Supplementary note · 24 October 2026

Reference geometry and recoverable relationships

A known reference can locate points, carry a finite formal record, and support an exact translation when the required data and the decoding rule are retained. The written arguments are supplied. They have not been reviewed as a proof.

Exact OMEGA scaling $a\mapsto\lambda a$ , where $0<\lambda<1$ , is injective at every finite step. Distinct inputs can become indistinguishable after integer quantization or an observation that discards distinctions. Retaining valid reconstruction witnesses reverses the integer transition.

The elementary reason is that $\lambda a=\lambda b$ implies $a=b$ when $\lambda$ is nonzero. The floor map $T(d)=\lfloor Qd/S\rfloor$ has multiple predecessors; its valid witness supplies the missing distinction.

Three noncollinear marks already in the construction locate a point of their plane from exact distances. A finite encoding can carry a finite record. The known shape does not supply the unknown measurements. A property survives an observation exactly when it is constant on every observation fiber. A linear history separates states only when the observation map has full column rank. Rank-one data in the plane cannot invent a second independent direction.

Planar inverse

Let $o$ be the center, $p_{\pm}=o+a(cu\pm Av)$ , and $P=o+Xv+Yu$ . For $R_0=\|P-o\|^2$ and $R_{\pm}=\|P-p_{\pm}\|^2$ ,

X=\frac{R_--R_+}{4aA},\qquad Y=\frac{2R_0+2a^2(c^2+A^2)-R_--R_+}{4ac}.

The denominators require $a>0$ and $0<\varepsilon<\pi$ . A candidate record must reproduce the three defining distances. The six baseline marks alone are collinear; the center supplies the noncollinear anchor here. Planar distances leave a perpendicular direction ambiguous.

The reference proofs are the general recovery account. The geometric arithmetic note supplies the scale-discrepancy identity and the integer-history inverse for this family. The reference language is the typed record for the scale, byte, and text subset. It does not silently serialize an unrecorded position. Observation-fiber recovery is the same idea as network witness completion: invertible mixing still needs a quantization witness, while a singular map can hide a kernel distinction that a remainder does not restore.

Proofs, checker, and this archive’s run

The complete proof text is PROOFS.md, 54,827 bytes, SHA-256 feb3393dd3ad738aee848143830db47b2e4268bb2c34c7d2d9037eac27e5d5fc. The restored checker is omega_reference_verify.py, 27,579 bytes, SHA-256 deb36432d39ec4ee12319364e5c24260968239aef3cbab27b6fa4c8c9b8fda55.

The named 18-page PDF, 470,210 bytes, SHA-256 a426efb4046ff7bae61bd0f6e11dba51c8e2f0e90259052845fbf018403e99c4, was not in the corrective file and is not hosted. PROOFS.md is the complete text. It is not a byte-identical regeneration of that PDF.

This archive ran the restored checker on Python 3.10.21, normal and python3 -O. Both exited 0. The two stdout files match each other, SHA-256 715938ddf609ba08a3c62688d144beb2e38b2f2b8ab690a241b889446920b20a. Both stderr files are empty. Nineteen groups passed. The Decimal/DFT group is numerical. The other groups use exact arithmetic. One checker line is named “uniform-contraction observation collision”: that is an observation-rank check, not a claim that nonzero scaling fails to be injective.

The conversation record remains Python 3.12.14. Its stdout differs from this run only in that version line, so the digests differ. Those are different executions. Written arguments, this implementation check, formal verification, and independent review stay distinct. No proof assistant. No peer review. This does not add a counted source paper.

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.