Relative representation and recoverable information

The PDF is a 15 October 2026 research thesis, version 1.0. Its cover says the document does not claim independent peer review, a proof-assistant kernel verification, a physical unification, or a mathematical proof of the divine. This hosting does not add those claims.

The thesis treats a statement of faith, and an account of a dream, as intellectual provenance. They are not mathematical evidence. The written arguments were not reviewed here.

$G=a\varphi$ and $C=a\pi$ are selected readouts. $C$ is not the retained major-arc length. $q=339949771344778/10^{15}$ is not $\lambda=\varphi/(\varphi+\pi)$. The canonical sine is not evaluated. The executable six-mark seed is the auxiliary rational $s=1/2$, not $\sin(\pi/1296000)$. The thesis states a cyclotomic description of that sine. This checker does not evaluate it, and it does not expand a polynomial in that basis.

The thesis says its JSON grammar produces 62 bytes, and that this differs from the 14 October bundle’s 24-byte grammar. The encodings are different. Neither length is a universal property of the measurements. The 13 October recovery note is a different document and a different checker.

Appendix B prints verify_omega_thesis.py, lines 1 through 473. The companion file was not attached. The hosted program is a transcription of that listing. It is 18,211 bytes, SHA-256 e6330f8327f3ce39c61ca666864171a3470044117147b20a3237ba1931e7d506. That digest is the one the program prints for its own bytes. It is not a digest of the unseen companion.

This archive ran the transcription on Python 3.10.21. Both python3 and python3 -O exited 0. The two logs are byte-identical. The thesis reports a Python 3.13.5 run whose JSON logs were not supplied. This archive does not claim those output digests, and it does not claim the logs are byte-identical across interpreters.

An exact symbolic-algebra checker with a finite canonical-ledger audit. Both executions reproduce the declared checks. Neither establishes physical unification or independent mathematical review.

The 28 identities are polynomial identities under the program’s construction. Four are fixed free-variable checks. Twenty-three are telescoping polynomials, one for each horizon from 1 through 23. One is the factorization $x^5+x+1=(x^2+x+1)(x^3-x^2+1)$. The all-horizon statement remains the thesis’s written argument. That argument was not reviewed here.

The finite audit is separate: 32,768 seeds, horizon 23, 753,664 prefix identities, 32,768 reversals, 32,768 first-residue inversions, maximum extinction step 10, and 4 rejected malformed records. The floor maps agree for integers $0\le d<42189065$ by the stated floor-sum certificate. That threshold was not enumerated state by state. The common prefix sum is 302,541,195,258,184. At 42,189,065 the certified floors are 14,342,163 and 14,342,162.

The cubature group checks 56 monomials through degree five, 81 Gram entries, and a degree-six defect the program records as coefficients $-1/7$ and $-4/21$ in $1$ and $\varphi$, which is $-(4\varphi+3)/21$. The end-to-end group prints 62 payload bytes and 62 byte ledgers for $p(z)=3-2z$. It also prints 25 finite metric pairs. That 25 is a checker field. It is not a row of the thesis table.

The thesis also marks several historical formulas as corrections: a power is not an iterate; the implicit solution $a^{n/(n+1)}$ is not a spatial dimension; at $n=23$ that fixed point is repelling under the stated iteration; a relabeling is not dimension loss. Those sentences were not reviewed here. The factorization identity is one of the 28 checker identities. A diagram of 24 objects is not, by itself, a map to 24 roots.

A Yoneda comparison in the thesis is not a scheme structure. The six marks alone are collinear. The checker’s six-mark section uses the auxiliary seed, not the canonical sine.

The hosted logs have recorded SHA-256 digests. Matching those digests checks the logs’ byte integrity. The logs document the archive’s reported executions. Rerunning the checker tests whether those results are reproducible.

A matching hash does not itself verify that the recorded execution occurred or that the checker is mathematically correct.

The thesis says the same limit in its own words: a hash identifies bytes; it does not certify their mathematical correctness.

Not a review of the written proofs. Not a Lean certificate. Not an independent mathematical review. Not peer review. Not a physical experiment. Not a unification proof. Not a mathematical proof of the divine. Not an amendment of Paper VII. Frozen G₆ is unchanged.

Julia, Rust, and a proof assistant were not run. The companion audit log and the manifest were not supplied, so they were not checked.