# 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.