OMEGA NAUGHT supplement 1
15 October 2026

This directory is a supplementary release. It does not change the 15 October 2026 thesis file, that thesis’s publication records, or the frozen G₆ bytes.

Mathematical contract

1. Canonical affine readouts.
   s is the name sin(pi/1296000). It is not evaluated.
   The marks are z = (-s-phi^2, -s-1, -s, s, s+1, s+phi^2), labeled d2, d1, d, b, b1, b2.
   A field p(z) = alpha + beta z has alpha and beta in Q(phi).
   Each reading is A + B s, stored as four reduced rational coefficients of 1, phi, s, and phi s.
   From the outer readings, alpha = (A- + A+)/2 and beta = (B+ - B-)/2.
   The record is accepted only when recalculating all six readings agrees exactly.
   The checker proves those formulas as linear forms in the four rational coordinates of alpha and beta.
   {1, phi, s, phi s} is a basis of a four-dimensional rational vector space. The product s*s leaves that span.
   The integer factorization m = 1296000 = 2^7 * 3^4 * 5^3 and Euler totient 345600 are checked.
   The classical formula [Q(cos(2*pi/m)):Q] = phi(m)/2 = 172800, for m >= 3, is cited and is not reproved.

2. Rank-losing integer layers.
   y = floor((W x + b) / s0) and rho = W x + b - s0 y, with 0 <= rho < s0, including for negative numerators.
   A Smith factorization U W V = D is accepted only after the matrix identity and det U, det V in {+1,-1} are checked.
   SageMath was not called.
   If r = rank W, the witness k is the trailing part of V^{-1} x. Decoding uses (y, rho, k).
   For W = (1 1), b = 1, s0 = 5, the inputs (4, 7) and (5, 6) share y = 2 and rho = 2.
   For the factorization produced here, k equals the second coordinate. Omitting k leaves the collision.
   The serialized records are 77 bytes each. That length is not an optimality claim.

3. Certified transfer of the existing twelve-node rule at harmonic degree at most 2.
   Nodes are the unnormalized icosahedral nodes of squared radius phi+2. Unit-sphere coordinates are not in Q(phi).
   The basis is 1, x, y, z, xy, xz, yz, x^2-y^2, 2z^2-x^2-y^2.
   Discrete Gram matrices equal the continuous Gram matrix G. D = G^{-1} M^T W and T(target<-source) = M(target) D(source).
   The second grid is the rotation R(x,y,z) = ((3x-4y)/5, (4x+3y)/5, z). The third grid is R twice.
   These are the same resolution, not a higher frequency.
   Basis vectors recover, constants are preserved, the selected round trip equals the model projection, and T(R^2<-R) T(R<-I) = T(R^2<-I).
   Different subdivision frequencies are not enabled.

Record schemas

CANONICAL_AFFINE_READOUT/1
  sine: sin(pi/1296000)
  frame: normalized horizontal marks
  labels: d2, d1, d, b, b1, b2
  field: affine
  readings: six objects, each with four reduced rational strings

INTEGER_LAYER_WITNESS/1
  s0, y, rho, and k as canonical decimal strings
  A record without k does not determine the input.

GEOMETRY_LEDGER_RECORD/1.0 rejects padded extinction histories.
A fixed 23-step byte ledger retains padded steps.
Those schemas are not this readout and are rejected by its loader.
The 399 tool is unchanged.

Example records

canonical-readout.example.json is p(z) = 3 - 2z.
Its outer readings are 5 + 2 phi + 2 s and 1 - 2 phi - 2 s.
witness-collision.example.json holds the two witnesses k = 7 and k = 6.

Verification

From this directory:

  python3 verify_supplement_1.py
  python3 -O verify_supplement_1.py

Both should exit 0 and write the same JSON. The browser tools repeat the same exact arithmetic. They are not a separate certificate.

Scope

Proof scope: the written classical degree formula was not reviewed as a proof and was not reproved.
Exact certificates: the checker’s identities, the Smith identity, and the Gram and transfer identities.
Finite tests: the collision, four signed recoveries, the 3-2z readout, one fractional readout, and the arbitrary-array residual.
Formal execution: not run. This release does not generate Lean or SMT files.
Items 4, 5, and 6 of the implementation directive are not in this release.
Not peer review. Not independent reproduction. Not an amendment of Paper VII. Not a physical unification.

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. A hash identifies bytes; it does not certify their mathematical correctness.