# Wave oscillation and higher spatial dimensions: calculation audit **Anthony Michael Collado's Theory of Reality project · 22 September 2026** **Inputs:** Theory of Reality v1.70, especially BH, BK, BO and BS; Wave Chemistry Lab build 9. **Status:** Added controlled-intervention detection limits (1,920 repeated trials), multi-source phase/force predictions (160 paired datasets), exact feedback equivalence and a phase-work identity. Previous research is preserved. No new physical measurements were supplied. ## Latest result: controlled interventions, visibility limits and linked phases This round asks what changes when an experiment actively perturbs the system, rather than collecting more of the same observations. It adds a conditional detection test for a newly coupled hidden oscillator and a separate test of whether many harmonics share one circular phase pattern. The new calculations use **1,920 activation trials across eight mechanical systems**, plus **160 phase/force scenario datasets across 40 source configurations**. All are synthetic; repeated noise trials and paired controls are not independent physical systems. Three results matter. A sufficiently coupled hidden oscillator becomes measurable, but weak coupling creates a steep information cost. Known changes to visible stiffness cannot distinguish models that already have the same complete visible response. A common phase pattern predicts held-out harmonics and forces, while an explicitly matching ordinary-space field model continues to reproduce the predictions. ### 1. Which interventions can distinguish the existing models? Let χ(s) map calibrated visible forces to visible displacements, with zero initial conditions. Adding a known visible stiffness ΔK changes the equations by the feedback force −ΔKq. Therefore \[ q=\chi(s)[f-\Delta Kq],\qquad \boxed{\chi_{\Delta K}(s)=[\chi(s)^{-1}+\Delta K]^{-1}.} \] If two models have identical χ(s), they also have identical χ_ΔK(s) wherever the inverses exist. This includes models with different hidden-coordinate bases and models with additional completely disconnected modes. A known additive change to visible stiffness does not open a coupling to those modes. The same reasoning extends to a prescribed well-posed visible feedback experiment: identical causal response operators, identical free outputs and the same feedback law give the same visible solution. This statement does **not** cover an intervention that also changes hidden couplings, sensing, masses or unmodeled forces. Whether a real actuator changes only K is a physical calibration question. The numerical check compared a five-coordinate model with an eight-coordinate counterpart containing three additional disconnected oscillators and a rotated hidden basis. Across eight systems, ten positive stiffness perturbations and twelve Laplace probes—**960 comparisons**—the maximum difference after perturbation was **8.89×10⁻¹⁶**, and the feedback identity error was **1.34×10⁻¹⁵**. This verifies the algebra on the tested matrices; the identity itself explains why more such measurements cannot discriminate them. ### 2. Making an invisible oscillator observable A different intervention changes the actual coupling. The test model is \[ \ddot q+Kq+\epsilon b w=f,\qquad \ddot w+\nu^2w+\epsilon b^Tq=0. \] Here q has three components, b is a known unit vector and ε is a calibrated reciprocal coupling. When ε = 0, w is invisible to these inputs and sensors. When ε is nonzero, eliminating w gives \[ \chi_\epsilon(s)^{-1}=s^2I+K- \frac{\epsilon^2bb^T}{s^2+\nu^2}. \] Writing χ₀ = (s²I + K)⁻¹, the Sherman–Morrison identity gives the exact change \[ \boxed{\chi_\epsilon-\chi_0= \frac{\epsilon^2\chi_0bb^T\chi_0} {s^2+\nu^2-\epsilon^2b^T\chi_0b}.} \] For a fixed finite observation setup and weak coupling, the visible signal is order ε². It must travel from the visible system into the hidden oscillator and back. Changing the sign of ε is equivalent to reversing the hidden coordinate, so the visible response cannot depend on that sign. For independent Gaussian measurement errors with known standard deviation σ, two fixed response means μ_ε and μ₀ have \[ D_{\rm KL}=\frac{\|\mu_\epsilon-\mu_0\|^2}{2\sigma^2}=O(\epsilon^4). \] Likewise the information about ν is \[ \mathcal I_\nu=\frac{\|\partial_\nu\mu_\epsilon\|^2}{\sigma^2} =O(\epsilon^4). \] Independent repetitions add information. Thus, in this weak-coupling limit with the same observation window, **halving ε requires approximately 16 times as many repetitions** to retain the same information. This is not a universal law for every instrument: direct hidden sensing, correlated noise, changing the observation duration or a different intervention can change the scaling. At fixed Laplace probes, the numerical log–log slopes were **2.000006** for signal size and **4.000025** for frequency information over ε = 10⁻⁴ to 10⁻². The exact susceptibility identity agreed to 8.89×10⁻¹⁶ absolute error. ### 3. A detection test with independent calibration and validation The activation protocol and identifier were hashed before execution. Eight randomly oriented three-coordinate systems have visible frequencies 0.61, 1.12 and 1.90 in model units. Their hidden frequency is drawn between 0.78 and 1.66. Six coupling commands range from 0.0001 to 0.03. Every setting receives twenty independent pairs of noisy measurements in each of two scenarios: - The command reaches the hidden oscillator and establishes the declared coupling. - The command does not reach it; the physical response remains the baseline. Each measurement contains three known impulse inputs, three displacement outputs and 256 samples at interval 0.30. Measurement noise is independent Gaussian, with standard deviation 0.1% of baseline response RMS. K, b, the intended coupling and the baseline prediction are treated as independently known. Uncertainty in those calibrations is not included in this pilot. Repeat A selects ν by a fixed 101-point search followed by local scalar refinement. The allowed search interval is 0.75–1.70. **Independent repeat B is never passed to the frequency fit.** Given A's prediction μ̂, define δ = μ̂ − μ₀ and \[ \boxed{Z=\frac{\delta^T(y_B-\mu_0)}{\sigma\|\delta\|}.} \] Under the specified no-change null, Z is standard normal conditional on A, for any nonzero chosen δ. Searching frequencies on A therefore does not invalidate B's one-sided 1% test. This is a per-trial conditional test; it is not a family-wise 1% bound across 1,920 decisions. A separate engineering gate requires the reserved mean-square error relative to noise variance to be at most 1.20. Both gates must pass for acceptance. A high Z can establish a change relative to this calibrated baseline while still leaving ν poorly determined. | Coupling ε | Hidden mode accepted / 160 | No-coupling control falsely accepted / 160 | Median relative frequency error among accepted active cases | |---:|---:|---:|---:| | 0.0001 | 0 | 2 | — | | 0.0003 | 1 | 2 | 21.65% (one case) | | 0.001 | 17 | 0 | 2.01% | | 0.003 | 152 | 2 | 1.49% | | 0.01 | 160 | 0 | 0.102% | | 0.03 | 160 | 0 | 0.0194% | Across the 960 no-coupling control trials, Z alone triggered seven times and the complete test falsely accepted **six times (0.625%)**. Those false positives remain in the results. At the weakest active coupling, a real hidden oscillator remained undetected in all 160 trials; that does not mean it did not exist. The coupling command has model-dependent units. This calculation does not establish that an actuator capable of reaching an extra spatial direction exists. The same equations describe an ordinary hidden mechanical component. Its practical contribution is an explicit detection threshold and information budget for a specified intervention. ### 4. One phase pattern must predict all harmonics The second experiment strengthens the preceding compact-circle calculation by using several sources and controlled phase changes. Let seven sources have phases θᵢ. For the nth pair of sine/cosine modes, the normalized cross-coupling matrix is \[ C^{(n)}_{ij}=\cos[n(\theta_i-\theta_j)] =u_i^{(n)}\cdot u_j^{(n)},\qquad u_i^{(n)}=(\cos n\theta_i,\sin n\theta_i). \] Each matrix is positive semidefinite, has diagonal one and has rank at most two. Those properties alone are insufficient: different modes can each have rank two while referring to incompatible phase patterns. The stronger relation is \[ \boxed{C^{(n)}_{ij}=T_n(C^{(1)}_{ij}),} \] where Tₙ is a Chebyshev polynomial. For example, \[ C^{(2)}_{ij}=2(C^{(1)}_{ij})^2-1,\qquad C^{(3)}_{ij}=4(C^{(1)}_{ij})^3-3C^{(1)}_{ij}. \] These powers act **entry by entry**, not as matrix multiplication. The identity checks agreed to 8.89×10⁻¹⁶. A shared phase organization therefore imposes many relations that arbitrary frequency lists or separate rank tests do not impose. The first-mode Gram matrix determines phases up to a common rotation and reflection. The common rotation has no effect on pairwise predictions. To resolve the reflection when making a signed intervention, the calibration includes one first-mode source-pair reading after a known positive shift of 0.37 radians. The held-out experiment then shifts source zero by either 0.37 or 0.83 radians and checks every source pair in modes 2–6. These inputs are hypothetical **normalized modal cross-couplings**, not raw atomic line positions or the existing animation's displayed dots. A real experiment would need calibrated source amplitudes, mode resolution and an independently characterized phase actuator. Interpreting a phase shift as literal motion through an extra spatial coordinate would require additional physical evidence. ### 5. Predicting forces after a source-phase change For the same illustrative scalar field on ordinary space times a circle, put u = r/R and θ = θᵢ − θⱼ. The angular Green function is \[ V(r,\theta)=-\frac A r P(u,\theta),\qquad P(u,\theta)=\frac{\sinh u}{\cosh u-\cos\theta} =1+2\sum_{n\ge1}e^{-nu}\cos(n\theta). \] This established potential is equation (11) of Floratos, Leontaris and Vlachos, [arXiv:1008.0765](https://arxiv.org/abs/1008.0765). The calculation here differentiates it and uses reconstructed phases to make reserved predictions. It is a specified scalar toy model, not a universal atomic or gravitational force law. The ordinary radial attractive-force component, normalized by A/r², is \[ \boxed{B(u,\theta)=P-u\,\partial_uP =\frac{\sinh u}{D}-u\frac{1-\cosh u\cos\theta}{D^2}, \qquad D=\cosh u-\cos\theta.} \] Knowing R, the long-range normalization and the reconstructed phases fixes this force curve for all source pairs. No force readings are used to infer the phases. The potential also predicts the generalized force conjugate to the controlled phase: \[ \tau_\theta=-\partial_\theta V =-\frac A r\frac{\sinh u\sin\theta}{(\cosh u-\cos\theta)^2}. \] For an ideal quasistatic source-phase change at fixed r, the required external work is \[ W_{\rm ext}=-\int_{\theta_a}^{\theta_b}\tau_\theta\,d\theta =V(r,\theta_b)-V(r,\theta_a). \] An independent integration at A = R = r = 1, from θ = 0.7 to 1.8, gave **0.8462291221202521** energy units, agreeing with the potential difference within 1.12×10⁻¹⁶. This is an additional analytic work prediction, not a measured-work benchmark. Only if θ is independently identified as a spatial coordinate w/R could τθ/R be interpreted as force along that direction; an ordinary internal phase has a conjugate generalized force as well. ### 6. What the phase experiment recovered and what survived it A separate protocol fixed forty source configurations, seven approximately evenly spaced phases with independent jitter of ±0.15 radians, calibration noise 0.0001 and reserved noise 0.01 in normalized observable units. A rank-two positive projection of the first-mode matrix estimates the phase vectors, followed by unit normalization and the single orientation-calibration reading. Each configuration reserves **210 higher-mode cross-couplings** and **1,344 normalized force readings**: two phase shifts, all 21 source pairs, modes 2–6, and 32 separations from r/R = 0.4 to 6. The second shift is absent from calibration. The first shift is calibrated only through one first-mode pair, while its higher modes and forces are withheld. Seven source phases contain **six continuous relative-phase parameters**, since a common rotation is unobservable; the orientation calibration fixes the remaining discrete reflection choice. Across the six tested modes (1–6), unrelated phase patterns would instead have 6 × 6 = **36** continuous parameters, holding radius and normalizations fixed. The shared rule therefore supplies a compact predictive description. This is a parameter-count calculation, not a fitted model-selection score. The matching ordinary-space tower uses the same six phases and gains the same compression, so this economy does not distinguish its spatial interpretation. The recovered phase RMS error had median **0.00337 degrees** and maximum **0.00921 degrees**. Those scores use truth only after fitting and compare phases relative to source zero. The inputs are deliberately well distributed and very low-noise; this does not establish recovery at arbitrary phase configurations or laboratory noise levels. Calibration consistency plus two reserved mean-square diagnostics determine acceptance. The reserved threshold is 1.25 times the stated noise variance. These are engineering gates; the score does not fully propagate uncertainty in the inferred phases and is not a calibrated discovery probability. All four scenarios within a configuration share calibration readings and noise draws. | Synthetic observation model | Passes all checks | Median reserved harmonic error / noise variance | Median reserved force error / noise variance | |---|---:|---:|---:| | Compact-circle scalar model | 39/40 | 1.030 | 1.015 | | Ordinary 3D field tower with matching rotating source couplings | 39/40 | 1.030 | 1.015 | | First-mode couplings copied unchanged to every higher mode | 0/40 | 9,834 | 41,857 | | Each higher mode has its own independent source phases | 0/40 | 9,905 | 10,356 | The independent-phase controls still have positive rank-at-most-two coupling matrices in every mode. They fail because the cross-mode and controlled-shift relations do not agree, demonstrating why a common phase test is stronger than a rank check. The one declined compact-model case had reserved harmonic score 1.275, above the declared 1.25 gate. Its force score was 0.984. No thresholds were changed to rescue it. The ordinary-space tower survives because its nth pair of fields couples to source i through (cos nθᵢ, sin nθᵢ). Its dot products, their controlled changes and its summed forces are exactly those of the compact description. A separately evaluated 300-term force sum agreed with the closed form within **2.31×10⁻¹⁴ absolute normalized-force units** across this benchmark. All forty paired acceptance decisions matched. The result adds a genuine consistency requirement to the wave model: infer one phase structure, then predict other harmonics, interventions and forces. It narrows the tested alternatives, but it does not uniquely identify spatial geometry. The most concrete next physical inputs would be calibrated intervention data and a justified observation model that distinguishes a spatial displacement from an ordinary internal phase. None were supplied in this round. ### Reproduction and provenance All new files are under `intervention_v6/`. From the archive root: ```bash OPENBLAS_NUM_THREADS=1 python intervention_v6/run_activation.py OPENBLAS_NUM_THREADS=1 python intervention_v6/run_phase.py OPENBLAS_NUM_THREADS=1 python intervention_v6/check_math.py OPENBLAS_NUM_THREADS=1 python intervention_v6/plot_results.py python intervention_v6/make_report.py python intervention_v6/verify_outputs.py ``` Both protocols and model files were written and hashed before their respective benchmarks. Seeds, full trial ledgers, summaries and predetermined examples are included. Activation examples use system zero and repeat zero; phase examples use configuration zero. The later `check_math.py` performs independent identity and scaling checks without modifying fit rules. The numerical outputs and previous report are preserved; `REPORT_after_response.md` is the snapshot before this addition. The atomic browser engine and theory manuscript were not changed. --- ## Previous multi-oscillator and force result: several hidden oscillators and a frequency–force prediction Two calculations have been added. First, the reconstruction now handles several hidden oscillators, including several acting through the same visible direction. Second, an explicit compact-space scalar-field model connects its wave spectrum to a separate force-versus-distance measurement. These are conditional mathematical models and synthetic tests, not new physical measurements or proof of a fourth spatial dimension. The new round contains **96 independently generated controlled-response systems** and **240 frequency–force datasets**. The latter use 40 independent parameter/noise draws across six paired observation scenarios; they are not 240 independent experiments. All earlier calculations are retained below. ### 1. More hidden waves than visible directions Let q contain three measured coordinates and w contain m hidden oscillator coordinates. Assume known unit masses, linear calibrated sensing, reciprocal conservative coupling, a repeatable equilibrium, and controlled visible forcing f: \[ \ddot q+Kq+Gw=f,\qquad \ddot w+Hw+G^Tq=0, \qquad M=\begin{pmatrix}K&G\\G^T&H\end{pmatrix}>0. \] Here K is 3×3, H is m×m and G is 3×m. The integer m counts internal mechanical degrees of freedom. These coordinates can all describe ordinary mechanisms in three-dimensional space. The visible residual is \[ r=\ddot q+Kq-f=-Gw. \] Consequently, its instantaneous rank cannot exceed rank(G), which cannot exceed three. That **does not limit the number of hidden oscillators to three**. In particular, \[ G=b\begin{pmatrix}g_1&g_2\end{pmatrix},\qquad H=\operatorname{diag}(\nu_1^2,\nu_2^2) \] has rank(G) = 1, but has two distinguishable hidden frequencies when both couplings are nonzero and the frequencies differ. This directly extends the previous single-hidden-oscillator calculation, which correctly declined this case because one oscillator could not close its equations. #### The memory of hidden motion Eliminating w gives \[ \ddot q+Kq=f-Gw_{\rm free}(t) +\int_0^t\Gamma(t-\tau)q(\tau)\,d\tau, \qquad \boxed{\Gamma(t)=G H^{-1/2}\sin(H^{1/2}t)G^T.} \] For zero initial conditions, the measured susceptibility obeys \[ \boxed{\chi(s)^{-1}=s^2I+K-G(s^2I+H)^{-1}G^T.} \] Thus multiple hidden oscillations can appear in the time dependence of a matrix that never has more than three rows. “Off the matrix” is better made precise as dynamics absent from an assumed state model, rather than a point literally leaving a matrix. An exact, noiseless order witness follows from the moment matrices \[ C_k=GH^kG^T,\qquad \mathcal B_0=\begin{pmatrix}C_0&C_1\\C_1&C_2\end{pmatrix} =\begin{pmatrix}G\\GH\end{pmatrix} \begin{pmatrix}G^T&HG^T\end{pmatrix}. \] Its rank can exceed rank(G). With \[ \mathcal B_1=\begin{pmatrix}C_1&C_2\\C_2&C_3\end{pmatrix}, \] whitening the nonzero range of B₀ and diagonalizing the corresponding B₁ recovers eigenvalues of H when the stacked coupling has full hidden rank. Larger block matrices are needed if two block rows do not suffice. The numerical identity checks found: | Constructed example | Rank of G | Rank of the two-block moment matrix | Observable hidden modes | |---|---:|---:|---:| | Two parallel hidden oscillators | 1 | 2 | 2 | | Four hidden oscillators | 3 | 4 | 4 | | Three hidden oscillators, one disconnected | 2 | 2 | 2 | The moment calculation uses known matrices to verify the algebra. It is **not** claimed as a stable algorithm for extracting high derivatives from noisy measurements. The actual blind benchmark uses time-domain responses instead. #### Reconstruction from controlled impulses Apply a known unit velocity impulse in each visible coordinate, with a zero-state reset between experiments. Record all three visible displacements for each input, giving a 3×3 response matrix R(t). In the assumed model, \[ R(t)=\sum_{j=1}^{N}p_jp_j^T\frac{\sin(\Omega_jt)}{\Omega_j}, \qquad \sum_jp_jp_j^T=I_3. \] For distinct, observable normal modes, each residue is symmetric, positive semidefinite and rank one. Assemble their factors into P = [p₁ … pₙ], so PPᵀ = I₃. Complete its rows to an orthogonal matrix O. Then \[ \boxed{\widehat M=O\operatorname{diag}(\Omega_j^2)O^T} \] provides a compatible full mechanical realization. Its upper-left block can be checked against independently calibrated K. The remaining blocks supply a candidate G and H. In this restricted fully observable model, the inferred number of hidden coordinates is N − 3; the corresponding first-order state has 2N components. For noisy data, the implementation projects fitted residues onto rank-one positive matrices, checks how large that correction was, normalizes the visible rows, and rejects substantial disagreement with known K. It does not silently assume that an arbitrary frequency fit is a physical mechanical realization. The hidden basis remains arbitrary: w′ = T w, G′ = GTᵀ and H′ = THTᵀ for any orthogonal T produce the same visible response. Unknown masses would permit further normalization ambiguities. Completely disconnected modes cannot be counted by this experiment. Degenerate poles, unresolved weak modes, finite record length, incorrect clocks, aliasing, damping and nonlinear sensing require additional modeling; this benchmark does not resolve every such case. ### 2. What was actually run The protocol and identifier were hashed before the run. Eight scenarios received 12 fresh systems each, with four systems at each of three response-noise levels: zero, 0.01% and 0.1% RMS. Each fit sees 768 samples spaced by 0.25, with nine response channels. The first 512 samples estimate candidate orders 3–8; the next 128 select order; the final 128 check continuation before the final refit. Another **384 future samples are withheld from identification**. The inputs supplied to the identifier are only the visible responses, clock spacing, known K, calibrated noise level and protocol. Hidden matrices and future observations are used only by the generator and scorer. Candidates start from a delay-matrix decomposition and refine sine frequencies by least squares. Selection uses a fixed noise allowance; validation, residue consistency, completeness, known-K agreement and refit stability have fixed 3% bounds. These are engineering checks, not calibrated confidence levels. | Synthetic system | Inferred observable hidden count | Accepted | What that means | |---|---:|---:|---| | No hidden oscillator | 0 in 12/12 | 12/12 | Correct minimal visible model | | Two coupled hidden oscillators | 2 in 12/12 | 12/12 | Both recovered | | Two hidden oscillators with parallel couplings | 2 in 12/12 | 12/12 | Time structure separates a rank-one residual | | Four coupled hidden oscillators | 4 in 12/12 | 12/12 | Hidden order can exceed the number of measured channels | | Three hidden oscillators, one disconnected | 2 in 12/12 | 12/12 | The disconnected coordinate remains undetectable | | One-way hidden driver, reset to zero | 0 in 12/12 | 12/12 | No visible input can excite this driver; its response equals the no-hidden case | | Damped response | Incorrect undamped order | 0/12 | The assumed conservative model is declined | | Incorrect independently supplied K | 2 in 12/12 | 0/12 | Good frequency recovery does not override the K mismatch | The one-way control is deliberately informative: passing a response fit does not validate unexcited internal mechanisms. Its hidden coordinates start at zero, receive no visible feedback and remain zero. The fitted ordinary visible model is valid for this experiment even though the generator contains an inaccessible driver. Across the four rows with observable coupled hidden motion, the correct observable count was recovered in **48/48 systems**. At **0.1% response noise**, the 16 corresponding systems had **0.00839% median future-response RMS error**, with a **0.01088% worst case**. At 0.01% noise the median was 0.000782%; in the noiseless cases the worst relative error was 3.31×10⁻¹⁵. These favorable results rely on long records of simple stationary signals and known calibrated impulses; they are not a guarantee for uncontrolled atomic measurements. The trial-1 noisy examples also recovered the hidden memory kernel, with relative errors between 0.00835% and 0.01949% over the checked interval. Independent matrix-exponential evaluation agreed with the modal response to 5.65×10⁻¹⁵ absolute error. Schur-complement and hidden-basis invariances were checked separately. Every trial, including declined cases, remains in the CSV and JSON outputs. ### 3. An independent prediction from an explicitly chosen spatial model To connect frequency to a physical observation, a model must specify the field, its dynamics, geometry and coupling to instruments. The following is an **added illustrative model**, not something uniquely derived from the atomic animations or already established by the Theory of Reality manuscript: - A massless scalar wave propagates on three ordinary spatial coordinates plus a circle of circumference 2πR. - The wave speed c is independently known and is the same in the ordinary and compact directions. - Sources are localized at the same compact coordinate and couple to this field with the assumed common mode weights. - Point-source forces are measured with a known long-distance normalization A. - Spectral lines are correctly labeled consecutive compact modes, at zero ordinary spatial momentum. The circle boundary condition makes the compact wave numbers n/R, giving \[ \Omega_n^2=c^2|k|^2+c^2n^2/R^2. \] At k = 0, adjacent positive-frequency modes have angular-frequency spacing \[ \boxed{\Delta\Omega=c/R},\qquad \Delta f=\frac{c}{2\pi R}. \] These are compact field modes, **not arbitrary atomic transition lines**. Identifying an atomic “base frequency” with this spacing would add an unsupported assumption. The same Green function fixes the static potential. Its image representation and mode representation agree: \[ 2R\sum_{\ell=-\infty}^{\infty} \frac{1}{r^2+(2\pi R\ell)^2} =\frac1r\coth\!\left(\frac{r}{2R}\right), \] \[ V(r)=-\frac A r\left[1+2\sum_{n=1}^{\infty}e^{-nr/R}\right] =-\frac A r\coth\!\left(\frac{r}{2R}\right). \] This potential and its compact-mode interpretation are established results, not a new discovery here; see Floratos, Leontaris and Vlachos, *Gravitational Atom in Compactified Extra Dimensions*, equation (12), [arXiv:1008.0765](https://arxiv.org/abs/1008.0765). Related compactification-dependent Yukawa corrections are treated by Kehagias and Sfetsos, [hep-ph/9905417](https://arxiv.org/abs/hep-ph/9905417). The calculation here uses a scalar-field toy model; it does not claim this formula applies to all gravitational theories, electromagnetic fields or atomic forces. Differentiating gives the attractive force magnitude \[ \boxed{F(r)=\frac A{r^2}\left[\coth x+x\,\operatorname{csch}^2x\right], \qquad x=\frac{r\Delta\Omega}{2c}.} \] **The spectral fit fixes the entire normalized force curve without fitting the force readings.** At r much greater than R, F approaches A/r². At r much smaller than R, F approaches 4AR/r³. Both follow from the same R. A useful additional numerical prediction comes from the local force exponent: \[ D_F(r)=1-\frac{d\log F}{d\log r} =3+\frac{2x^2\operatorname{csch}^2x\coth x} {\coth x+x\operatorname{csch}^2x}. \] It runs from 4 at short distances to 3 at large distances in this model. The point where F has local power r⁻²⋅⁵ satisfies \[ \boxed{\frac{r_*\Delta\Omega}{c}=4.1311190512.} \] This is a concrete cross-observable relation. D_F is an effective **force exponent**, not an independent measurement of the number of spatial coordinates. An ordinary-space interaction can have the same exponent. As a unit illustration only, if c is 299,792,458 m/s and R is 1 micrometre, the predicted spacing is 47.713 THz in cycles per second, and r* is 4.131 micrometres. Neither scale was inferred from physical data here. ### 4. Testing the frequency–force relation without fitting the force Before generating these datasets, the separate compact-model protocol fixed seed 20260930, 40 parameter/noise draws, a radius uniform from 0.6 to 1.4 in arbitrary length units, c = 1, six spectral calibration modes, six reserved spectral modes, and 96 reserved force separations from 0.1 to 20. Spectral Gaussian noise has standard deviation 0.2% of the fundamental gap; force noise has independent 1% relative standard deviation. Absolute noise levels are assumed independently calibrated. For calibration mode numbers n, estimate \[ \widehat{\Delta\Omega}=\frac{\sum n\Omega_n}{\sum n^2},\qquad \operatorname{Var}(\widehat{\Delta\Omega})=\frac{\sigma_\Omega^2}{\sum n^2}, \qquad \widehat R=c/\widehat{\Delta\Omega}. \] All 96 force readings remain reserved. The score includes their independent noise plus the common uncertainty propagated from the fitted gap. The covariance has form diagonal noise + vvᵀ Var(gap), evaluated with the analytic derivative of F and inverted by the Sherman–Morrison identity. This is a first-order uncertainty approximation appropriate to the small calibration error in this pilot, not an exact general nonlinear likelihood. Three standardized mean-square diagnostics—calibration spectrum, reserved spectrum and reserved force—must each be at most 1.75. This is a declared consistency gate, not a 95% or 99% discovery threshold. Each scenario shares its parameter and noise draws with the alternatives to make comparisons controlled. | Observation scenario | Passes all checks | Median force residual / modeled noise variance | |---|---:|---:| | Compact scalar field model | 31/40 | 0.998 | | Ordinary 3D fields with the matching infinite mode tower | 31/40 | 0.998 | | Matching spectral ladder, inverse-square force only | 0/40 | 193,208 | | Spectral and force radii differ by 35% | 0/40 | 457 | | Matching ladder, only the first Yukawa force correction | 0/40 | 83,409 | | Spectrum has a quadratic distortion | 0/40 | 74.0 | The spectral estimator recovered R with **0.01055% median relative error** and **0.04732% worst error** in the compact-model cases. All 40 passed the reserved force check. Nine failed one or both spectral noise gates: 33/40 passed the calibration-spectrum gate and 37/40 passed the reserved-spectrum gate. These declines are retained; no thresholds were relaxed after inspecting the results. ### 5. The exact alternative that remains In ordinary three-dimensional space, take one massless scalar field and an infinite sequence of massive fields with inverse interaction ranges n/R and source weights giving the factor two. Their force is \[ F_{\rm tower}(r)=\frac A{r^2}\left[1+2\sum_{n=1}^{\infty} \left(1+\frac{nr}{R}\right)e^{-nr/R}\right]. \] It is identically the compact model's force. At zero ordinary momentum the same fields have the same mode frequencies. Thus these two descriptions give the same observables in this experiment. A finite 600-term sum agreed with the closed form to **4.84×10⁻¹⁶ relative error** across all benchmark parameter draws; paired noisy observations differed only by floating-point rounding, at most 1.43×10⁻¹⁴ in the dimensionless force ratio. Their acceptance decisions were identical in all 40 pairs. The second calculation therefore supplies a way to **reject inconsistent frequency/force combinations** and to estimate a common scale under a stated model. It does not uniquely identify geometry. Nor does detecting a fourth internal oscillator establish a fourth spatial direction. Progress toward physical evidence requires measured data plus additional independently justified restrictions or observables that actually distinguish the alternatives under comparison. No new physical measurements were supplied in this round. ### Reproduction and provenance All new scripts are in `response_v5/`. From the archive root: ```bash OPENBLAS_NUM_THREADS=1 python response_v5/run_multi.py OPENBLAS_NUM_THREADS=1 python response_v5/run_compact.py OPENBLAS_NUM_THREADS=1 python response_v5/check_identities.py OPENBLAS_NUM_THREADS=1 python response_v5/force_slope.py OPENBLAS_NUM_THREADS=1 python response_v5/plot_results.py python response_v5/make_report.py ``` The two protocols and identifiers retain their pre-run SHA-256 hashes. `multi_trials.csv` contains all 96 systems, `multi_diagnostics.json` the blind identifier diagnostics, and `multi_examples.json` the fixed trial-1 examples. `compact_trials.csv` contains all 240 scenario rows and `compact_examples.json` the fixed trial-0 examples. Generators and seeds reproduce the full inputs. These scripts use the earlier pinned NumPy/SciPy/Matplotlib dependencies and the existing modal initializer without modifying it. The first compact-run export stopped on a duplicate `true_R` keyword in an example dictionary, before the benchmark completed. The export statement alone was corrected; the original log is retained as `compact_first_export_error.log`. Neither the frozen model nor the protocol changed. `force_slope.py` and `check_identities.py` are explicitly subsequent analytic checks, not new tuned acceptance tests. The prior full report is preserved as `REPORT_after_geometry.md`. The browser lab and theory manuscript were not modified. --- ## Previous joint-geometry result: testing whether hidden motion predicts spatial distances The calculation now connects the reconstructed hidden waves to a separate distance observable. It aligns independently recovered hidden coordinates, calibrates one common metric scale, and predicts ranges that were not used for calibration. This is a stronger consistency test than detecting another oscillator or observing an apparent fourth matrix rank. Across **20 fresh eight-marker wave ensembles**, the procedure recovered all 20 relative sign configurations and the common scale with **0.473% median relative error** and **1.941% maximum error**. The complete test accepted 18/20 declared 4D cases. Two cases were declined by preceding hidden-wave checks; their good distance fits were not used to override those checks. The same procedure accepted **18/20 ordinary-3D cases with a specifically constructed shared sensor bias**, because those cases produced exactly the same observations. This exposes the remaining identifiability limit: extra spatial geometry and a suitably coupled measurement mechanism cannot be distinguished by the present readings alone. All data added in this round are synthetic. The benchmark has **160 marker wave fits and 120 geometry datasets**, with six observation scenarios sharing each of the 20 wave ensembles. It has not measured a fourth physical spatial dimension. ## 1. The joint hypothesis For marker i, let qᵢ(t) be its calibrated three-dimensional position and ĥᵢ(t) its independently reconstructed hidden oscillator coordinate. The previous reconstruction leaves an arbitrary sign for each marker. Under the new spatial hypothesis, \[ w_i(t)=a\,s_i\widehat h_i(t),\qquad a\ge0,\quad s_i\in\{-1,+1\}, \] with one common spatial conversion a and a fixed zero-equilibrium convention. The predicted full distance is \[ \boxed{\widehat D_{ij}^2(t)=\|q_i(t)-q_j(t)\|^2 +a^2\left[s_i\widehat h_i(t)-s_j\widehat h_j(t)\right]^2.} \] This specifies what a distance instrument is assumed to measure. It is not implied by finding a hidden oscillator. In particular, it assumes the ranging interaction measures the full metric; a field confined to a lower-dimensional subspace need not do so. This is one proposed observation model, not a generic prediction of every extra-dimensional theory. The common scale and zero offsets are substantive restrictions. Allowing arbitrary per-marker scale, time-dependent mappings or freely chosen offsets would create a different, more flexible hypothesis and require new reserved-data tests. A genuine extra-dimensional system outside the present family could be declined. ### Relation to the distance-matrix calculation If independently measured squared distances satisfy the boxed relation, define \[ \Delta_{ij}=D_{ij}^2-\|q_i-q_j\|^2,\qquad B_{\rm residual}=-\tfrac12J\Delta J,\qquad J=I-\tfrac1N\mathbf1\mathbf1^T. \] For the noiseless one-coordinate model, \[ B_{\rm residual}=a^2JShh^TSJ, \qquad S=\operatorname{diag}(s_1,\ldots,s_N). \] It is positive semidefinite with rank at most one. The added requirement is that its coordinate must agree with the wave-derived h using the same signs and scale across times, marker pairs and spatial scales. The Gram matrix and pairwise prediction are algebraically related views of the same distance readings; they must not be counted as two independent experimental confirmations. ## 2. Calibration and reserved readings were kept separate Each ensemble has eight independently generated visible/hidden oscillator systems, using the earlier frozen wave identifier and fresh random draws. The known visible stiffness and mass-normalization assumptions of that identifier remain in force. Each has 720 wave observations at Δt = 0.25. The eight mean visible positions are cube corners, with a small superposed visible displacement. The wave data alone produce ĥᵢ(t). The true hidden trajectories are used only by the synthetic distance generator and scorer, never as inputs to the joint decoder. - **Calibration:** instrument A supplies 252 distances: the first 14 of 28 lexicographic marker pairs, at 18 early times and scale 1. These pairs connect all eight markers. - **Reserved test:** instrument B supplies 2,016 distances: all 28 pairs at 18 later times, at scales 1/8, 1/4, 1/2 and 1. This includes 14 marker pairs and three scales absent from calibration. - **Noise:** each range has independent zero-mean Gaussian error with known standard deviation 0.003 in arbitrary length units. The two instruments have independent random errors. Some controls deliberately share a systematic bias. Visible spatial positions are treated as exactly calibrated in this conditional test. Noisy wave channels are separate inputs. A physical application must include tracking uncertainty, correlated range errors, clock alignment and a propagation model for any travel-time measurement. The wave reconstruction uses the full 720-sample visible history, including times at which geometric readings are later withheld. Thus this is a prediction of a separate withheld measurement channel, **not a claim to forecast future positions from earlier waves alone**. ### Infer the relative orientation and metric scale Fix s₁ = +1 because a simultaneous sign flip cannot change distances. The eight-marker pilot explicitly searches the remaining 2⁷ = 128 sign assignments. This is a small exhaustive search, with exponential sign-search scaling; no general efficient solver is claimed. For each assignment, fit the nonnegative squared scale λ = a² on A only. For a measured range Y = D + ε with known noise standard deviation σ, \[ \mathbb E(Y^2-\sigma^2)=D^2, \qquad \operatorname{Var}(Y^2)=4D^2\sigma^2+2\sigma^4. \] The fit regresses Y² − σ² − D₃² on the squared signed hidden-coordinate difference. Weights use the visible D₃ in the displayed variance formula, a fixed approximation for calibration. The selected signs and scale are frozen before inspecting B. The later conditional test remains valid under its null regardless of how this independent calibration chose its prediction direction. ## 3. A precisely stated statistical test Let d₀ be the vector of flat-3D distances for reserved observations, d̂ the prediction fixed by instrument A, δ = d̂ − d₀, and y the reserved instrument-B readings. Under the specified null \[ H_0:\quad y=d_0+\varepsilon,\qquad \varepsilon\sim N(0,\sigma^2I), \] and conditional on the wave data and independent calibration, \[ \boxed{Z=\frac{\delta^T(y-d_0)}{\sigma\|\delta\|}\sim N(0,1)} \] for nonzero δ. The sign/scale search does not use y, so selecting among 128 calibration fits does not invalidate this conditional distribution. If δ is zero, the procedure does not claim a signal. The one-sided 1% critical value is 2.32635. The complete predeclared acceptance rule requires all of the following: 1. Every marker passes the earlier hidden-wave identification checks. 2. The reserved conditional null p-value is at most 0.01. 3. The fixed prediction improves the reserved Gaussian log likelihood relative to d₀. 4. The mean squared standardized reserved residual is at most 1.5. The last requirement checks adequacy of the particular prediction. Rejection of a flat-3D, unbiased-instrument null alone does not select a fourth spatial dimension. The p-value is not a probability that space is 3D or 4D. Its null includes the stated instrument and calibration assumptions. Log p-values are retained to avoid numeric underflow; very small synthetic values are not experimental significance claims. For a known non-diagonal Gaussian noise covariance C, the corresponding statistic would use \[ Z_C=\frac{\delta^TC^{-1}(y-d_0)}{\sqrt{\delta^TC^{-1}\delta}}. \] That covariance generalization is derived here, but the executed benchmark uses σ²I. Unknown or estimated error correlations need separate calibration; treating correlated readings as independent can overstate evidence. ## 4. Results across the six observation models | Synthetic observation model | Rejects calibrated flat-3D null | Passes the complete one-coordinate test | Median reserved error / noise variance | |---|---:|---:|---:| | Flat 4D metric | 20/20 | 18/20 | 1.005 | | Ordinary 3D, internal hidden oscillators | 0/20 | 0/20 | 1.004 | | Curved intrinsic 3D geometry | 20/20 | 0/20 | 39.790 | | 3D, state-dependent bias only in instrument A | 0/20 | 0/20 | 13.252 | | 3D, shared state-dependent instrument bias | 20/20 | 18/20 | 1.005 | | Two independent hidden metric coordinates | 20/20 | 0/20 | 2.344 | An error/noise-variance ratio near 1 means residuals are comparable to the imposed random measurement noise. The ordinary internal-motion model fits its near-zero extra metric contribution adequately but provides no independent signal, so it is declined as an additional spatial coordinate. Curved 3D and the two-hidden-coordinate model strongly violate the flat-3D null yet fail the specific one-coordinate distance prediction. The wave stage accepted 158/160 marker fits. One marker in ensemble 5 and one in ensemble 12 failed rank-one/closure or stability checks. Both entire joint cases were declined, even though their geometric predictions fit within noise. All 20 diagnostic 4D sign sets were correct up to the unavoidable global sign. Scale statistics include all 20 diagnostic fits, not just the 18 accepted cases. The geometry controls are specific examples, not an exhaustive survey: - Curvature uses geodesic distances on an intrinsically three-dimensional sphere of radius 5, represented by a three-coordinate chart. Its extra apparent distance is not the recovered oscillator coordinate. - Instrument-A bias adds the exact state-dependent range shift that would be produced by the proposed extra metric term. Instrument B is unbiased, so this artifact fails the reserved test. - Shared bias gives both instruments that same shift. Independent random noise does not remove their common systematic response. - The two-coordinate metric adds another independent oscillating contribution to distance. Declining it means the one-coordinate hypothesis is inadequate; it does not demonstrate that no hidden metric contribution exists. ## 5. The exact ordinary-space counterexample In ordinary 3D space, let an instrument report \[ Y_{ij}=D_{3,ij}+b_{ij}(h)+\varepsilon_{ij}, \qquad b_{ij}(h)=\sqrt{D_{3,ij}^2+a^2(h_i-h_j)^2}-D_{3,ij}. \] Then \[ Y_{ij}=\sqrt{D_{3,ij}^2+a^2(h_i-h_j)^2}+\varepsilon_{ij}, \] which is exactly the declared 4D observation law. The bias was constructed to test identifiability; the calculation does not assert that a real instrument has this response. Using the same random noise draws, the 4D and shared-bias datasets were **byte-identical in all 20 ensembles**, and all fitted parameters, statistics and decisions agreed. This equality holds algebraically for any trajectories covered by the two observation laws, not merely at the sampled times. Their observation distributions have zero Kullback–Leibler divergence under the same noise law. Additional readings of the same kind cannot distinguish them. This does not show that every calibrated physical experiment is incapable of distinguishing spatial hypotheses. It shows why an instrument's coupling law must be independently constrained. If two models make different independently testable calibration predictions, those additional observations can break the equivalence. We have not supplied such observations here. ## 6. A quantitative requirement on systematic-error calibration A post-benchmark sensitivity calculation uses the predetermined ensemble-0 example. No fitted model or primary threshold was changed. Suppose an independent calibration establishes a per-reading systematic range bound |bᵢ| ≤ η. The largest shift in the displayed Z under a biased flat-3D null is bounded by \[ \frac{|\delta^Tb|}{\sigma\|\delta\|} \le\frac{\eta\|\delta\|_1}{\sigma\|\delta\|}. \] A conservative statistic is therefore \[ \boxed{Z_{\rm robust}(\eta)=Z- \frac{\eta\|\delta\|_1}{\sigma\|\delta\|}.} \] For this synthetic example, the 1% threshold is retained only while the assumed per-reading bound is below approximately **0.01212 length units**, or **4.041 times** the random range-noise standard deviation. The value is in arbitrary units, not metres. It is a requirement for this particular bounded-bias test, not a universal instrument specification or a sufficient proof of 4D geometry. The bound must come from independent calibration. It cannot be justified by the same distance residual under investigation. Repeating measurements reduces random error; it cannot by itself constrain an unmeasured coherent bias or distinguish identical observation laws. ### Even weak correlation can change a nominal significance level For an equicorrelated Gaussian error model with unchanged marginal variance, \[ C=\sigma^2[(1-\rho)I+\rho\mathbf1\mathbf1^T], \] the naive independent-error statistic has variance \[ \operatorname{Var}(Z)=1-\rho+ \rho\frac{(\mathbf1^T\delta)^2}{\|\delta\|^2}. \] For the same predetermined example, a common pairwise error correlation of only **ρ = 0.01** changes the nominal one-sided 1% Z threshold into an actual **17.76% false-alarm probability** under that correlated null. At ρ = 0.001 it becomes **3.01%**. These are exact Gaussian calculations for the Z component, not simulated false-acceptance rates of the complete gated procedure. Correctly using a known C restores the standard-normal statistic. This is why an empirical noise covariance must accompany a real ranging test. ## 7. What would turn this into a physical evidence test The pipeline now provides a concrete test interface: calibrated visible dynamics and positions, reconstructed hidden states, a separate range-calibration set, reserved ranges from another measurement channel, and an explicit error model. It can test whether a single wave-derived coordinate consistently predicts a metric contribution. The missing physical input is an interaction model specifying how the ranging signal couples to that coordinate, together with independent calibration data that constrain state-dependent sensor effects. Spatial measurements must also have documented units, position uncertainty, timing and error covariance. The earlier curvature controls remain relevant. Without those inputs, declaring the computed hidden coordinate to be a new spatial direction would assume the interpretation the test is meant to establish. No real spatial-distance dataset was analyzed in this round. The previously archived LIGO strain segment is not a collection of calibrated inter-marker spatial distances and cannot be substituted for one. The current result is a reproducible conditional measurement test and an explicit demonstration of its remaining ambiguity. ## Reproduction From the archive root, with the existing pinned dependencies: ```bash OPENBLAS_NUM_THREADS=1 python geometry_v4/run_joint.py python geometry_v4/bias_sensitivity.py OPENBLAS_NUM_THREADS=1 python geometry_v4/plot_joint.py python geometry_v4/make_report.py ``` The joint protocol and decoder were fixed and hashed before the benchmark. The earlier hidden-wave identifier is also checked against its original hash. The first complete computation encountered a NumPy-scalar JSON-export error; only serialization was corrected, and replayed numerical CSV ledgers were verified byte-identical to that run. The original export-error log and ledger hashes are retained for provenance. No result-driven model or threshold adjustment was made. `geometry_trials.csv` retains all 120 cases, `wave_trials.csv` all 160 preceding fits, and `examples.json` the predetermined ensemble-0 inputs and outputs for every observation scenario. The six scenarios share each wave ensemble, so results are not 120 independent hidden-wave experiments. `bias_sensitivity.py` is explicitly a later analytic sensitivity calculation. The browser lab and theory manuscript were not modified. --- ## Previous hidden-wave result: reconstructing an omitted coupled wave The user's “off-matrix” proposal has a precise, useful implementation: measure the force residual left by a calibrated visible model, test its low-dimensional structure, and solve for a missing dynamical coordinate. **In the declared synthetic family, the new identifier recovered one coupled hidden coordinate in all 30 strong-coupling cases.** It inferred coupling and frequency from visible observations and predicted a new applied-impulse response. The primary experiment contains **240 cases across eight scenarios**. An adaptive follow-up used **31 new controlled-response datasets** from already fitted cases, including the one independent driver mistakenly accepted as reciprocal feedback. All 31 follow-up comparisons favored the correct feedback model. This is conditional system identification in simulated systems, not a measurement of a fourth spatial direction. Four ordinary oscillator displacements produce exactly the same observable equations. The current atom layer was also checked at **424,800 scalar density samples across all 118 element models**, reserving half for scoring. Its extra temporal component is accounted for by ordinary pairwise interference; no unexplained component remains above numerical error in these tests. ## 1. Define the matrix and its omitted motion Let q(t) have three calibrated displacement channels. A concrete extension of the visible model is \[ \ddot q+Kq+g\,w=f(t),\qquad \ddot w+\omega_h^2w+g^Tq=0. \] Here K is an independently calibrated 3 × 3 visible stiffness matrix, g an unknown three-component coupling, and w an unobserved scalar coordinate. This is an undamped, reciprocal linear oscillator model with a known mass-normalized kinetic metric and equilibrium. It is an explicit conditional extension being tested; it was not derived uniquely from the existing atom equations. During fitting f = 0. Define the “off-matrix” residual as \[ \boxed{r(t)=\ddot q(t)+Kq(t)=-g\,w(t).} \] This is a departure from the visible model's dynamics, not a coordinate literally moving outside a numerical array. All residual vectors should point along the same visible-channel direction if a single scalar w with constant g explains them. Stack the vectors into a matrix and use singular value decomposition to test this rank-one property. Write g = αb with \|b\| = 1 and α > 0, and let a(t) = bᵀr(t). Then \[ a=-\alpha w,\qquad \boxed{\ddot a=\alpha^2 b^Tq-\omega_h^2 a.} \] The second equation supplies the missing scale and bare frequency under the reciprocal-coupling assumption. Rank-one residual forcing alone does not do that. **Conditional identifiability statement.** With exact continuous data, known K and mass normalization, nonzero rank-one residual, linearly independent functions a(t) and bᵀq(t), and the displayed reciprocal oscillator equations, α² and ω_h² are uniquely determined by a two-column linear regression. If both coefficients are physically admissible, g and w are recovered up to the joint sign change (g,w) → (−g,−w). If the regression columns are dependent, or coupling is zero, this conclusion does not follow. Positive definiteness of the assembled stiffness additionally checks stability of this conservative model. This is a statement about the chosen model, not about all possible hidden-variable theories. Observability—recoverability of a state from outputs—is the standard systems concept behind such a test. See the [MathWorks observability documentation](https://www.mathworks.com/help/control/ref/statespacemodel.obsv.html). We do not count numerically tiny singular values as physical dimensions. ## 2. Avoid differentiating noisy measurements directly The identifier fits a shared temporal mode expansion using only the observed history: \[ q(t)=\sum_j[u_j\cos(\Omega_jt)+v_j\sin(\Omega_jt)]. \] A delay-matrix fit proposes one to six frequency pairs. The first half of the 720-sample history fits candidates; the next quarter chooses order; the final quarter checks a refit. Uniform spacing is Δt = 0.25 arbitrary time units. Selection, calibration and refit-disagreement checks each have a 5% threshold. For either sine or cosine coefficient cⱼ, differentiation is algebraic: \[ R_j=(K-\Omega_j^2I)c_j,\quad a_j=b^TR_j,\quad s_j=b^Tc_j, \qquad \boxed{\omega_h^2a_j-\alpha^2s_j=\Omega_j^2a_j.} \] The algorithm solves the boxed coefficient equation after the residual rank test. It checks regression conditioning, positive coupling, positive hidden stiffness, positive full-system stiffness, and agreement of parameters between refits. It requires at least 2% residual forcing relative to the visible stiffness force, at most 5% rank-one residual error and at most 5% closure error. The exact rules are in `protocol.json`. Every normal mode is deliberately excited in this benchmark. The input interface receives **only** noisy q samples, Δt, calibrated K, the forecast horizon and fixed thresholds. It never receives the hidden trajectory, true g, hidden frequency or generator order. Those quantities enter the scorer afterward. The prototype is therefore blind to the planted hidden values, while still relying on explicitly supplied physical calibration. ## 3. What was recovered | Measurement noise | Coupling-vector error | Bare-frequency error | Hidden-history reconstruction error | Visible impulse-forecast error | |---|---:|---:|---:|---:| | Zero | 2.17 × 10⁻¹²% | 9.87 × 10⁻¹⁴% | 4.43 × 10⁻¹²% | 9.84 × 10⁻¹²% | | 0.1% | 0.0825% | 0.0016% | 0.1046% | 0.0945% | | 1% | 0.7411% | 0.0081% | 0.8286% | 0.6652% | Entries are median percentage errors over ten strong-coupling systems per noise level. The zero-noise row reflects floating-point precision, not experimental accuracy. Coupling error is relative vector error after the unavoidable joint sign alignment; bare-frequency error is relative scalar error. Hidden-history error is normalized by actual hidden-history RMS for scoring. Visible errors are normalized by observed channel fluctuation RMS. The hidden-history diagnostic was added in the declared adaptive follow-up without changing the original identifier. The applied impulse and its response were not used to fit any hidden parameter. A representative predetermined example at 0.1% noise recovered: | Quantity | Planted value | Recovered value | |---|---:|---:| | Bare hidden angular frequency | 1.360213424 | 1.360236908 | | Coupling g₁ | 0.104992202 | 0.104846129 | | Coupling g₂ | 0.164971845 | 0.164805686 | | Coupling g₃ | −0.089115515 | −0.089069035 | The frequency is radians per arbitrary time unit. Couplings have inverse-time-squared units in the declared coordinate normalization. These are not physical atomic base frequencies. **Retained limitations:** passing the checks did not guarantee every forecast error stayed below 5%. In the original run, integrating the approximate recovered physical model all the way from time zero produced up to 15.40% visible future error and 27.24% hidden future error at 1% noise. Restarting at the observation boundary from the fitted state, evaluated afterward, reduced the worst errors to 3.25% visible and 5.64% hidden. Both sets of scores remain in the ledgers. The maximum new-impulse forecast error in the strong-coupling primary cases was 3.52%. ## 4. Competing explanations were tested | Generator | Accepted as one reciprocal hidden coordinate | Interpretation | |---|---:|---| | One reciprocal hidden oscillator | 30/30 | Correct model family | | No hidden oscillator | 0/30 | Correctly declined | | Weak reciprocal coupling | 6/30 | Only coupling norm 0.1 cleared the chosen visibility floor | | Incorrect visible calibration | 0/30 | Calibration error was not accepted as hidden feedback | | Two independently coupled hidden oscillators | 0/30 | One-coordinate hypothesis declined | | Two hidden oscillators with parallel coupling | 0/30 | Rank-one forcing alone did not pass the dynamics check | | Independent external driver | 1/30 | One false acceptance of reciprocal feedback | | Nonlinear sensor, three actual oscillators | 0/30 | Sensor distortion was not accepted as hidden feedback | The weak-coupling sweep used norms 10⁻⁴, 10⁻³, 10⁻², 0.05 and 0.1, six cases each. Only the 0.1 cases passed the declared visibility floor. This abstention policy is a chosen operating threshold, not a proof that every smaller nonzero coupling is fundamentally undetectable. The fully decoupled case is fundamentally different: no observation of q can reconstruct a coordinate that has no effect on q. Two controls matter especially: - **Incorrect K:** a rank-one error in visible stiffness also creates rank-one residual forcing. Its closure regressors become dependent in exact arithmetic, preventing separate identification of the proposed coupling and frequency. The model therefore cannot treat every residual direction as a new hidden coordinate. - **Parallel hidden couplings:** two genuine hidden coordinates can produce rank-one forcing. The one-frequency hidden dynamics then fail the closure test. Residual rank counts independent forcing directions, not necessarily the number of missing variables. One 1%-noise external-driver case nevertheless passed all primary checks. It is retained as a false acceptance of reciprocal feedback. An autonomous driving wave really is unobserved in that generator, but it does not respond to the visible coordinates as the assumed reciprocal model requires. ## 5. An intervention distinguished driving from feedback After observing that failure, a separate response protocol was written before running its new data. It tested all 30 strong hidden-coupling fits plus the accepted external-driver fit. It did not change the original thresholds or refit the original parameters. For each visible axis, apply a known unit velocity impulse and subtract a matched no-impulse run from the same initial state. Average ten independent paired repetitions. The differenced noise standard deviation is √(2/10) times the individual measurement noise. These resets and noise assumptions are part of the synthetic protocol. The subtraction removes an unchanged autonomous driver. A reciprocally coupled hidden oscillator, however, changes in response to the visible impulse and contributes to the differential response. The two frozen predictions are: 1. The inferred coupled model. 2. The calibrated visible K with no hidden feedback in the differential response. All **31/31** held-out response comparisons favored the correct feedback model. In the earlier false acceptance, mean squared error was **2.4080 × 10⁻⁴** for the inferred feedback model and **2.6227 × 10⁻⁵** for the independent-driver response. This resolves that example under the declared controls. It is not a population error bound or a claim that 31 new independent hidden systems were trained: the follow-up reused 31 earlier fits and generated new response observations. An ordinary internal oscillator responds reciprocally in exactly the same way. The intervention establishes a coupling distinction, not an extra-space distinction. ## 6. The missing coordinate appears as memory in the visible equations Eliminating w exactly gives \[ w(t)=w(0)\cos\omega_ht+\frac{\dot w(0)}{\omega_h}\sin\omega_ht -\int_0^t\frac{\sin[\omega_h(t-s)]}{\omega_h}\,g^Tq(s)\,ds, \] and hence \[ \ddot q+Kq=f-gw_{\rm free}(t)+\int_0^t\Gamma(t-s)q(s)\,ds, \qquad \boxed{\Gamma(t)=\frac{gg^T}{\omega_h}\sin(\omega_ht).} \] This provides a precise version of a hidden wave feeding back into the visible matrix: the visible dynamics acquire a memory term. The numerical check reproduces the explicitly evolved hidden coordinate within **4.32 × 10⁻¹⁴** absolute error. With zero initial conditions and Laplace variable s, the effective visible response has inverse \[ \chi(s)^{-1}=s^2I+K-\frac{gg^T}{s^2+\omega_h^2}. \] Thus a calibrated frequency-response experiment could test a rank-one, frequency-dependent feedback term. Several hidden oscillators would generally produce a sum of such terms. The expression follows by eliminating the hidden equation; it is not an experimentally discovered new law. Poles of a response function describe dynamics, and do not by themselves specify spatial geometry. ## 7. What the existing atomic equations contain For the current three-state engine, \[ \Psi(\mathbf r,t)=\sum_j A_j(\mathbf r)e^{-i\omega_jt}, \] the intensity is \[ |\Psi|^2=\sum_j|A_j|^2+ 2\operatorname{Re}\sum_{j