweak/strong sectors and gravity cannot be silently
replaced by a scalar harmonic field. CERN's overview explicitly distinguishes
the Standard Model's three included interactions from gravity [3]. The table
does not treat unresolved quantum gravity or a specific dark-matter mechanism
as established equations that can simply be copied into this engine.

### CG.3 — Conditional electromagnetic coupling on a grid

#### Assumptions and orientation

Take a finite ring of N >= 3 sites. A complex amplitude psi_j carries cell
probability n_j = |psi_j|^2, with sum_j n_j = 1. Time is continuous; outputs can
be sampled at fixed ticks. This is a single-particle, spinless, nonrelativistic
model with prescribed external fields. It is neither a many-body field theory
nor a claim of fundamental spacetime discreteness.

For each undirected edge {j,k}, choose one orientation and let U_jk transport
the amplitude at k to j. Require |U_jk|=1 and U_kj=U_jk*. Let w_jk>0 have units
of energy, V_j be real with units of energy, and hbar>0. Define

\[
\mathcal E[\psi]=\sum_{\{j,k\}}w_{jk}|\psi_j-U_{jk}\psi_k|^2
+\sum_j V_j|\psi_j|^2=\psi^\dagger H\psi.
\]

Each undirected edge is counted once. Consequently

\[
H_{jj}=V_j+\sum_{k\sim j}w_{jk},\qquad
H_{jk}=-w_{jk}U_{jk}\quad (j\sim k).
\]

Postulate the matter action

\[
S_m=\int dt\left[
\frac{i\hbar}{2}(\psi^\dagger\dot\psi-\dot\psi^\dagger\psi)
-\psi^\dagger H\psi\right].
\]

Variation with respect to psi* with vanishing endpoint variations yields

\[
\boxed{i\hbar\dot\psi=H\psi.}
\]

This is a derivation **from the supplied action**. The action, complex state
space and Born interpretation have not been derived from higher-dimensional
strings. Hermiticity follows from the edge conjugation rule and real V; hence
the evolution preserves the norm. For fixed H, exp(-iHt/hbar) is exact evolution
of this finite model, up to numerical evaluation error. It does not identify
a fundamental tick or impose strict relativistic causal propagation.

#### Static and time-dependent gauge covariance

Changing the local phase convention by G_jj=exp(i chi_j) requires

\[
\psi'=G\psi,\qquad U'_{jk}=e^{i\chi_j}U_{jk}e^{-i\chi_k}.
\]

Then psi'_j-U'_jk psi'_k = exp(i chi_j)(psi_j-U_jk psi_k), so the kinetic
quadratic form is invariant. For a time-independent change, H'=G H G† and
the evolved state transforms by the same G. Cell probabilities do not change.

For time-dependent G, directly differentiating psi'=G psi gives instead

\[
\boxed{H'=G H G^\dagger+i\hbar\dot G G^\dagger
=G H G^\dagger-\hbar\operatorname{diag}(\dot\chi).}
\]

Omitting this scalar-potential term is wrong. It is retained and independently
tested in this release. Static convention covariance alone would not verify
this time-dependent equation.

#### Probability and charge continuity

Define the outward probability current

\[
J_{j\to k}=-\frac{2}{\hbar}\operatorname{Im}
(\psi_j^*H_{jk}\psi_k)
=\frac{2w_{jk}}{\hbar}\operatorname{Im}(\psi_j^*U_{jk}\psi_k).
\]

Direct differentiation of n_j gives

\[
\boxed{\dot n_j+\sum_kJ_{j\to k}=0,\qquad
J_{j\to k}=-J_{k\to j}.}
\]

The product psi*_j U_jk psi_k is unchanged by the phase convention, so this
current is gauge invariant. Multiplying by the supplied charge q gives charge
per time through an edge. Converting cell charge into a spatial density also
requires the cell volume; these finite probabilities are not already SI charge
densities. The continuity proof remains valid for time-dependent Hermitian
links and real potentials.

#### Continuum relation and closed-loop phase

In a smooth continuum comparison with spacing a, set

\[
w=\frac{\hbar^2}{2ma^2},\qquad
U_{jk}=\exp\left(\frac{iq}{\hbar}\int_{x_k}^{x_j}\mathbf A\cdot d\mathbf l\right),
\qquad V_j=q\Phi(x_j)+W(x_j).
\]

For smooth fields and wavefunctions resolved by the grid, Taylor expansion
of the covariant differences recovers

\[
\boxed{i\hbar\partial_t\psi=
\left[\frac{(-i\hbar\nabla-q\mathbf A)^2}{2m}+q\Phi+W\right]\psi.}
\]

The corresponding continuum convention is psi'=exp(iq Lambda/hbar)psi,
A'=A+grad Lambda and Phi'=Phi-partial_t Lambda. Mass, charge, hbar and the
remaining potential W are inputs. For the electron-charge magnitude e, writing
alpha=e^2/(4 pi epsilon_0 hbar c) defines a dimensionless coupling; it does not
derive its observed value or its scale dependence from geometry or octaves.

On the implemented ring U_j,j+1=exp(i theta/N), with indices modulo N, the
closed-loop product equals exp(i theta) and is gauge invariant. An arbitrary
nontrivial loop phase cannot be removed by a single-valued site gauge. With
uniform w and V=0, the discrete eigenvalues are

\[
E_m=2w\left[1-\cos\left(\frac{2\pi m+\theta}{N}\right)\right],
\quad m=0,\ldots,N-1.
\]

Their multiset is periodic under theta -> theta+2pi, but a general change in
theta changes the spectrum. Our links run from j+1 to j, so theta has the
opposite sign to q/hbar times the forward-oriented magnetic flux. No sign
convention changes the spectrum as a multiset here. This is a finite analogue
of the phase sensitivity discussed by Aharonov and Bohm [2], not a new
measurement of their effect. Link-based gauge constructions have an established
precedent in lattice gauge theory [1]; this module does not implement Wilson's
full quantum gauge-field theory.

The continuum check holds mode m=2, length 2pi, theta=0.8, hbar=m_mass=1
fixed while refining N. It tests that one smooth low-frequency mode. High
frequency modes near the lattice cutoff do not have this same small-error
continuum approximation.

### CG.4 — A field law still needs its own action

Gauge covariance of matter coupling does not specify how A and Phi evolve.
For example, postulate the classical vacuum field and external-source action

\[
S_{EM}=\int dt\,d^3x\left[
\frac{\epsilon_0}{2}|\mathbf E|^2
-\frac{1}{2\mu_0}|\mathbf B|^2
-\rho\Phi+\mathbf J\cdot\mathbf A\right],
\quad
\mathbf E=-\nabla\Phi-\partial_t\mathbf A,\quad
\mathbf B=\nabla\times\mathbf A.
\]

Assume sufficiently smooth fields, positive constant epsilon_0 and mu_0, and
boundary conditions that remove the integration-by-parts boundary terms.
Variation in Phi yields epsilon_0 div E-rho=0. Variation in A yields
epsilon_0 partial_t E-(1/mu_0)curl B+J=0. The definitions of E and B supply
the remaining identities. Thus

\[
\nabla\cdot\mathbf E=\rho/\epsilon_0,\quad
\nabla\times\mathbf B=\mu_0\mathbf J+
\mu_0\epsilon_0\partial_t\mathbf E,\quad
\nabla\cdot\mathbf B=0,\quad
\nabla\times\mathbf E=-\partial_t\mathbf B.
\]

Under a gauge transformation the source action changes by a boundary term
minus integral Lambda(partial_t rho+div J). Conserved sources are therefore
required. In source-free vacuum these equations give waves with speed
1/sqrt(mu_0 epsilon_0). This written calculation is standard conditional
mathematics: it assumes the Maxwell quadratic action and its constants.

To couple the sectors self-consistently, derive the matter current by varying
the matter action with respect to the potentials and use it as the field
source, without double-counting the interaction. That coupled field evolution,
its constraints and its energy accounting are **not implemented in 0.9.0**.
Classical Maxwell–Schrodinger coupling would still not be full QED.

Why symmetry alone is insufficient: an extra gauge-invariant diagonal W_j
changes the Hamiltonian without violating covariance. The finite counterexample
below checks this explicitly. These are different supplied potentials, not a
claim that two fundamental theories were experimentally distinguished.
More generally, adding a term proportional to (F_mu_nu F^mu_nu)^2 to a field
action also preserves gauge invariance while changing field dynamics. Additional
assumptions or measured constraints are needed to restrict such terms. Gauge
symmetry alone therefore neither forces the pure Maxwell action nor determines
all the interaction constants.

### CG.5 — Frozen computation and independent verification

Protocol: `GAUGE_PROTOCOL_0.9.json`, fixed before the first execution.

SHA-256: `aeb3703be7925de8c7eb4941e742508c2b45cd1cf0e8e32fb0d52ec0e4529575`.

The production implementation assembles an incidence-matrix energy and evolves
by a matrix exponential. Three synthetic rings have N=5,9,16, fixed seeds,
real potentials, unit-modulus links and normalized complex initial states.
The ring spectrum and refinement examples are specified separately. No measured
data or fitted physical parameter is used. Units are dimensionless with hbar=1.

| New verification | Executed result |
|---|---:|
| Production acceptance gates | 8/8 pass |
| Entire package unit suite | 120/120 pass |
| Independent numerical checks | 37/37 pass |
| Independent failure-control cases | 9/9 detected |
| Largest static gauge covariance error in H | 8.89e-16 |
| Largest evolution covariance error | 9.13e-16 |
| Largest probability-continuity residual | 1.12e-16 |
| Largest time-dependent equation residual | 4.58e-16 |
| Independent direct-ODE versus production evolution | 1.20e-12 maximum component error |
| Independent time-dependent gauge ODE comparison | 1.35e-13 maximum component error |

The audit imports no production module. It assembles entries using edge loops,
integrates the equations with an adaptive ODE solver, computes currents from
the link formula, tests continuity by central finite differences, and checks
plane-wave eigenvectors directly. This is an independent implementation in the
same numerical environment, not an independent laboratory or formally verified
floating-point proof. The written identities provide the mathematical claims;
the finite checks assess this implementation.

| Continuum refinement | Computed energy | Error from continuum value 2.26275360363842 |
|---|---:|---:|
| N=32 | 2.23004531373974 | 0.03270828989868 |
| N=64 | 2.25454079574145 | 0.00821280789697 |
| N=128 | 2.26069816166778 | 0.00205544197064 |

The successive error ratios are 3.9826 and 3.9956, consistent with the written
second-order expansion for this mode. They do not establish a physical lattice
spacing or a universal continuum limit for every state and interaction.

The negative results are part of the release:

- Rephasing only the state while leaving the links unchanged changes final
  site probabilities by maxima 0.11192, 0.24974 and 0.17001 in the three cases.
  This is a physically changed setup, not a valid mere gauge relabeling.
- Omitting the scalar term under a time-dependent relabeling produces equation
  residuals 0.48037, 0.36063 and 0.37385.
- Adding 1.7 energy units to the first site's potential still passes gauge
  covariance, but changes final probabilities by 0.08500, 0.07941 and 0.03349.
  Passing covariance therefore does not identify the Hamiltonian.

No production model, seed, tolerance or physical parameter was changed in
response to these results. Any future change to a physical model requires a
newly frozen validation design; old cases remain regression tests.

### CG.6 — Release boundary and next work

This release advances PC-05 and documents the conditional action behind PC-06.
It supplies a local continuity law and a consistent charged-wave connection.
It does not derive the Standard Model, general relativity, all material
properties, the fine-structure constant, or a unique higher-dimensional origin.
There is no new measured physical evidence in this release.

The earlier hydrogen measurement comparison remains open. The immediate
integration gate is a self-consistent field–matter benchmark with a declared
action, conserved charge, tracked total energy, Gauss-constraint monitoring and
grid/time convergence. Freeze that design before generating its validation
cases. A subsequent physical comparison must use independent calibrated data,
not merely the same imported equations as its reference.

The complete 1.83 paper and previous README, claim ledger and VERSION are
preserved under `history/pre_0.9.0`. The old VERSION file still said 0.7.0 while
the prior README and release were 0.8.0; this metadata inconsistency is corrected
to 0.9.0 without changing earlier records. New files are additive. All other
previous archive files must match byte for byte before release.

All prior boundaries remain: complete deterministic closure for a fixed
transition family is idempotent; integer-tick point capture does not enforce
continuous non-crossing; angular impulse is r cross J without an extra time
factor; finite prefixes do not identify arbitrary generators. The 2K-sample
continuation guarantee assumes a noiseless homogeneous constant-coefficient
linear recurrence of order at most K from the initial state over the declared
field. Unconstrained residuals are not identified with God; no claim of P=NP,
seven physical dimensions or unbounded computation is introduced.

### CG.7 — Sources and provenance

1. K. G. Wilson, **Confinement of quarks**, Physical Review D 10, 2445,
   published 15 October 1974. Primary precedent for exact gauge invariance with
   lattice link variables. Our finite matter calculation is narrower than
   this gauge-field theory. [Publisher and DOI](https://journals.aps.org/prd/abstract/10.1103/PhysRevD.10.2445).
2. Y. Aharonov and D. Bohm, **Significance of Electromagnetic Potentials in the
   Quantum Theory**, Physical Review 115, 485, published 1 August 1959. Primary
   theoretical reference for phase effects of electromagnetic potentials.
   [Publisher and DOI](https://journals.aps.org/pr/abstract/10.1103/PhysRev.115.485).
3. CERN, **The Standard Model**, official overview, accessed 21 September
   2026. Used only for the established distinction between the Standard Model's
   included interactions and gravity. [CERN](https://home.cern/science/physics/standard-model/).

The link identities, action variation, finite current and ring dispersion are
written explicitly above and checked where indicated. The cited constructions
are established work; this release claims their scoped integration and audit,
not a new discovery of gauge theory. Primary-source pages were checked on
21 September 2026. No new physical discovery was found or inferred from them.

## Appendix CH — A self-consistent gauge–matter ring

### CH.1 — Bounded question and assumptions

Appendix CG left one precise gap: its gauge field was prescribed, so matter did
not source the field that acted on it. This appendix asks whether a minimal
finite Hamiltonian can close that feedback loop while retaining local charge
continuity, total-energy accounting, gauge covariance and a Gauss constraint.
It does not ask whether this model is fundamental or whether space is discrete.

The declared system is a periodic one-dimensional ring. Site `j` carries a
complex matter amplitude `psi_j`; the oriented link from `j` to `j+1` carries
a compact phase `a_j` and canonical electric variable `E_j`. Positive `w_j`
and real static `V_j` are supplied. Units are dimensionless with `hbar=q=1`.
The Hamiltonian is

\[
H=\sum_j w_j|\psi_j-e^{ia_j}\psi_{j+1}|^2
 +\sum_jV_j|\psi_j|^2+\frac{\kappa}{2}\sum_jE_j^2 .
\]

Its equations are

\[
i\dot\psi=\frac{\partial H}{\partial\bar\psi},\qquad
\dot a_j=\kappa E_j,\qquad
\dot E_j=-J_j,
\]

where

\[
J_j=2w_j\operatorname{Im}(\bar\psi_j e^{ia_j}\psi_{j+1}).
\]

The field therefore responds to the matter current, while its link phase acts
back on matter. A periodic Gauss law cannot carry nonzero total charge. The
model explicitly supplies a uniform neutralizing background and defines

\[
G_j=E_j-E_{j-1}-(|\psi_j|^2-1/N).
\]

That background is an assumption, not a discovered physical constituent.

### CH.2 — Written identities

For any time-independent site phases `chi_j`,

\[
\psi_j\mapsto e^{i\chi_j}\psi_j,\qquad
a_j\mapsto a_j+\chi_j-\chi_{j+1},\qquad E_j\mapsto E_j
\]

leaves every edge difference covariant and the Hamiltonian invariant. Direct
differentiation of the matter density gives

\[
\frac{d|\psi_j|^2}{dt}+J_j-J_{j-1}=0.
\]

Consequently `dG_j/dt=0` when `dot E_j=-J_j`. Global phase symmetry preserves
`sum |psi_j|^2`, and the autonomous Hamilton equations preserve the displayed
total `H`. These are exact claims about the semidiscrete continuous-time ODE.
They do not mean a generic time-stepping algorithm preserves energy exactly.
The implementation therefore reports drift and refinement instead of silently
promoting floating-point tolerance to a theorem.

### CH.3 — Frozen computation and counterexample

Protocol `SCGM-1-RING-2026-09-22-v1` was fixed before execution; SHA-256 is
`8ad5db32322480dfd10376835307de3018f7352bd857f00c4c09afb73e8e4eac`.
Three seeded rings with `N=6,9,12`, random normalized matter states, real
potentials, positive hopping weights and exactly initialized Gauss data were
integrated with DOP853. No measured value or physical parameter was fit.

| Frozen check | Executed result |
|---|---:|
| Entire package unit suite | 130/130 pass |
| Maximum norm drift | 2.91e-12 |
| Maximum total-energy drift | 8.77e-12 |
| Maximum Gauss residual | 9.29e-13 |
| Maximum local-continuity residual | 2.78e-16 |
| Maximum gauge-related trajectory error | 1.95e-12 |
| Independent scalar-loop RK4 final-state error | 2.18e-13 |
| RK4 error ratios on step halving | 15.81 to 16.46 |

The independent program imports no production module. It separately constructs
the link forces with scalar loops and evolves them using fixed-step RK4. The
near-sixteen refinement ratio is consistent with fourth-order time convergence
on these cases; it is not a spatial continuum result.

The decisive failure control holds `dot E=0` while the moving link still drives
matter. In the same cases, the maximum Gauss residual becomes 0.156 to 0.369
and total-energy drift becomes 0.0179 to 0.0471. Thus gauge-covariant matter in
a supplied field is not automatically a self-consistent field–matter theory.

### CH.4 — What changed, and what did not

PC-06 advances from a written Maxwell action to one executable feedback and
constraint benchmark. It does not close PC-06. A one-dimensional ring has no
magnetic plaquettes or three-dimensional radiation, and this release performs
no spatial refinement. The action, charge normalization, hopping weights,
potential and background are assumed. Classical field–wave coupling is not
full QED, and matching its own conservation identities is not external physical
validation.

No evidence for a physical fourth, fifth or seventh dimension is obtained. No
fine-structure constant, elemental property, fundamental grid spacing or tick
is derived. An unconstrained discrepancy would remain model inadequacy, not a
unique hidden variable or theological identification. Revision 1.14's
idempotent fixed-law closure and all revision 1.15 boundaries remain intact,
including the restricted noiseless order-at-most-K assumption behind the 2K
linear-recurrence continuation result.

The next test is a frozen two-dimensional plaquette model with magnetic energy,
spatial and temporal refinement, and a validation record reserved before any
model change. Only after that should a separate calibrated measurement be used;
agreement with equations imported into both generator and reference would not
be independent evidence.

### CH.5 — Primary-method sources

1. J. Squire, H. Qin and W. M. Tang, **Geometric integration of the
   Vlasov–Maxwell system with a variational particle-in-cell scheme**, 2014.
   Discrete gauge symmetry yields exact Gauss preservation.
   [arXiv](https://arxiv.org/abs/1401.6723).
2. M. Kraus et al., **GEMPIC: geometric electromagnetic particle-in-cell
   methods**, 2017. The work distinguishes exact constraint and charge
   preservation from excellent long-time energy behavior.
   [arXiv](https://arxiv.org/abs/1609.03053).
3. J. Xiao et al., **A lattice Maxwell system with discrete space-time symmetry
   and local energy-momentum conservation**, 2017.
   [arXiv](https://arxiv.org/abs/1709.09593).
4. P. Chen et al., **A unified Hamiltonian solution to Maxwell–Schrödinger
   equations for modeling electromagnetic field–particle interaction**, 2016.
   [arXiv](https://arxiv.org/abs/1609.02246).

These primary works establish relevant methods and conservation distinctions.
They do not validate this toy ring as nature or as evidence for extra dimensions.

## Appendix CI — Magnetic plaquettes and the limits of Gauss preservation

### CI.1 — Declared two-dimensional Hamiltonian

Appendix CH closed the smallest feedback loop on a one-dimensional ring, which
has no magnetic plaquette. The next bounded question is whether a two-dimensional
link field can carry magnetic energy while matter, electric links and magnetic
curl exchange energy without violating the Gauss constraint.

Let a periodic `N_x` by `N_y` grid carry complex site amplitudes `psi`, real
oriented link potentials `a_x,a_y`, and conjugate electric variables `E_x,E_y`.
The oriented plaquette is

\[
B_{x,y}=a^x_{x,y}+a^y_{x+1,y}-a^x_{x,y+1}-a^y_{x,y}.
\]

With the same gauge-covariant matter-edge energy as Appendix CH, the declared
dimensionless Hamiltonian is

\[
H=H_{\rm matter}+\frac{\kappa}{2}\sum(E_x^2+E_y^2)
 +\frac{\beta}{2h^2}\sum B^2.
\]

This is a noncompact classical U(1) link model in temporal gauge, not compact
quantum lattice gauge theory. The residual gauge map is

\[
\psi\mapsto e^{i\chi}\psi,\quad
a_x\mapsto a_x+\chi-\chi_{+x},\quad
a_y\mapsto a_y+\chi-\chi_{+y}.
\]

Every plaquette is invariant. Hamilton's equations set `dot a=κE`; the electric
equations contain both minus the matter current and the signed adjacent-
plaquette difference. The same explicit uniform background as Appendix CH makes
the periodic total charge zero.

### CI.2 — Written constraint and conservation results

Matter evolution gives the exact lattice continuity equation

\[
\dot\rho+\Delta_x^-J_x+\Delta_y^-J_y=0.
\]

The divergence of the magnetic plaquette force vanishes because the two mixed
finite differences cancel. Therefore

\[
G=\Delta_x^-E_x+\Delta_y^-E_y-\rho
\]

is constant when initialized to zero. Gauge invariance supplies this constraint
structure; the autonomous Hamiltonian supplies exact continuous-time total-
energy conservation. These are semidiscrete mathematical identities. The
floating-point integrator is evaluated by drift and convergence, not described
as exactly energy preserving.

### CI.3 — Frozen computation and independent replay

Protocol `P2D-1-MAGNETIC-2026-09-23-v1` was frozen before execution. Its SHA-256
is `f9d347c0e2a9b5477a5b55b6c30fb5d898b2e3fa1bb5bd4490e4a2d8445ac65a`.
Three seeded `4x5`, `5x6` and `6x7` coupled grids use positive hopping weights,
static real potentials and exactly initialized periodic Gauss data. No physical
parameter or measurement was fitted.

| Frozen check | Executed result |
|---|---:|
| Entire package unit suite | 142/142 pass |
| Maximum norm drift | 2.39e-12 |
| Maximum total-energy drift | 1.20e-11 |
| Maximum Gauss residual | 7.18e-13 |
| Maximum continuity residual | 1.81e-16 |
| Maximum gauge-related trajectory error | 1.01e-12 |
| Magnetic plaquette change | 0.236–0.672 |
| Independent final-state error | 1.76e-13 |
| RK4 time-refinement ratios | 15.51–16.44 |

The independent program imports no production module. It assembles every edge
and plaquette contribution with separate component loops and evolves the system
with fixed-step RK4.

For a linear vacuum magnetic mode `(m_x,m_y)=(2,1)` on a `2pi` square, the
discrete frequencies for `N=16,32,64` are `2.18764249`, `2.22388190` and
`2.23301646`, approaching the continuum value `sqrt(5)=2.23606798`. The error
falls by factors 3.97384 and 3.99345, as predicted by the second-order sine
dispersion expansion. A separately assembled operator Rayleigh quotient agrees
with all three frequencies within 4.45e-16.

### CI.4 — Counterexamples and correction

Two omissions distinguish the conservation requirements:

- With matter-current feedback removed, the maximum Gauss residual is
  0.0806–0.1460 and energy drift is 0.00228–0.00841.
- With magnetic-curl feedback removed, energy drift is 0.00714–0.00938. The
  omitted magnetic force has zero discrete divergence, so this omission need
  not reveal itself through Gauss-law failure.

The resulting correction is substantive: **preserving Gauss's law is necessary
but not sufficient to certify correct total field–matter dynamics.** Energy
exchange and the complete declared equations must be checked separately.

### CI.5 — Evidence boundary and next gate

The spatial refinement above covers one linear vacuum mode. It does not prove
spatial convergence of the interacting matter–field system. The model is 2D,
nonrelativistic and classical in its field variables; it has no 3D radiation,
spin, pair creation, photons, renormalization or calibrated physical constants.
It neither derives QED nor identifies a fundamental grid, tick or extra
dimension. Matching an imported continuum formula is implementation validation,
not new measured evidence.

The next test is a preregistered manufactured interacting solution that forces
both matter and link sectors and measures joint spatial convergence. Only after
that model is frozen should new calibrated measurements be reserved. All prior
corrections remain: fixed-law deterministic closure is idempotent; integer-tick
capture is not continuous non-crossing; angular impulse has no extra time
factor; arbitrary generators are not identified by finite prefixes; and the
2K continuation result assumes a noiseless homogeneous constant-coefficient
linear recurrence of order at most K valid from the initial state.

### CI.6 — Primary sources

1. K. G. Wilson, **Confinement of quarks**, *Physical Review D* 10, 2445
   (1974). [DOI](https://doi.org/10.1103/PhysRevD.10.2445).
2. J. Kogut and L. Susskind, **Hamiltonian formulation of Wilson's lattice
   gauge theories**, *Physical Review D* 11, 395 (1975).
   [DOI](https://doi.org/10.1103/PhysRevD.11.395).
3. J. Squire, H. Qin and W. M. Tang, **Geometric integration of the
   Vlasov–Maxwell system with a variational particle-in-cell scheme** (2014).
   [arXiv](https://arxiv.org/abs/1401.6723).
4. J. Xiao et al., **A lattice Maxwell system with discrete space-time symmetry
   and local energy-momentum conservation** (2017).
   [arXiv](https://arxiv.org/abs/1709.09593).

These primary sources establish relevant link, plaquette and structure-
preservation methods. They do not supply experimental validation of this model.


## Appendix CJ — Corrected magnetic frequency and interacting spatial consistency (revision 1.87)

### CJ.1 — Correction to the revision 1.86 table

The Hamiltonian in Appendix CI has electric energy `kappa/2 sum E^2` and magnetic energy `beta/(2h^2) sum B^2`; Hamilton's equations give `a_dot=kappa E` and, in vacuum, `E_dot=-(beta/h^2) curl*B`. Consequently, for Fourier mode `(mx,my)` on a square lattice,

`omega_h = sqrt(kappa beta) (2/h) sqrt(sin^2(mx h/2)+sin^2(my h/2))`.

Revision 1.86's frequency list omitted `sqrt(kappa beta)` and therefore applies only to the auxiliary unit-speed case. Its published error *ratios* remain correct because a common factor cancels. At frozen `kappa=0.7,beta=0.6`, the N=16,32,64 frequencies are **1.41775437, 1.44124019, 1.44716007**, approaching **1.44913767**, rather than the old numbers approaching `sqrt(5)`. The program now accepts explicit coefficients and an independently assembled Fourier curl-curl operator confirms these values. The old report and implementation remain in the historical archive.

### CJ.2 — Frozen smooth coupled spatial test

Freeze `L=2pi`, `epsilon=0.19`, `A_x=0.08 cos y`, `A_y=0.07 sin x`, `V=0.08 cos(x-2y)`, `kappa=0.7`, `beta=0.6`, and the normalized matter wave `phi=[1+epsilon exp(i(x+y))]/[L sqrt(1+epsilon^2)]`. On grids N=12,24,48 use `psi=h phi`, `a=h A` evaluated at link midpoints, hopping `1/h^2`, and nonzero electric links satisfying the discrete Gauss constraint exactly. The electric field has the correctly normalized sinusoidal density divergence; analytic continuum targets use the covariant derivative `D=partial+iA` and current `J=2 Im(conj(phi) D phi)`. Compare all five components of the semidiscrete time derivative, each divided by `h`, to the independently differentiated continuum derivative at matching points.

The frozen acceptance range for each adjacent maximum-error ratio was [3.1,4.9]. Matter errors were 0.00179400, 0.000452067, 0.000113241; electric-x errors 0.00134705, 0.000340125, 0.0000852427; link errors 0.0000361663, 0.00000922473, 0.00000231762. Electric-y errors also fell with ratios 3.9597,3.9899. All five components passed, with minimum ratio 3.9206 and maximum 3.9921. Gauss residual stayed below 6.1e-18. Removing current or magnetic force leaves respective electric RHS changes 0.01655 and 0.04793 at N=48, so the test exercises both couplings. An independent Fourier eigenvalue calculation and centered-difference continuum derivatives checked the material analytical terms. All 142 inherited package tests passed.

### CJ.3 — Claim boundary and next experiment

This is **written conditional mathematics and finite synthetic equation verification**. Exact Gauss compatibility follows algebraically for the chosen initial data. The measured convergence is an *instantaneous spatial consistency* check, not convergence of the evolved nonlinear interacting trajectory and not a full forced manufactured solution. The continuum U(1) matter model is an assumption, without measured calibration or demonstrated microscopic emergence; no fundamental spatial grid, clock, extra dimension or theological interpretation follows. Next freeze a smooth forced coupled trajectory and independently verify time and spatial convergence before fitting new measured data. The fixed-transition complete closure theorem remains idempotent. Integer-tick point capture cannot imply continuous non-crossing, angular impulse has no additional time factor, arbitrary generators cannot be inferred from finite prefixes, and the 2K continuation theorem assumes a noiseless homogeneous constant-coefficient recurrence of order <=K from the initial state.

Methodological primary source: P. J. Roache, *Code Verification by the Method of Manufactured Solutions*, Journal of Fluids Engineering **124** (2002), DOI [10.1115/1.1436090](https://doi.org/10.1115/1.1436090). The present test is a restricted RHS consistency check rather than a full manufactured-solution implementation.


## Appendix CK — Frozen evolving coupled spatial check (revision 1.88)

Starting from the same smooth nonzero matter, Gauss-compatible electric and magnetic initial fields as Appendix CJ, freeze grid sizes 6/12/24, `t in {0,0.015,0.03}`, DOP853 tolerances `rtol=1e-11,atol=1e-13`, and the [2.5,5.5] error-ratio acceptance gate before execution. Compare normalized site wavefunctions at common coordinates and average adjacent fine edge values to the coarse midpoint for all gauge and electric links. This alignment is essential because edge variables occupy staggered positions.

At `t=0.03`, the coarse-medium to medium-fine maximum-error ratios are 3.81562 for matter; 4.15575 and 4.09895 for horizontal/vertical gauge links; 4.11243 and 4.01129 for electric links. The worst Gauss residual is under 2.35e-17 and norm drift under 2.23e-16. A separately vectorized bond-gradient implementation checks the RHS at an initial and independently perturbed state; a fixed-step RK4 replay at N=12 differs from production DOP853 by 1.11e-16 in maximum component. The full inherited 142 tests also pass.

**Scope:** This is finite synthetic *short-time three-grid self-convergence* for one chosen smooth initial condition and imported nonrelativistic U(1) equations. Agreement with second-order scaling is not an analytic stability or convergence theorem for arbitrary initial conditions, and it is not verification against an exact evolved continuum trajectory. No new measured evidence, fundamental time tick, spatial lattice, extra physical dimensions, QED derivation or theological identification is supplied. Next freeze a genuinely manufactured forced trajectory with independently computed solution and test longer times and less smooth initial conditions, then reserve independent measurements. The v1.14 fixed-family closure and v1.15 noncrossing, angular impulse, generator and restricted 2K recurrence corrections remain intact.

Related primary research: Christiansen (2023), [*Convergence of a discretization of the Maxwell–Klein–Gordon equation based on finite element methods and lattice gauge theory*](https://doi.org/10.1002/num.23008). Its model and proof differ from these calculations.


## Appendix CL — An exact driven evolution and a sampling counterexample (revision 1.89)

For a dimensionless periodic square of length `L=2pi`, fix a normalized unit-magnitude matter wave `phi=L^-1 exp(i[0.23 sin(x+y)+0.65t])`, nonzero gauge links `A_x=0.08 cos(0.9t) cos y`, `A_y=0.07 sin(0.9t+0.37) sin x`, and electric fields `F=partial_t A/0.7`. Let `V=0.08 cos(x-2y)` and `beta=0.6`. Gauss's law holds for this exact continuum state because density is uniform and each electric component has zero divergence. Add explicitly declared external terms `S_psi=phi_t+i H(A)phi` and `S_F=F_t+J+beta curl*B`, where `H=-D^2+V`, `D=partial+iA`, `J=2 Im(conj(phi)Dphi)` and `B=partial_x A_y-partial_y A_x`. Then the continuum functions are an exact solution of the *driven* equations by direct substitution. The forcing can exchange charge with matter, so the unforced discrete Gauss invariant need not hold at each finite grid size.

**Retained failed test:** Frozen v1 sampled `J` at cell centers even though discrete currents and `S_F` live on oriented link midpoints. In v1 the matter error ratios were 3.68/3.90 but the link and electric ratios were only 2.07–2.37, below the frozen minimum 2.5. Its protocol, code and results are preserved under `history/failed_mms_0.14_v1`. This is a counterexample to inferring spatial order without aligning staggered observables.

**Corrected frozen test:** v2 moves the prescribed current to the appropriate midpoint. At grid sizes N=8,16,32 and t=0.12 the ten adjacent error ratios for matter, both gauge links and both electric links range **3.67699–3.98003**, inside the frozen [2.5,5.5] interval; maximum normalized component error at N=32 is 4.35e-5. Independent centered differences verify nine analytical source components within 1.13e-8 against a 5e-7 gate. This is finite computation plus a written exact-solution construction, not new measured evidence. The v1.86 frequency correction and all historical negative results remain. The next test should examine longer evolution and a case with physically sourced charge under the **unforced** equations, keeping validation data fresh.

Mathematical and scientific boundaries: The sources are imposed, not a derivation of electromagnetism or of a physical lattice. The result does not prove uniform stability or general convergence of unforced coupled trajectories, and does not evidence additional spatial dimensions or a theological mechanism. Fixed-family complete deterministic closure remains idempotent. Integer-tick point capture is not continuous non-crossing; angular impulse has no extra time factor; arbitrary generators are not identified by finite prefixes; the 2K continuation statement is restricted to noiseless homogeneous constant-coefficient recurrences of order at most K from the initial state.

Primary method reference: Roache, [*Code Verification by the Method of Manufactured Solutions*](https://doi.org/10.1115/1.1436090), *Journal of Fluids Engineering* 124 (2002).


## Appendix CM — Longer unforced coupled evolution (revision 1.90)

Use the existing smooth periodic charged matter, nonzero electric links satisfying Gauss and nonzero magnetic plaquettes from Appendix CJ. Freeze lattice sizes 6/12/24, source-free Hamiltonian dynamics, `kappa=0.7,beta=0.6`, checkpoints `t=0,0.25,0.5,0.75,1.0`, DOP853 tolerances `rtol=1e-11, atol=1e-13`, and a spatial self-error ratio interval [2,6] for all five matter, gauge and electric components at both t=0.5 and t=1.0. Fine link values are averaged to the coarse link midpoint; matter values are compared at coincident sites. Thresholds for Gauss, norm and energy drift are 1e-8; independently disable each coupling as a failure control.

At t=0.5 the ten reported ratios over five fields run 3.3659–3.9397, and at t=1.0 they run 3.2231–4.1802, inside the frozen interval. The maximum Gauss residual is 9.70e-14, norm drift 3.25e-13 and energy drift 1.05e-12 across the fifteen states. At grid n=12, removing current feedback produces 0.001134 energy drift; removing magnetic feedback yields 0.002263, each exceeding the frozen 1e-5 failure-control gate. Independently vectorized bond-gradient equations advanced for 1,500 RK4 steps agree with production DOP853 at t=1 within 3.21e-12 in the maximum packed component. All 142 inherited tests passed.

This is **finite synthetic evidence of spatial self-consistency for one unforced evolving state through t=1** under an assumed dimensionless nonrelativistic gauge-matter Hamiltonian. It is not an exact comparison with the continuum trajectory, a theorem on arbitrary initial states or long-time stability, nor empirical evidence of a fundamental grid, physical extra dimensions or theology. The prior driven exact-solution verification, its failed source-placement trial and the v1.86 magnetic-frequency correction remain in the history. Next test a different smoother and a deliberately less smooth initial condition, varying the time solver tolerances independently and reserving fresh data for any later model alteration.

Related primary research: Christiansen et al., [*Convergence of a discretization of the Maxwell–Klein–Gordon equation based on finite element methods and lattice gauge theory*](https://doi.org/10.1002/num.23008), published 8 March 2023. Its proof pertains to different equations and discretization. The v1.14 fixed-family complete deterministic closure is still idempotent; v1.15 point capture still lacks continuous noncrossing, angular impulse has no extra time factor, finite prefixes do not identify arbitrary generators, and 2K continuation assumes a noiseless homogeneous constant-coefficient recurrence of order at most K valid from its initial state.


## Appendix CN — Regularity limits on grid claims (revision 1.91)

The prior smooth examples suggested second-order spatial self-consistency for specific initial fields. Freeze a paired test on grids n=8,16,32 at t=0,0.1,0.25: one original smooth charged state; one normalized profile `psi proportional to 1+0.22|sin x|+0.15 i sin y`, with exactly solved discrete periodic Gauss field, the same smooth gauge links and potential, and no external forcing. Preserve the same coarse-site and midpoint-averaged-link comparison. Before running, propose that at t=0.25 at least one cusp ratio be below 2.5 while every smooth ratio remains above 2.5; also require norm, Gauss and energy drifts below 1e-8.

At t=0.25 the smooth matter maximum-error ratio is **3.65792**, versus **1.76565** for the cusp matter. Smooth gauge and electric ratios are 3.872–3.885; cusp gauge and electric ratios remain 3.083–4.288. All frozen safety checks pass, including maximum cusp energy drift 2.67e-12. An independently vectorized bond-current solver and RK4 reproduce the n=16 cusp state at t=0.25 within 3.62e-13 in maximum packed-state difference. Thus the specific universal extrapolation of the smooth near-four ratio to rough matter is contradicted. **A two-grid self-error ratio is not a theorem giving the asymptotic order**; neither the rough profile's exact continuum error nor general convergence has been measured.

This is a finite synthetic negative control. Conservation laws do not alone certify spatial convergence. The analytic continuum field and its smoothness class must be stated before making accuracy claims; no measured spectrum or material constant has yet been predicted independently. This model cannot be identified as a physical grid merely by converging to the continuum Hamiltonian it imports. The next test should compare a prespecified physical observable against calibrated measurements with independent parameter fixation and a competing conventional model, or derive a general stability bound under explicitly stated regularity assumptions. Earlier 1.86 frequency correction and 1.89 failed midpoint test remain recorded.

Related primary mathematical work: Ji, Ostermann, Rousset and Schratz, [*Low Regularity Full Error Estimates for the Cubic Nonlinear Schrödinger Equation*](https://doi.org/10.1137/23M1619617). It proves regularity-dependent rates for a different equation, norm and scheme; it does not prove the present result. Complete closure for fixed transition laws remains idempotent; integer-tick capture still does not enforce continuous non-crossing, angular impulse still has no extra time multiplier, arbitrary generators are not inferred from finite prefixes, and 2K continuation assumes noiseless homogeneous constant-coefficient linear recurrence order at most K from the initial state.


## Appendix CO — Linear dispersion can distinguish equations, not discrete space (revision 1.92)

### CO.1 — Restricted conditional experiment

On the periodic 2D noncompact U(1) vacuum sector described earlier, linearizing the magnetic and electric link equations yields the frequency

`ω_h(m)=sqrt(κβ) (2/h) sqrt[sin²(m_x h/2)+sin²(m_y h/2)]`, with `h=2π/N`, `κ=0.7`, `β=0.6`.

Freeze a local isotropic nondispersive continuum comparator `ω_c(m)=c|m|` and calibrate its only speed `c` to mode `(1,0)`. With mode `(2,0)` held out, its prediction is `2 ω_h(1,0)`, whereas the frozen lattice predicts `2cos(π/N)ω_h(1,0)`. Consequently, the relative discrepancy is exactly `1−cos(π/N)`. At N=16,32,64 it equals 1.92147%, 0.48153%, 0.12045%. A synthetic 0.1% *relative ratio* tolerance distinguishes the declared N=32 alternative from this particular comparator, but the discrepancy falls to 0.1% or less at every integer N≥71. Equal-magnitude `(5,0)` and `(3,4)` modes provide a directional test: their relative lattice frequency differences at N=16,32,64 are 7.83310%, 1.87617%, 0.46422%, whereas the calibrated restricted continuum comparator predicts equality. Each claim assumes assigned mode labels, a fixed spatial length L, and a predeclared model family, not calibrated real-world units.

### CO.2 — Constructive continuum counterexample

Define a **continuous** field `f(x,y)` on the 2π-periodic torus, with evolution `f_tt=−L_h f` and translation operator

`L_h f=(κβ/h²)[4f(x,y)−f(x+h,y)−f(x−h,y)−f(x,y+h)−f(x,y−h)]`.

On each continuous Fourier mode `exp(i[m_x x+m_y y])`, the eigenvalue is exactly `ω_h(m)²`. This follows by substituting the exponential and using `2−2cos θ=4sin²(θ/2)` in each coordinate. **No restriction to lattice sites is needed**: the function and operator act at every real `(x,y)`. This written countermodel rules out inferring ontological spatial discreteness from the linear dispersion alone if continuous nonlocal spatial interactions are admitted. It is a scalar linear vacuum counterpart and does not reproduce the full nonlinear charged-matter equations or establish their physical adequacy.

### CO.3 — Verification and next falsifier

The frozen study computes all four specified modes at N=16,32,64. A separately assembled lattice curl-curl operator and independently evaluated continuum shift operator agree with the formula within 2.89e-13; a second scalar audit checks all 12 cases, the `1−cos(π/N)` identity and N=71 cutoff. These are *written mathematics and finite synthetic calculations*, not measured evidence. The next physical claim must fix the spatial scale and free parameters independently, predeclare a standard and a continuous-interaction alternative, derive a differing observable of the **complete** field–matter model, and reserve measured data to test it. Absent such a test, retain the continuum and lattice interpretations as observationally indistinguishable at this level. Fusi et al. (2026), [arXiv:2601.10184](https://arxiv.org/abs/2601.10184), develop related linear lattice/continuous correspondences; this does not transfer a theorem to the nonlinear engine.

All previous corrections remain in force: complete deterministic closure for fixed transitions is idempotent; integer-tick point capture is not continuous non-crossing; angular impulse has no extra time factor; finite observed prefixes do not identify arbitrary generators; and 2K continuation assumes a noiseless homogeneous constant-coefficient linear recurrence of order at most K from the initial state. Neither mathematical ambiguity nor any unmeasured residual identifies God, seven dimensions or P=NP.


## Appendix CP — Exact nonlinear equivalence at designated sampling nodes (revision 1.93)

### CP.1 — Declared models and theorem

Take any fixed odd N and the full six-real-component finite-dimensional vector field `du/dt=F_N(u)` of Appendix CI: real and imaginary charged matter, two real gauge-link angles, and two real electric-link fields at each site of an N×N periodic torus. Keep every existing nonlinear bond phase `exp(i A)`, matter current, magnetic plaquette, and parameter fixed. Let `S_N` evaluate each component at the designated positions `(2πj/N,2πk/N)`. Let `I_N` be the *unique* trigonometric interpolant with frequencies `-(N-1)/2,…,(N-1)/2` in each direction, componentwise, of those six sampled arrays. Discrete Fourier orthogonality gives the exact algebraic identity `S_N I_N=Id`.

Construct a continuously defined real field `U(x,y,t)` in this finite Fourier subspace, including values at all *off-grid* positions, by the explicitly declared evolution

`∂_t U = I_N F_N(S_N U)`, with `U(0)=I_N u(0)`.

**Conjugacy proof (written mathematics):** Applying `S_N` gives `d(S_N U)/dt=S_N I_N F_N(S_N U)=F_N(S_N U)`; the common initial condition and local uniqueness of the finite smooth ODE imply `S_N U(t)=u(t)` wherever its solution exists. Conversely `I_N u(t)` solves the constructed continuous-field equation. Thus *every* observation consisting solely of the designated samples at any times—including nonlinear charged-matter interactions and correlated magnetic effects—is identical for these two descriptions. Nothing in the argument identifies a physical ontology. The continuous field uses N and the chosen sites in its law, has only `6N²` real degrees of freedom, and is generally nonlocal; it is not an independently derived local Maxwell/QED continuum field theory. In particular, a function `sin(Nx)` vanishes at all x-grid nodes but not between them: allowing independent off-grid degrees of freedom breaks the constructed uniqueness, so it is **not** claimed that the two descriptions match every possible off-grid measurement.

### CP.2 — Frozen finite checks and counterexample

The preregistered file `NONLINEAR_CONJUGACY_PROTOCOL_0.18.json` fixes N=5,7, seeds 20260924 and 20260925, `κ=0.7`, `β=0.6`, checkpoints 0/.04/.08 and tolerances before scoring. Four source-free nonlinear initial states have nonzero magnetic plaquette and matter current. The maximum sampled trajectory discrepancy between independently integrated lattice-state and Fourier-coefficient ODEs is **1.55e-12** (threshold 1e-9); maximum sample inversion error **1.67e-16**; maximum sampled RHS conjugacy error **6.67e-16**. An independently assembled vectorized bond-current/magnetic RHS agrees within **2.23e-16**, and its 400-step RK4 replay agrees with a separate DOP853 path within **5.04e-14**. Independent explicit Fourier summation checks the interpolation. The three additional regression tests pass, as do all 142 inherited tests. The exact equivalence is a theorem; the listed tolerances and outcomes only check its particular implementation. Off-grid values in this test are chosen by interpolation, not measured.

The construction is a direct application of trigonometric collocation, related to [Tadmor's Fourier pseudospectral notes](https://www.math.umd.edu/~tadmor/lecture-notes/hyperbolic/node30.html) (1998). [Fusi et al.](https://arxiv.org/abs/2601.10184) (revised 12 March 2026) study *linear* lattice–continuum correspondences; neither reference provides a physical identification theorem for this nonlinear engine. Thus Appendix CO's linear nonlocal shift counterexample is strengthened in one precise direction: even full nonlinear **sampled** dynamics cannot identify a fundamental spatial grid against arbitrary engineered continuous interpolation laws. The constructed law has no independent simplicity/locality advantage and does not predict observed data. No measured evidence, derived fine-structure constant, fourth/fifth dimension, seven physical dimensions, P=NP, or theological cause follows.

**Next test:** independently specify a physically motivated *local* continuous gauge–matter comparator with the same calibrated parameters and measurement operator; find a distinctive observable before collecting new measured data, including an independently measured off-grid/scale control. Exact agreement inside the interpolation class is expected and cannot be treated as validation of either ontology. All earlier negative results and v1.14/v1.15 corrections, including the initial-state noiseless order-at-most-K assumption for the 2K recurrence continuation theorem, remain unchanged.


## Appendix CQ — A local continuum comparator and an off-site observable (revision 1.94)

### CQ.1 — Independent equation and observation definitions

On a periodic square of side `L=2π`, declare complex matter field `φ`, real vector potential `A=(A_x,A_y)`, conjugate electric field `E=(E_x,E_y)`, scalar potential `V(x,y)=0.08 cos(x−2y)`, covariant derivative `D=∇+iA` and magnetic field `B=∂_x A_y−∂_y A_x`. The independently specified *local* continuum equations in temporal gauge are

`i φ_t = (−D_x²−D_y²+V)φ`; `A_t=κ E`; `E_t=(−j_x−β ∂_y B, −j_y+β ∂_x B)`; `j=2 Im(φ*Dφ)`, with `κ=0.7` and `β=0.6`. On smooth periodic fields, integration by parts gives formal conservation of the declared Hamiltonian `H=∫(|Dφ|²+V|φ|²+κ|E|²/2+β B²/2)dxdy`, and `∂_t|φ|²+∇·j=0`. These are conditional statements from a postulated nonrelativistic continuum action; they are not independently measured laws derived from a string grid. The equations contain no N or finite translation by h. The Fourier implementation approximates their derivatives numerically; the PDE itself is spatially local.

At t=0 fix `φ=[1+0.19 exp(i(x+y))]/[L sqrt(1+0.19²)]`, `A_x=0.08 cos y`, `A_y=0.07 sin x`, `E_x=E_y=0.19 sin(x+y)/[L²(1+0.19²)]`. The continuum initial Gauss law `∇·E=|φ|²−L^-2` holds. Sample the same functions as site matter amplitudes `ψ_jk=h φ(x_j,y_k)`, link angles `h A_i` and discrete-Gauss-compatible link electric values in the existing lattice engine (`h=L/N`, exact discrete correction for E). This correction is declared, not fitted to final outcomes.

**Fixed off-site observation map:** at physical coordinate `(x,y)=(0.73,1.11)` and dimensionless time `t=0.12`, compare gauge-invariant matter probability density `|φ|²` and oriented magnetic flux density B. For the lattice, use unique band-limited trigonometric interpolation of `ψ/h` at sites and of the oriented plaquette angle divided by `h²` at plaquette centers. The interpolation is an explicit observation assumption; it does not imply the lattice has physical off-site matter. Initial density agrees exactly between the declarations at this point. The initial magnetic readouts differ because a finite plaquette is not an infinitesimal curl.

### CQ.2 — Frozen computation, numerical audit and scope

The SHA-256-fixed `LOCAL_COMPARATOR_PROTOCOL_0.19.json` specifies lattice N=8/16/32, independently solved continuum reference grids 32/48, one off-site point and t=0/.12, DOP853 tolerances, two readouts and success gates. No fitting uses t=.12 outputs. The continuum references agree to `2.63×10^-13` in density and `1.53×10^-16` in magnetic field at t=.12. The 48-point reference reads **density 0.0252434373445820; B 0.1234163057828462**. Lattice outputs read, respectively, **(0.0251077069609426, 0.120286642983818)** for N=8, **(0.0252093513758201, 0.122629478336591)** for N=16, and **(0.0252348926354291, 0.123219322375936)** for N=32. The finite lattice and local PDE thus make different conditional predictions, though lattice off-site absolute density errors shrink by 3.9820 and 3.9891 and magnetic errors by 3.9776 and 3.9944. A numerical ratio in one smooth synthetic case is not a general convergence theorem or an empirical grid spacing.

The separately frozen audit derives the initial continuum RHS directly from analytic trigonometric fields (maximum difference `9.13×10^-15`), replays t=.12 with independent 240-step RK4 (off-site readout difference `5.86×10^-14`) and checks relative continuum energy drift `2.23×10^-16`. Maximum continuum numerical Gauss residual is `2.93×10^-13`; lattice Gauss and norm residuals stay below the frozen `10^-8` gates. All inherited tests and three added regression tests pass. These are written conditional mathematics and finite synthetic implementation checks, **not measured evidence**.

### CQ.3 — Physical-data applicability gate

The user's supplied ATLAS/HEPData [clockwork-gravity search](https://arxiv.org/abs/2305.10894) reports limits from 13 TeV proton collisions in the `k–M_5` parameter plane; it does not measure this model's off-site 2D charged-field probability density or plaquette flux. The [Fermi GRB 090510 observation](https://arxiv.org/abs/0908.1832) tests a relativistic photon propagation delay; this engine does not specify a 3D relativistic photon, emission model, calibrated length h or map from dimensionless time to seconds. Its bound cannot honestly be imported as an exclusion of the current lattice. [Ma, Zhang and Cao](https://doi.org/10.1093/imanum/drab076) (published 26 October 2021) analyze a Maxwell–Schrödinger temporal-gauge discretization; their theorem concerns their own equations and does not prove these code outputs.

**Decision:** the frozen synthetic experiment discriminates the *two declared equation families* at finite N with an assumed off-site observation map; it cannot prefer either as a physical theory with the currently available measurements. Revision 1.93's engineered nonlocal continuous exact counterpart remains observationally equivalent at chosen sample sites, so permitting an unrestricted continuum law restores ambiguity. A physical claim requires independent units and grid-scale calibration, a 3D/relativistic electromagnetic extension if using real photons, a validated detector map with uncertainties, and a held-out measured observable that the prespecified competing models predict differently. No extra dimension, seven-dimensional ontology, P=NP or theological cause is inferred. Fixed-law closure idempotence, the v1.15 angular-impulse and sampled-collision corrections, the finite-prefix ambiguity, and the noiseless homogeneous order-at-most-K initial-state premise for 2K recurrence continuation all remain unchanged.

## Appendix CR — Mirrored four-dimensional geometry, inaccessible coordinates and causal shadows (revision 1.95)

### CR.1 — Which matrix is the right-glove operation?

Write a **hypothetical fourth *spatial* coordinate** as `w`, separately from physical time `t`. A mirror across `w=0` is `R=diag(1,1,1,-1)`, with determinant `-1`; the 3D observation map is `P=[I_3  0]`. Therefore `PR=P`. Reflection changes a signed oriented four-volume `det[v_1-v_0,...,v_4-v_0]/4!` to its negative. This is the numerical analogue of swapping a handed glove: the **sign of an orientation**, not an observed scalar 1 literally becoming a physical object −1. In four spatial dimensions, flipping *all* coordinates (`-I_4`) has determinant `+1`, so it is not an orientation reversal. A single coordinate sign reversal has determinant `-1`.

The full frozen unlabelled example consists of five vertices in `(x,y,z,w)`:

`(0,0,1,0), (1,0,1,0), (0,1.2,1,0), (0,0,2.4,0), (0.21,0.35,1.26,1.1)`.

The oriented four-volumes are exactly `+77/1000` and `−77/1000` after applying `R`. An independent rational-arithmetic enumeration of all `5!=120` permutations of their squared-distance matrix found **one** automorphism: the identity. Consequently the unlabelled simplices cannot be related by an orientation-preserving 4D rigid motion. Indeed such a map, followed by `R^−1`, would be an orientation-reversing symmetry of the original simplex; trivial distance automorphism forces it to fix all five affinely independent vertices, so it would have to be the identity, a contradiction. Distinguish *physical translation/rotation of an unchanged object* from an imposed reflection, which is an orientation-reversing operation. This statement is Euclidean geometry conditional on the existence of the assumed four spatial coordinates.

### CR.2 — Shadows that cannot tell the pair apart

**Exact identifiability theorem (written proof):** Let an observation under each permitted lamp/instrument setting `u` be `O_u(X)=F_u(PX)` for any deterministic or random 3D transport-and-readout law `F_u` with the same conditional distribution when `PX` is the same. Since `PR=P`, `O_u(RX)=O_u(X)` for every such `u` and every 4D shape `X`. Additional ordinary 3D lighting angles can improve a reconstruction of `PX` but cannot recover the discarded `w` values. The theorem does not cover a coupling to `w`, a different observation map, or a true 4D light-transport interaction.

The frozen protocol `4D-MIRROR-SHADOW-IDENTIFIABILITY-2026-09-24-v2` (SHA-256 `2159b94aed213620806aee8bf70b280ef635d3b77a216d65b60f5b12febea519`) tests 16 point-light positions and 96×96 screen intensities under one declared 3D Gaussian opacity model. The largest pixel difference is exactly zero; a `+0.10` change to a *visible* x coordinate instead changes one pixel by `0.3918635931328874`. Both are **synthetic checks** of an exact matrix result, not independent evidence for an extra dimension. A hypothetical `P_θ(x,y,z,w)=(x cosθ + w sinθ,y,z)` gives an x-separation `2(1.1)sin(30°)=1.1` at the last vertex. Building a physical `P_θ` is an unsolved prerequisite, not something achieved by rotating a 3D lamp.

**Preserved failed first trial:** A previous labelled-only simplex had equal-length coordinate edges and could be relabelled into its reflected copy by a proper 4D rotation. It demonstrated *labelled* orientation reversal but not an unlabelled left/right-glove pair. Its original frozen protocol, code, figure, outputs and negative audit remain in `history/prototype_0.20_v1/`; the asymmetric example above fixes that specific loophole without asserting any physical fourth spatial coordinate.

### CR.3 — A shadow under a sky shining from every direction

Take an ideal opaque sphere of radius `a`, whose centre is height `a+h` above a flat table. At the table point directly under its centre, it occludes an upper-hemisphere cone of half-angle `θ_0=arcsin[a/(a+h)]`. For uniform radiance from all *upper* directions and a horizontal receiving surface, incident power is weighted by the cosine of incidence. The directly blocked irradiance fraction is therefore

`[∫_0^θ0 cosθ sinθ dθ]/[∫_0^(π/2) cosθ sinθ dθ] = sin²θ_0 = [a/(a+h)]²`.

At `a=1 cm` and `h=0`, `1 cm`, `9 cm`, the fractions are respectively `1`, `1/4`, `1/100`; independent angular quadrature agrees within `5.56×10^−17`. The corresponding *raw solid-angle* fraction at `h=a` is `1−sqrt(3)/2≈0.134`, which is different from the cosine-weighted **25% energy deficit**. This local 3D shadow can exist even if its contrast is hard to see from a particular eye position. Diffuse table reflection, light passing through the object, finite aperture, air scattering, diffraction and thermal changes can all alter the actual appearance; the calculation does not assert a hidden physical shadow between atoms.

For an ideal point lamp distance `D` before a small planar blocker and a screen distance `L` behind it, similar triangles give silhouette magnification `M=1+L/D`; moving a *nearby* point lamp changes the shadow size considerably. A real extended source replaces the sharp edge with a penumbra. In the small-angle, far-source approximation, a blocker transverse radius `a` retains a full umbra only while approximately `L<a/θ_source`, where `θ_source` is the source's angular radius. An illustrative **assumed** `θ_source=0.005 rad` gives `L<2 m` for `a=1 cm`; this is geometry, not a newly measured solar value or a claim that a small blocker extinguishes light everywhere behind it. The blocker intercepts only the rays that hit it; illumination from other directions still arrives.

### CR.4 — Does the old shadow remain when the object leaves?

For a single ideal collimated ray in vacuum with blocker-to-screen path length `d`, let removal occur at time zero and use the idealized screen illumination `I(t)=I_0 H(t−d/c)` after accounting for the pre-removal dark reference, where `H` is a step function and `c=299792458 m/s` exactly in SI. The dark spot can persist **temporarily** for `d/c`, because the first unblocked photons have not reached the screen. Calculated delays are `3.33564 ps` at 1 mm, `333.564 ps` at 0.1 m, `3.33564 ns` at 1 m, `33.3564 ns` at 10 m. Ordinary moving shadows carry information, not an independent dark substance; once light arrives there is no persistent optical imprint in this ideal model. A material might retain a separate photochemical/thermal afterimage, which would require its own dynamics. Lifting a massive blocker faster than light is neither needed nor a physically available operation in this model.

The independently repeated flight calculation and exact rational 4D distance/volume audit are in `independent_mirror_shadow_audit.py`; the 3D angular quadrature is in `shadow_gap.py`. The frozen 3D gap protocol SHA-256 is `dbf997889b27930ffed1cb30e9adf8551667970ba867e0a7c3b22907afa2f96f`. Existing *measured* time-resolved optics—[Velten and colleagues' femto-photography, published 1 July 2013](https://www.media.mit.edu/publications/femto-photography-capturing-and-visualizing-the-propagation-of-light-2/) and [Heide and colleagues' partial-occluder experiment, first posted 20 November 2017](https://arxiv.org/abs/1711.07134)—motivates testing finite-speed transport and distinguishable 3D occlusions; neither measures this proposed 4D pair. This revision adds no new physical measurement.

### CR.5 — What could actually separate models?

A measured observer that changes with `w` would evade the exact 3D-projection alias, **if** its response law and calibration were specified *before* examining measurements. Such a response must then defeat explicit ordinary 3D alternatives, such as unmodelled reflectance, thickness, orientation, time-of-flight scattering and material response. Current theory supplies no derivation of such an instrument or a measured fourth/fifth spatial coordinate. The next bounded test is a frozen multi-distance, time-resolved shadow experiment against ordinary 3D transport and material-memory alternatives; it can check causal optics, but by itself cannot prove a higher dimension. The status of previous conditional physics and the v1.14 idempotent complete closure, v1.15 sampled-collision/angular-impulse/generator corrections, and initial-state homogeneous order-at-most-K restriction of the 2K recurrence result are unchanged.

## Appendix CS — Multi-distance shadow timing cannot locate an optical tail (revision 1.96)

### CS.1 — Frozen question and declared observation

Revision 1.95 proposed varying blocker-to-screen distance to challenge the ideal instant-clearance approximation. Consider a synchronized removed blocker, a calibrated scalar direct-light channel and a known causal normalized detector impulse response `g_a(s)=1_{s≥0} exp(−s/a)/a`. Here `a=0.15 ns`; the three path distances are `d=0.1,1,10 m`; `c=299792458 m/s`. Define relative time `s=t−t0−d/c`, with a common trigger offset `t0`. The question is **not** whether `d/c` is causal—it is whether a measured tail after this arrival identifies its physical source. What follows assumes linear, time-invariant, causal single-channel transport and normalized brightness, and does not describe all real scattering media.

### CS.2 — Exact mechanism counterexample (written proof)

Model A has a local material or detector-adjacent exponential waiting time `b=0.8 ns`, with kernel `m_b(u)=1_{u≥0}exp(−u/b)/b`. The observed photon-arrival density after removal is `g_a*m_b(s)`.

Model B has **no** such material wait, but an extra optical path length `ℓ≥0`, with density `q(ℓ)=exp[−ℓ/(cb)]/(cb)` (here `b` must be expressed in **seconds** when multiplying by `c`, giving mean extra length `cb=0.2398339664 m`). Its arrival density is

`∫_0^∞ q(ℓ) g_a(s−ℓ/c) dℓ`.

Substituting `u=ℓ/c` transforms this integral *exactly* into `g_a*m_b(s)`. Therefore all ideal arrival densities, their integrated shadow-clearance curves, and any observations whose conditional distribution depends only on those curves agree at **every** distance `d` and at every time—not just at three sampled distances. In particular, equal-intensity Poisson photon-count models give identical likelihoods for every possible count record; collecting more of the same observations cannot choose between these mechanisms. This is a mathematical non-identifiability result **inside the declared model classes**, not evidence that the actual material has an afterglow or that real scattering must be exponential.

For the integrated normalized brightness fraction after the direct arrival, both models give, for `s≥0` and `a≠b`,

`F(s)=1−[b exp(−s/b)−a exp(−s/a)]/(b−a)`.

A separately calibrated direct-path-only model instead gives `F_direct(s)=1−exp(−s/a)`. Multiple distances can verify the predicted `d/c` shift, and calibrated response can detect **that some additional tail exists**, but the tail's location is not identified by scalar timing alone. If detector response `g_a` itself is not independently calibrated, even direct-only transport with a broadened detector kernel `g'=g_a*m_b` produces the same trace. If `t0` is unknown, a constant extra delay likewise cannot be assigned a unique origin from center timings.

### CS.3 — Exact versus finite verification

Frozen `OPTICAL_TAIL_IDENTIFIABILITY_PROTOCOL_0.21.json` has SHA-256 `d1c10434531860fff55c139c94f05bf26c1812aa32bf7d549cf95e5fcb7726f1`. The three direct-arrival times are `0.333564095198 ns`, `3.335640951982 ns` and `33.356409519815 ns`. At **1 ns after direct arrival**, each rival tail model predicts brightness `0.6476723962800752`; calibrated direct-only transport predicts `0.9987273661986602`. The two separately computed material-time and extra-length quadratures agree with the closed form across 24 frozen distance/time pairs to at most `2.22×10⁻¹⁶`. A wholly separate pure-Python midpoint/survival-probability audit reproduces the 24 pairs to at most `4.07×10⁻⁹`, with exact agreement for the reported 1-ns check. Changing only the stipulated path-delay constant to `0.4 ns` changes the 1-ns predicted brightness by `0.22175518620249057`; this **synthetic intervention** is not a real-world operation proven to isolate scattering. All **158** inherited and new tests passed after this model addition, and **324** inherited package files outside three expected metadata updates remained byte-identical. The identity in CS.2 is the proof; numerical agreement is only a finite implementation check.

As experimental context, [Velten et al.'s 2013 femto-photography](https://www.media.mit.edu/publications/femto-photography-capturing-and-visualizing-the-propagation-of-light-2/) measured time-resolved light transport, while [Jeon et al.'s 2023 polarimetric iToF study](https://arxiv.org/abs/2306.17618) built an additional optical channel to handle scattering-induced multipath. These primary studies motivate independent scattering controls; neither studied this exact mathematical pair or established an extra spatial coordinate. No physical measurement was acquired in this revision.

### CS.4 — Correction, scope, next test

Correct any interpretation of CR.5 suggesting that **multi-distance timing alone locates a delayed shadow's physical cause**. It cannot distinguish these two explicit causal countermodels, no matter the number of noiseless scalar distance traces. An independently validated intervention must alter one kernel while leaving the other unchanged—for example a calibrated optical-path isolation/control and a material-swapping control, with sensitivity and instrumental responses measured separately—then freeze predictions and test held-out data. Merely switching to an additional uncalibrated sensor can reproduce the same ambiguity. A higher-dimensional account must also make a distinct, prespecified measurable prediction after accounting for these 3D alternatives. Fixed-law complete closure remains idempotent; sampled integer-tick coincidences do not enforce continuous non-crossing, angular impulse carries no extra time factor, finite prefixes do not identify arbitrary generators, and the 2K recurrence-continuation guarantee continues to assume a noiseless homogeneous constant-coefficient recurrence of order at most K from the initial state. No seven physical dimensions, P=NP or theological identification is inferred.

## Appendix CT — What a zero-lifetime reference can and cannot identify (revision 1.97)

### CT.1 — Frozen paired-reference model

Let `G`, `P` and `M` be normalized nonnegative causal delay kernels for the detector, optical path and sample material. Let `*` denote convolution. A reference with negligible material lifetime and a test sample are observed as

`Y_0 = G*P`, and `Y_1 = G*P*M`.

This formulation assumes linear time-invariant transport, independent additive delays, the same emission/detection channel, and **exactly the same detector and path kernels** in the two measurements. These are model assumptions; they are not consequences of using a reference vial. The frozen numerical instance uses exponential detector, common-path and material means `0.15 ns`, `0.30 ns` and `0.80 ns` respectively. Protocol `REFERENCE_CALIBRATION_PROTOCOL_0.22.json` has SHA-256 `a94480497f8caac62f9027b82624e7b00c02970012eaababcee58826bb73cc53`.

### CT.2 — Positive identifiability theorem under exact path matching

For `s>=0`, take the Laplace transform `L_K(s)=integral_0^infinity exp(-st)K(t)dt`. A normalized nonnegative kernel has `L_K(s)>0` at every finite `s`. Therefore

`L_Y1(s)/L_Y0(s) = [L_G(s)L_P(s)L_M(s)]/[L_G(s)L_P(s)] = L_M(s)`.

The ratio identifies the material transform, and uniqueness of Laplace transforms identifies `M` up to equality almost everywhere, **within these assumptions**. This is a written mathematical proof. It is stronger than merely subtracting fitted lifetimes, but it does not certify that an experimental reference and sample actually share `P` and `G`.

### CT.3 — Exact path-mismatch counterexample

Now permit a sample-only optical-path kernel `Delta`, so `Y_1=G*P*Delta*M`. The same ratio becomes `L_Delta L_M`, identifying only the convolution `Delta*M`. In particular:

- Model A: `M` is exponential with mean `0.8 ns` and `Delta` is a point mass at zero.
- Model B: `M` is a point mass at zero and `Delta` is exponential with mean `0.8 ns`.

Both models have the same reference law `G*P` and exactly the same sample law `G*P*Exp(0.8 ns)` for all times. Thus even noiseless infinite paired-reference data cannot say whether this delay is material residence time or sample-specific multipath. The reference control is mechanism-selective only after path equality is established independently.

The first moments make the same limitation transparent:

`mean(Y_1)-mean(Y_0)=mean(M)+[mean(P_sample)-mean(P_reference)]`.

If an external calibration justifies `|mean(P_sample)-mean(P_reference)|<=epsilon`, then the material mean lies within `epsilon` of the observed mean difference. This is a useful sensitivity bound, not distributional identification. In the frozen calculation the observed difference is `0.7972492522 ns`; an **assumed** `epsilon=0.05 ns` gives `[0.7472492522,0.8472492522] ns`. The interval is conditional on the stated bound; no real apparatus supplied that bound here.

### CT.4 — Finite checks and research boundary

At transform points `s=0.1,0.5,1,2 ns^-1`, production code gives material ratios `25/27,5/7,5/9,5/13`. An independent exact-rational implementation reproduces those values and the analytic means `9/20 ns` for the reference, `5/4 ns` for the sample and `4/5 ns` for their difference. A separately seeded simulation of 200,000 reference/sample paths gives maximum transform error `6.86e-4` between the two exactly aliased sample mechanisms and estimates the material mean as `0.7972492522 ns`. Four targeted tests pass; the complete package suite passes after this addition. The theorem and counterexample carry the result; Monte Carlo is only a finite synthetic implementation test.

[Gomez-Sanchez et al. (accepted 20 March 2024)](https://pmc.ncbi.nlm.nih.gov/articles/PMC11161395/) write the fluorescence measurement as a convolution with an instrument-response function and document that this response can vary across measurements, compromising reconvolution. [Trinh and Esposito (2021)](https://pmc.ncbi.nlm.nih.gov/articles/PMC7878588/) likewise describe measuring an instrument response with a known or zero-lifetime reference and show how photon statistics and response width limit lifetime resolution. These primary studies support reference calibration and its caveats; they do not validate the frozen exponential kernels, path equality, an optical afterglow, or an extra spatial coordinate.

**Classification:** CT.2 is conditional written mathematics; CT.3 is a written exact non-identifiability counterexample; CT.4 is finite synthetic computation. There is no new measured evidence, no physical fourth/fifth-dimension inference, and no theological interpretation. The next empirical step is to preregister a same-wavelength, same-geometry paired TCSPC measurement with independently measured detector response and an explicit upper bound on sample/reference path mismatch, then test held-out traces against both material and multipath models. All v1.14 and v1.15 corrections remain in force, and the 2K continuation result still assumes a noiseless homogeneous constant-coefficient linear recurrence of order at most K valid from the initial state.