Self-similarity, numerical difference, and exact recovery

The note fixes the center, frame, opening, and labels of the six-mark kernel. It takes $G(a)=a\varphi$ and $C(a)=a\pi$, and multiplies the scale by $G/(G+C)$. For the canonical $Q=339949771344778$ and $S=10^{15}$, it states five conclusions.

1. The geometric evolution is an exact scaling family, conjugate to $\Omega(v)=1+\lambda(v-1)$ on $v>1$.
2. The integer transition is a quantized rational homothety at every nonnegative integer, and a quantized $\lambda$-homothety on the stated floor-agreement domain.
3. The one-step and finite-horizon differences separate into approximation and quantization terms.
4. A valid finite trajectory and its terminal state plus ordered remainders are in bijection. A global identity alone does not establish that validity.
5. If the initial scale is $\delta d_0$, a validated record determines the ideal scales and affine coordinates as exact expressions.

The note says its proofs establish all five conclusions, and that the formula applies only to a validated record. That review has not been done here. The note is not an amendment of Paper VII.

The hosted checker is 16,815 bytes, SHA-256 e044133aceefb442bfa226bdf61359a58bca413e12196f090ff19eef3ef84ab5. The supplied Python 3.12.14 record names that hash. This archive reran the same bytes on Python 3.10.21. Both modes exited 0. The two logs are byte-identical, and they match the body of the supplied record. Matching pass lines supports execution agreement.

Passes 1 and 4 are symbolic. The gaps, seed radius, and Gram minor are identities in $\mathbb Q(\varphi)[s]$. The accumulated-difference checks use an indeterminate $X$, not a floating approximation to $\lambda$. Fifty-five seed and horizon cases were checked. The canonical sine is not evaluated. Positivity uses the note’s geometry assumptions.

Passes 2 and 3 are rational certificates: the Machin enclosure $\ell<\lambda<u$, the truncation $Q<S\lambda<Q+1$, and the first-disagreement denominator 42,189,065. The checker does not enumerate every integer below that threshold.

Pass 5 rejects a record whose global numerator is divisible by $Q^N$ while one backward step is $1/2$. Pass 6 checks fiber sizes, the ordered-remainder example, the bound $Q^{10}\cdot 32767<S^{10}$, and the split between the repeated path $26\to 8\to 2$ and $\lfloor 26\lambda^2\rfloor=3$.

Pass 7 is the finite audit: 32,768 states, horizon 23, 753,664 prefix invariants, and maximum extinction step 10. It is the same finite domain as the transition-invariant ledger audit. It does not extend the symbolic identities.

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.

Not a Lean certificate. Not an independent mathematical review. Not peer review. Not a physical experiment. Not a unification proof. Not an amendment of Paper VII. Exact algebra does not establish noise stability or a real calibration. Frozen G₆ is unchanged.

The later integration ledger is a different document and a different checker. The 15 October thesis is a later document and a later checker. Its written arguments have not been reviewed here either.