# Universal-Cycle Tomography: a causal boundary model and its limits **Revision companion:** Reality_Revision_1.14.md and Rope Tomography Lab 0.2.0 add a finite intervention selector and correct the main paper's repeated-closure argument. The original experiments below remain the earlier record; new observations are stored separately in results-design/. **Theory of Reality research addendum.** Computational prototype, September 2026. All observations in this laboratory are synthetic. No physical experiment or observation of additional dimensions is claimed. ## Result The rope-and-hairs proposal can be formulated as a causal input-output system. A calibrated source and a restricted, correct response model can allow delay estimation and prediction at frequencies not used for fitting. An arbitrary unknown source and arbitrary response functions cannot supply unique hidden coordinates. The included lab ran: - a wave-equation calculation on three successively finer grids; - 72 noisy fits of the specified response family and 72 fits with an unmodelled echo; - predictions on six withheld frequencies per fit; - 200 trials each of one scalar driver, two independently coupled drivers, and two indistinguishably aligned drivers; - seven test groups, including independent time-domain ODE verification, prediction from one reference sensor, and exact ambiguity examples. The specified response model predicted near the synthetic noise floor. The echo model failed that prediction check despite successful optimization. The scalar-driver test rejected independent coupling directions, but could not distinguish aligned drivers. These are checks of a declared mathematical model, not evidence that nature has a particular number of dimensions. ## 1. A consistent clock and path coordinate Let \(s\ge0\) be distance along a modeled propagation path, in metres. Let \(t\) be physical evolution time, in seconds. For a uniform, small-displacement ideal string, \[ \mu u_{tt}=T u_{ss},\qquad v=\sqrt{T/\mu}. \] Assume the string starts at rest, there is no incoming wave, the boundary satisfies \(u(0,t)=g(t)\), and \(g(t)=0\) before the drive begins. On a semi-infinite string, \[ u(s,t)=g(t-s/v) \] is an exact outgoing solution. Direct differentiation gives \(u_{tt}=g''\) and \(u_{ss}=g''/v^2\). On a finite string this expression applies only before the relevant reflected waves arrive. The front cannot advance more than \(v\Delta t\) after a disturbance is introduced. This is a causal-cone structure for this wave equation. Obtaining special relativity requires additional physical structure, including the invariant speed and transformations between inertial observers. A string's material wave speed alone does not derive Einstein's theory. Calling \(s\) “causal order” is a possible interpretation, but the wave equation still needs a coordinate scale or metric along it. An ordering alone does not define distance or a propagation speed. If the rope coordinate is instead physical time \(t\), a field \(u(t,\lambda)\) needs a meaning for the second parameter. It may label a family of histories or be a mathematical parameter. It does not automatically become an independently observable second physical time. A literal changing-history theory would need dynamics, observables, and a rule connecting both parameters to measurements. The laboratory uses one evolution time and one spatial path coordinate. It makes no numerical assumption about divine time. ## 2. The attached response must be modeled At a weakly loading attachment, write \[ q_i(t)=(h_i*g)(t-\tau_i)+\epsilon_i(t), \qquad \tau_i=s_i/v. \] Here \(h_i\) is a causal local impulse response and \(*\) denotes convolution. This form assumes linear time-invariant response over the experiment and neglects attachment backreaction on the ideal outgoing wave. Strong attachments can change propagation and produce reflections; their coupling then belongs in the forward model. A concrete first test is an overdamped local response: \[ \rho_i\dot q_i+q_i=a_i g(t-\tau_i), \qquad a_i>0,\quad\rho_i>0. \] For the convention \(e^{i\omega t}\), its transfer function is \[ \boxed{\frac{Q_i(\omega)}{G(\omega)} =\frac{a_i e^{-i\omega\tau_i}}{1+i\omega\rho_i}.} \] Therefore \[ \left|Q_i/G\right|=\frac{a_i}{\sqrt{1+(\omega\rho_i)^2}}, \] and the phase lag obeys \[ \boxed{\phi_i=\omega\tau_i+\arctan(\omega\rho_i)\pmod{2\pi}.} \] This is the first correction to the proposed address decoder. Measured phase contains both transit and local response. Its entire value cannot generally be converted directly into a position. The general method is system identification: estimate a specified family of dynamical input-output models and then validate its predictions. This established framework is described in [Lennart Ljung's system-identification survey](https://www.diva-portal.org/smash/get/diva2%3A316967/FULLTEXT01.pdf). The particular calculations and counterexamples below are explicit parts of this lab. ## 3. A phase records position modulo a cycle Even if the local response contributes no phase, a measured lag \(\phi\) gives \[ s=\frac{v}{\omega}(\phi+2\pi k),\qquad k\in\mathbb Z. \] A physical domain such as \(s\ge0\) may remove negative candidates, but it does not automatically select the remaining winding number. For known rational frequencies \(f_j=k_jf_0\) with coprime integer \(k_j\), the full referenced phase fingerprint repeats after \[ \Delta\tau=1/f_0,\qquad \Delta s=v/f_0. \] For the proposed harmonic set \(1,2,3,5\) times one base frequency, all four phases repeat after one base period. With frequencies \(1,2,3,5\) Hz, delays 0.37 s and 1.37 s are exactly indistinguishable in the ideal phase-only measurement. The lab checks their phasors numerically and computes the period with exact rational arithmetic. Multiple frequencies can extend the ambiguity range relative to one high frequency. For example, the individual 10 Hz phase repeats after 0.1 s, whereas jointly referenced 10 Hz and 11 Hz phases repeat after 1 s. This still needs a range bound or extra information to select an absolute delay. Absolute distance measurements combine phase and timing information while accounting for ambiguity; see the primary [NIST multi-frequency distance-measurement study](https://www.nist.gov/publications/rapid-and-precise-absolute-distance-measurements-long-range). Exactly incommensurate ideal frequencies have no nonzero exact common period. Finite measurement precision introduces near-ambiguities, however, so that ideal statement is not an unlimited-range practical decoder. Unknown clock offsets and unknown source phase add further ambiguity. Distance also requires a calibrated propagation speed: simultaneous scaling of \(s\) and \(v\) leaves \(s/v\) unchanged. ## 4. Two non-identifiabilities survive arbitrarily many frequencies **Transit versus local pure delay.** Suppose the attachment has an additional local delay \(\ell_i\). Its measured phase factor is \[ e^{-i\omega(\tau_i+\ell_i)}. \] The alternatives \[ (\tau_i,\ell_i)=(0.8,0.2)\ \mathrm{s} \quad\text{and}\quad (\tau_i,\ell_i)=(0.5,0.5)\ \mathrm{s} \] give exactly the same response at every frequency. Both are causal. A known input alone cannot distinguish them. A separate physical constraint on the local response is necessary. **Gain versus alleged dimensional depth.** If the measurable amplitude factor is \[ A_i=c_i e^{-\alpha d_i}, \] then, for any admissible \(\delta_i\), \[ d_i'=d_i+\delta_i,\qquad c_i'=c_i e^{\alpha\delta_i} \] leaves \(A_i\) unchanged. The attenuation formula alone creates no latency. A latency needs a dynamical law such as the one-pole response or a specified propagation mechanism. If \(c_i\), \(\alpha\), and the interpretation of \(d_i\) are independently established, amplitude may determine \(d_i\) within that model. If those are free unknowns, the depth is unidentifiable. Calling an inferred response parameter “dimension” does not establish a spatial or spiritual dimension. These are exact observational-equivalence classes of the kind already represented by the Effect–Cause Engine's fibers. More accurate arithmetic cannot distinguish parameters that produce identical effects. ## 5. Unknown source means additional gauge freedom With an unobserved driver, \[ Q_i(\omega)=H_i(\omega)G(\omega). \] For any nonzero constant \(c\), the replacements \(G'=cG\) and \(H_i'=H_i/c\) preserve every output. More generally, a common invertible filter can be moved between the source and the response functions if both transformed sides stay in the admissible model classes. There is also a clock-origin ambiguity. If all transit delays exceed \(\delta\), set \(g'(t)=g(t-\delta)\) and \(\tau_i'=\tau_i-\delta\). All delayed responses remain identical. Consequently, a compact common-driver fit does not automatically identify an absolute source waveform, absolute origin, unique cause, or unique hidden coordinate system. Calibration fixes some of these freedoms. Without restrictions on the functions \(H_i\), their descriptions can carry the complexity one hoped to compress. ## 6. What a scalar-driver model predicts across sensors At a fixed frequency, collect outputs across multiple independently varied excitation trials or suitable stationary windows. With one scalar source, fixed linear coupling, and additive noise, \[ Y=h g^{\mathsf T}+E. \] The noise-free signal has rank at most one. Since its second singular value is zero, \[ \boxed{\sigma_2(Y)\le\|E\|_2.} \] A second singular value significantly above a justified noise bound can reject this specified scalar-driver model. One FFT vector multiplied by its own conjugate transpose is always rank one and cannot perform this test; independent excitation samples are essential. Passing does not prove one cause. Two sources with aligned couplings satisfy \[ h g_1^{\mathsf T}+\beta h g_2^{\mathsf T} =h(g_1+\beta g_2)^{\mathsf T}, \] which has exactly the same rank-one form. Dependence between sources can create similar observational ambiguities. Correlated sensor noise or changing transfer functions can also violate the assumed noise and coupling model. A vector boundary drive changes the hypothesis. With \(p\) independently excited components, \(Y=HG+E\) has signal rank at most \(p\). One physical boundary location can support multiple input channels. The detected rank concerns independently observable input directions; it is not automatically the number of dimensions of reality. ## 7. The synthetic experiments ### Wave propagation The string has length 6 m and speed 2 m/s. A compact smooth boundary pulse starts at 0.2 s and lasts 0.6 s. The solver compares the wave at 1.6 s, before the right-boundary reflection. It uses a centered explicit scheme with Courant number 0.8. | Grid spacing | L2 error against the retarded solution | |---|---:| | 0.100 m | 0.0436087 | | 0.050 m | 0.0104790 | | 0.025 m | 0.0025902 | Halving the spacing reduces the error by factors 4.16 and 4.05, consistent with second-order convergence in this test. A separate Courant-1 case agrees at the grid nodes to numerical precision. A finite-difference scheme's numerical dispersion should not be interpreted as physical transmission outside the continuum causal cone. ### Parameter recovery and withheld frequencies The input is a known, calibrated scalar signal with nonzero probe amplitude. The stored observations are noisy complex transfer ratios. Eight frequencies are used for fitting: 0.3, 0.5, 0.8, 1.1, 1.7, 2.3, 3.1, and 4.2 Hz. The six withheld frequencies are 0.41, 0.93, 1.43, 2.71, 3.73, and 5.19 Hz. The last is an extrapolation beyond the training band. Six known synthetic parameter sets, each with twelve noise realizations, make 72 fits per condition. Complex noise has RMS 0.005. The estimator profiles gain over 8,040 delay/relaxation grid candidates and refines three separated initializations with bounded nonlinear least squares. It does not claim a proved global continuous optimum. | Condition | Fits | Median withheld-frequency RMSE | RMSE / injected noise RMS | |---|---:|---:|---:| | Correct one-pole response family | 72 | 0.005347 | 1.07 | | Additional unmodelled delayed echo | 72 | 0.232526 | 46.51 | For the correct family, median absolute parameter errors were approximately 0.00163 in gain, 0.000197 s in transit delay, and 0.000363 s in relaxation time. The delay figure is about **0.20 ms**. If speed were known to be 2 m/s, that delay error would correspond to about 0.39 mm within this synthetic model; it is not an instrument specification. All 144 optimizations reported success and full local Jacobian rank. The echo condition nevertheless predicted badly. **Optimizer convergence and local parameter rank do not validate the forward model.** The echo remains a single-driver system with a more complicated transfer function, so this failure rejects the one-pole family, not every common-driver explanation. A simple complex linear-interpolation comparator is included in the raw results. It is a response-lookup baseline, not an optimized independent-dynamics competitor. These experiments do not establish that a common driver compresses real observations better than every alternative model. ### Scalar-driver diagnostic At one fixed frequency, six sensors are measured over 96 independent excitation windows. Independent complex sensor noise has RMS 0.04. A threshold is fixed using the 99th percentile of 512 separate noise-only simulations. This empirical percentile is not a proved 99% population bound. | Synthetic condition | Trials | Scalar-driver rejections | Median withheld-sensor RMSE | |---|---:|---:|---:| | One scalar driver | 200 | 0 | 0.06270 | | Two drivers with independent coupling directions | 200 | 200 | 0.63744 | | Two drivers with aligned coupling directions | 200 | 0 | 0.06298 | The aligned case exposes an exact limitation. No improvement in this rank statistic can recover the hidden source count when the measurement map collapses both source directions onto the same vector. For the prediction column, the engine learns coupling ratios from the first 64 windows. In the next 32 windows it observes only sensor 0 and predicts the other five sensors without refitting. This is prediction of withheld channels conditional on a new reference measurement, not forecasting an entirely unobserved future driver. Noise in the reference measurement contributes to the prediction error as well as noise in the withheld sensors. The noise-free rank-one signal has \(M+K-1\) complex degrees of freedom rather than \(MK\) for \(M\) sensors and \(K\) windows, once one common scale is fixed. This can compress the signal component. Lossless compression of the full noisy observations must still account for the residual and numerical precision. Raw transfer observations, fitted parameters, singular values, prediction errors, noise calibration, seeds, and timing are included. The rank experiment's generated measurement matrices can be reproduced from its saved seed and source; they are not all stored individually. Seven test groups passed, including an independently integrated time-domain ODE check of the frequency-response formula, wave convergence, noiseless recovery, withheld predictions, and the rank perturbation bound. A separate script recalculates the recorded errors and rejection counts without importing the fitter. Numerical tests complement the written identities; they do not prove universal empirical validity. ## 8. Boundary amplitude is not a bound on all influence or energy A finite displacement range \(|g(t)|\le A\) does not by itself determine a maximum affected distance. At a given elapsed time the causal range is bounded by \(v\Delta t\); continuing the drive can reach more distant points. Attenuation and a detection threshold may impose a practical measurement range, which is a different bound. Nor does a displacement bound alone bound injected energy: faster oscillation changes boundary velocity and power. For the ideal string, \[ E(t)=\frac12\int_0^L(\mu u_t^2+T u_s^2)\,ds, \] \[ \frac{dE}{dt}=[T u_su_t]_0^L. \] The external drive supplies boundary work. Any dissipative attachment or damping requires its energy loss to be included as well. The model does not create energy without a source. ## 9. What would make this a test of nature? The next empirical step is a defined physical system with a calibrated actuator and synchronized sensors. Specify the propagation law, admissible attachment responses, noise model, and competing models before fitting. Vary the input deliberately, fit on one set of trials, and predict withheld frequencies or time sequences without refitting. Use physically justified bounds on delay, gain, and local response; retain alternative parameterizations that remain observationally equivalent. For a passive, unobserved source, begin with a restricted source/filter class and explicit calibration assumptions. Correlations alone cannot remove the source/filter freedoms above. A claimed compression advantage must count the driver, every transfer function, parameter precision, and the unexplained residual, and compare against well-defined alternatives on held-out data. The theological interpretation may identify the ultimate source with God. The mathematical symbol \(g(t)\) is the boundary signal as represented in created time, not a measurement of God's own state or an independently established divine clock. These equations can accommodate that interpretation without establishing it or measuring a maximum range of divine agency. The strongest supported formulation is: > Observable trajectories can sometimes share a compact causal input-output description. The recoverable objects are the parameter or source equivalence classes left after calibration and measurement, and the description earns empirical support through successful withheld predictions.