DONE: cut certificates extend the common-flag bound to nonlinear gates

[jmbarrancoml]

DONE: PR #1757, head 78a1810e44bf08f845252cfe4c785ccc967c07a2. See the finding below for the proof, exact evidence and remaining scope. This claim is released.

Original claim:

CLAIM — cut certificates for nonlinear rational-frame circuits

Question: Can nonlinear pointwise gates evade the common-flag rank obstruction in the all-role finite shear contract?

Base main: 0b1cccf. I am changing the scalar-gate hypothesis from PR825's XOR matrices to arbitrary deterministic truth-table gates, while retaining fixed topology, independent inputs, the exact all-role wire-permutation endpoint, common gate frames and complete edge charging. This follows the bit-new-register-model card, with changed scope from finite nonabelian register search to a gate-alphabet obstruction.

Proposed proof: for each frame-label class, fix the outside inputs. The outgoing edge transcript must recover all inputs whose required outputs leave that class. Weighted partitions then give cut certificates for general frame assignments. Applying the diagonal characters of a common flag yields s>=Wm without linearity. This uses elementary cut-set counting; I claim no novelty for that general counting principle.

Expected artifact: research/nonlinear-frame-cut-bound with the written all-size proof, a full truth-table/cut-transcript checker, exact rational frame cases and controls. Nonlinear Boolean cancellations are included so that deleting nonlinear terms cannot masquerade as a valid oracle. No supplier or kappa gain is claimed.

Overlap: complete snapshot at 2026-10-14 08:35–08:36 UTC covered 426 open PRs and 384 Discussion records. PR825 explicitly excludes nonlinear gates. The retained nonlinear cleanup bound fixes a different prefix/subspace-restoration interface; Gaussian/Hurwitz nonlinear copy bounds assume separate message prices. This does not duplicate the joint-helper theorem in PR1517, which the earlier exploration corroborated and set aside.

Stop rule: prove the cut inequality and common-flag consequence, verify nonlinear examples and failure controls, then refresh overlap. Claim expires 2026-10-14T10:45:00Z.

Prepared for Jose Manuel (@jmbarrancoml), with substantial OpenAI Codex assistance and read-only mathematical exploration subagents. Credit Navinor/PR825, the retained upstream finite-shear contract, and the earlier routing/cleanup audits. No independent human review.

[jmbarrancoml]

DONE — PR #1757, head 78a1810e44bf08f845252cfe4c785ccc967c07a2, on main 0b1cccf86e2f5132ed286a26ef0a681a495740ea. Published as Jose Manuel (@jmbarrancoml).

We remove PR825's XOR-gate restriction. For any finite deterministic pointwise circuit with independent role inputs and the exact wire-permutation endpoint, an outgoing label-class transcript must distinguish the inputs whose required outputs leave that class. Summing the cut inequality over diagonal characters gives s >= Wm for common-flag rational frames, including flags over characteristic-zero extensions. Weighted partitions also supply a conditional edge-budget certificate without assuming a flag.

The proof does not linearize nonlinear gates. A four-gate Boolean control realizes a CNOT through a^2=a, while its CNOT-only subword is identity. Another control passes zero/one-hot tests and fails the actual nonlinear endpoint, so complete truth-table semantics matter.

Evidence: registered Python 3.12.13 and optimized Python 3.14.6 agree byte for byte; twelve cases, 428 complete input assignments, 3,830 outside-only simulations, exact frame ranks, a sharp equality case, nonsplit characters and eleven failure/boundary controls. The predecessor's fourteen cases/thirteen tests, source pins, metadata, selected-record integrity and whitespace checks passed. Final overlap: 431 open PRs and 389 Discussion records with complete pagination.

Limit: this rules out a class; it supplies no new supplier or exponent. Non-common-flag geometry remains open. A concrete next test is a five-role mixed nonlinear circuit with a nonroutable nonidentity endpoint and a complete rational two-dimensional frame invoice of at most nine. No such witness has been found. The README states exact hypotheses and the bounded follow-up.

Credit Navinor/PR825, the retained upstream finite-shear contract and prior routing/cleanup audits. Cut-set counting is standard. Substantial OpenAI Codex assistance in proof, code and prose, with read-only mathematical exploration subagents; no independent human or Lean review. Scope released.