# Theory of Reality — revision 1.14

5 September 2026 · Anthony Michael Collado's research framework  
Companion: Rope Tomography Lab 0.2.0  
Evidence: written mathematical arguments and synthetic experiments.

## 1. A retracted argument and its replacement

The previous paper's Theorem 17.1 required seven strict successive refinements, each the coarsest complete deterministic refinement for the **same** transition family. Those requirements are incompatible.

Write \(C_{\mathcal A}(\mathcal P)\) for the coarsest dynamically sufficient refinement of an observation partition \(\mathcal P\), with a fixed family \(\mathcal A\) of deterministic transitions on a finite state set. Then

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

**Proof.** Put \(\mathcal Q=C_{\mathcal A}(\mathcal P)\). Every transition already descends to \(\mathcal Q\), so \(\mathcal Q\) itself is an admissible sufficient refinement of \(\mathcal Q\). It is therefore its own coarsest sufficient refinement. Repeating complete closure cannot add a strict layer.

The old conditional statement had no instance satisfying all its premises. It consequently supplied no evidence for a seven-stage physical hierarchy. It is replaced in the main paper; the previous version is preserved in the archive's history directory.

A valid nearby construction observes the least significant bit of a seven-bit register, then right-shifts it repeatedly. Looking zero through six steps ahead produces

\[
2,\ 4,\ 8,\ 16,\ 32,\ 64,\ 128
\]

distinct observation-history classes. These are successive **finite-horizon** refinements. Complete closure already distinguishes all 128 states; closing it again changes nothing. An \(m\)-bit register similarly gives \(m\) levels. Seven is a chosen register size, not a dimensional theorem.

The exact implementation checks 3,984 partition/transition instances with one control through four states, and 3,678 with two controls through three states. Each check tests idempotence, stability, and agreement with a separate action-word trace calculation. All **7,662** pass. The general proof is the argument above, not the finite enumeration.

## 2. Define what a measurement can identify

Let \(e\in\mathcal E\) be an allowed experiment, \(\theta\in\Theta\) a modeled cause, and \(F_e(\theta)\) its predicted response. For deterministic predictions with a common specified noise law, define

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

For general stochastic models, use equality of full response distributions, not just means. Recoverable causes are classes \(\Theta/{\sim_{\mathcal E}}\).

Adding allowed experiments can split these classes:

\[
\mathcal E\subseteq\mathcal F\quad\Longrightarrow\quad
\sim_{\mathcal F}\ \subseteq\ \sim_{\mathcal E}.
\]

Equality on all of \(\mathcal F\) includes equality on its subset \(\mathcal E\). This proves the inclusion, but does not guarantee a strict split.

With repeated conditionally independent measurements in the implemented noise model, causes with identical responses under every allowed experiment generate identical data distributions. Even adaptive experiment choice cannot distinguish them: inductively, equal past-data laws induce equal laws for the next selected experiment and its response. A new representation cannot extract an absent distinction from the same data.

Structural identifiability and practical recovery under noise are established research topics. A 2025 primary research preprint studies choosing control inputs to improve identification and model discrimination. Our finite catalogue approach implements a limited version of that broader research direction; it does not claim the paper's optimal-control method or mathematical novelty. [Liu, Maini and Baker](https://arxiv.org/html/2506.11311v1)

## 3. What the selector optimizes

Each experiment supplies fixed candidate predictions \(\mu_{ei}\), known independent real Gaussian noise standard deviations, positive cost, and intervention assumptions. Complex responses are stored as real and imaginary coordinates. Noise deviations are **per real coordinate**, unlike the earlier lab's complex RMS noise convention.

For normalized candidate weights \(w_i\), common covariance \(\Sigma_e\), and cost \(c_e\), the objective is

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

The selector maximizes this weighted pairwise Gaussian KL criterion over affordable catalogue entries. It sees the predictions and weights, not the next true label or observation. It returns no selection when every affordable score is numerically zero, at the declared \(10^{-12}\) tolerance. That status is not a proof about experiments outside the catalogue.

This objective is not exact Bayesian information gain or a global optimum over every feasible apparatus. Generating the candidate hypotheses and establishing intervention feasibility remain outside the selector. Its pairwise implementation costs \(O(|\mathcal E|M^2r)\) arithmetic operations for \(M\) candidates and at most \(r\) coordinates, apart from computing the predictions.

Weights update by the declared Gaussian likelihood. Separately, the engine tests absolute compatibility:

\[
r_i^2=(y-\mu_{ei})^\top\Sigma_e^{-1}(y-\mu_{ei}).
\]

It retains candidates below the appropriate chi-square 99th percentile. If none survive, the result is **all candidates rejected**. This assumes fixed predictions, known noise, and no parameter fitting to that observation. It is a per-observation test; repeated adaptive use requires separate sequential error control.

## 4. Three ambiguities and their interventions

| Ambiguity | What cannot resolve it | Added experimental assumption |
|---|---|---|
| Delays 0.37 s and 1.37 s | Referenced phase at integer Hz | A noninteger-frequency probe separates this pair |
| Path/local delays (0.8, 0.2) s and (0.5, 0.5) s | More frequencies with the same total-delay response | Double path delay while preserving speed and the local attachment |
| Equal \(c e^{-0.4d}\) for different depth/coupling pairs | More of the same uncalibrated amplitude | Independently set coupling to one without changing depth |

These are hypothetical calibrated interventions. Their availability in a real system has not been demonstrated.

For the delay split, let \(\tau\) be propagation delay and \(\delta\) local pure delay. If an intervention scales only the path by known \(\lambda\ne1\), unwrapped total delays satisfy

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

Thus

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

This is exact under those assumptions. Phase-only readings still require cycle resolution before being interpreted as unwrapped delays. The simulation discriminates two fixed candidates; it does not recover arbitrary continuous delays from one wrapped phase observation.

Similarly, calibrated gain can separate the two modeled depth values. Calling this fitted depth an extra physical or spiritual dimension would require an independent operational definition and new evidence.

## 5. Executed evidence

The catalogue, noise, seed, budget, and scoring rule were written to the protocol file before scored observations were generated. Each case has 2,000 synthetic trials with balanced true labels. These cases expose known ambiguities; they are not representative of all inverse problems.

| Scenario | Result | Closest-mean errors / 2,000 | All-candidate rejections / 2,000 |
|---|---|---:|---:|
| Integer-frequency phase only | All posterior decisions remain tied | Not applicable | 15 |
| Expanded phase catalogue | 1.5 Hz selected; 0.5 Hz is equally optimal mathematically | 12 | 17 |
| More frequencies, unchanged path | All remain tied | Not applicable | 16 |
| Known path doubling | Pair distinguishable | 16 | 15 |
| Same uncalibrated gain | All remain tied | Not applicable | 22 |
| Independently reset coupling | Pair distinguishable | 37 | 8 |

The error column scores forced closest-mean decisions, including trials on which the adequacy check rejects both candidates. It is not the error rate among accepted decisions. False rejections can occur even when a candidate is true because the cutoff is statistical.

For two equal-prior candidates with common Gaussian covariance and Mahalanobis separation \(D\), the ideal forced-decision Bayes error is \(\Phi(-D/2)\). The informative cases give approximately 0.621%, 0.711%, and 2.485%. Observed errors are finite samples, not general guarantees.

The missing-model control is instructive. Candidates predicted means 0 or 1, noise deviation was 0.1, and the generating mean was 3. In **all 2,000 trials**, a candidate received more than 99% normalized posterior probability while the absolute check rejected **both**. High relative probability can identify the less wrong candidate.

The independent audit recalculated all **14,000** observation records, catalogue scores, decisions, and counts without importing the selector or simulator. All **15** unit-test groups pass, including the earlier wave and transfer tests. A first test incorrectly required 0.5 Hz instead of allowing equally optimal 1.5 Hz; that expectation was corrected without changing the selector, scored observations, or protocol.

These are synthetic tests and exact finite computations. This revision adds no real physical measurements.

## 6. Other corrections and continuing research

The main paper now distinguishes a finite-diagram identity from physical completeness, full local sensitivity rank from global uniqueness, stochastic outcomes from different future distributions, encoding ULPs from physical resolution, and raw bit edits from valid state transitions.

The claim ledger records status, assumptions, evidence, and failure conditions. Earlier claims are retained in history. The continuing workflow is to specify alternatives, establish their observational equivalences, declare an experiment catalogue, freeze predictions, collect new outcomes, compare absolute and relative fit, and revise with provenance. A counterexample or precise unresolved equivalence class counts as progress. Adding a layer solely to absorb a residual does not.

The next physical step needs a calibrated actuator, known path modification, synchronized sensors, and an attachment whose response remains unchanged by that modification. Multiple path lengths would test the delay-separation formula. Drift and intervention side effects must be considered before interpreting a discrepancy as new physics.

For RRLS, identical compiled invariants paired with opposite verified SAT answers rule out a universal decoder using just those invariants. A better representation helps only when it retains the needed distinction and its full computational cost is accounted for. P versus NP remains listed as unsolved by the [Clay Mathematics Institute](https://www.claymath.org/millennium/p-vs-np/). This revision proves neither equality nor separation.

God as the ground of reality remains an optional theological interpretation. A boundary-drive equation, unexplained residual, or fitted latent coordinate does not identify or measure divine action.