Transition invariants

The hosted checker is 12,112 bytes, SHA-256 e94400b6e136bed009388c5d17e4f1c698235b7b938f75c266bbaf363a15a0ca. The supplied Python 3.12.14 record names that hash, so the record and the hosted source are the same bytes. This archive executed those bytes on Python 3.10.21. Both modes exited 0, and the pass lines agree. The two logs differ only in the printed optimization mode. Matching pass lines supports execution agreement.

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 symbolic checks use free variables. A zero normal form is an identity under the stated assumptions, not a check of one numerical example. They include the five gaps $a(\varphi,1,2s,1,\varphi)$, the selected-path ratio $G/(G+C)=\varphi/(\varphi+\pi)$, the projective cross-ratio and its reference specialization, recovery of the six coefficients after a change of reference length, both characteristic invariants for arbitrary degree-at-most-five data, the one-step identity $Qd=Sy+\rho$ with its induction step, and the dimensionless ratio $\chi/m^2$ for one solution written at two lengths. $G$ and $C$ are the declared paths, not the whole outline. The ratio check does not derive that solution from the marked kernel.

The ledger audit is separate. It covers seeds 0 through 32767 and horizons 1 through 23: 753,664 prefix invariants. In that finite domain every seed reaches 0, and the longest time to 0 is step 10. Seeds 32767 and 32768 share the quotient 11139 and do not share the residual. That pair is a collision witness inside the audit, not a symbolic identity.

The drawings on the geometry page, the 399 floor page, and the sampling page illustrate these checks. The drawings are not extra theorems. The 13 October note uses the same finite ledger domain and states a broader recovery theorem. Its written arguments have not been reviewed here.

Pass 8 requires five quantities to change. A raw length changes under rescaling. The reference cross-ratio changes when $s$ changes from 1/2 to 2/3. A fixed-position wave reading changes with time. The two residuals above differ. A degree-six field that vanishes on the six nodes $h z_i$ at $s=1/2$ and $h=1/4$ does not stay zero at a later reading.

Those failures are part of the declared check. They are not defects in the run.

Not a Lean certificate. Not an independent mathematical review. Not a physical experiment. Not a unification proof. Exact algebra does not establish noise stability or a real calibration. The canonical one-arcsecond sine is kept symbolic. The finite ledger audit does not extend past seeds 0 through 32767 or horizons 1 through 23. Frozen G₆ is unchanged.