[petalas]

Summary

Scoped mathematical proof with an exact finite certificate:

S = {0,I,J,J+I,B,B+I}
J = [[0,1],[-1,0]], B = [[0,0],[-1,1]]
two binary cuts <= each edge's rational rank
all identity-shift demands cross both cuts
therefore s >= 2W, sharply; five roles need at least 10

Jose Manuel / jmbarrancoml's PR #1757 proposes the non-common-flag matrices J and B as a seed for a five-role nonlinear permutation circuit of cost at most nine. This new certificate excludes that target throughout the six-frame alphabet above, for every finite acyclic topology and number of deterministic gates satisfying the source's all-role endpoint contract. The source's weighted partition theorem is cited and applied; predecessor results are not re-audited. The search stopped at this first verified improvement.

The new proof and checker were produced by Nick Petalas. Credit PR #1757 for the nonlinear cut theorem, weighted criterion and seed question; Navinor / #825 for the earlier common-flag obstruction; CubeJerry / #907 for the rank-metric/cut-metric distinction; and the retained finite-shear, routing and cleanup lineage. Classical cut counting and rational rank retain their credit. Independent human and formal proof-kernel review are unclaimed.

Evidence

Before, the proposed seed escaped the common-flag premise: det(JB-BJ)=-2. After, its explicit two-cut certificate forces the sharp floor 2W without that premise. This is a new scoped obstruction, not a search failure or a complete multiplication theorem.

Local CPython 3.12.15 checks pass:

• all 36 ordered rank pairs by determinant and separate rational elimination;
• all three identity-shift pairs and the complete 4096 two-bit label assignments, with sixteen valid certificates;
• eleven certificate rejection controls, plus unexpected-argument and actual source-corruption rejection;
• normal and optimized warning-strict direct and registered replays, with identical generated hash-bound receipts;
• committed bytes/modes and declared source history, official export integrity, and normal/optimized read-only Git-free exported replays with empty PATH.

The two registered argv commands were timed directly at 0.022 and 0.026 seconds; the complete registered wrapper took 0.310 seconds. These are replay-budget measurements, not a performance claim. Only research/six-frame-seed-cut/ is added.

From the repository root, reproduce with the interpreter available in the shared environment:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/six-frame-seed-cut/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -O -W error -B research/six-frame-seed-cut/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/six-frame-seed-cut
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/six-frame-seed-cut --check-source-history

On another machine, substitute an absolute CPython 3.12 interpreter path. Expected markers are PASS_SIX_FRAME_SEED_CUT, PASS_REGISTERED_RESEARCH_PACKAGE and PASS_RESEARCH_COMMIT_BYTES. Source hashes, the historical card and the written proof are in the package. Base main is 19fdb086ea96af9a9f87936a2b888e10af4ed810, priority eight. Advance CLAIM.

Limitations and unrun obligations

Frames must belong to the six-element alphabet. All input symbols are independent; every required output is the specified role permutation, all physical edges including terminals are charged, every gate uses one common frame, and each demand has output-minus-input frame I. Additional frames, general rational combinations, changed endpoints, side channels and different input promises are not excluded. Sharpness is over the allowed circuit class, not every fixed topology or the nonroutable subclass.

The finite checker certifies the new matrix/cut certificate. The all-topology implication is the written application of PR #1757's cut theorem. No predecessor circuit census, Lean proof, native/physical supplier, complete implementation bill, uniform compiler, selected cold replay or complete multiplication theorem was run or supplied. No new kappa or selected-frontier change.

Open next questions

1. Add one rational frame and all its identity-shift demands. Determine whether a weighted partition certificate still excludes cost nine before searching circuit words. Losing these two particular cuts alone does not prove feasibility.
2. Produce a complete nonlinear permutation circuit outside this alphabet with s<2W, including the full truth table, DAG and every charged frame edge. Physical implementation and a complete supplier invoice remain separate.

No follow-up is reserved.

Related open work

The completion refresh examined all twelve other open account PRs created since the advance CLAIM: #5945, #5946, #5947, #5948, #5949, #5950, #5951, #5952, #5953, #5954, #5956 and #5957. None studies this six-frame seed or computes its minimum rank cost, and no conflicting shared number was identified. No additional computations were run to check these contributions.

• Prove sharp spectral conditioning for signed rank transport (no κ) #5952 sharpens a signed rank-potential increment in a finite-group formal register model. It does not compute this rational-frame circuit's rank-mass minimum.
• Prove a fourth dirty role keeps the S3 xyx double at six calls (no new κ) #5947 studies four dirty roles in an S3 group-algebra call alphabet. Its call-count minimum is a different quantity from this fixed six-frame cut obstruction.

The source PR #1757 and metric-scope PR #907 are merged predecessors. The earlier full open-PR search likewise found no other submission on this seed. Unpublished work and arbitrary code diffs remain outside overlap coverage.

CI notes

Both hosted checks passed for head 07d644c8c798e3d3fb676a1451b08b41afd54b02 in run 40530378398: plan in 9m38s and replay (research/six-frame-seed-cut) in 1m36s. The completed check list was read once, and the run head equals the PR head.

The package replay executes both registered normal and optimized commands. This certifies the bounded automated checks, not independent human or formal proof-kernel acceptance of the written all-circuit theorem. All local scientific, committed-byte/source-history and exported replay checks listed above also passed. No CI failure remains.

Merge danger

Door: two-way. Blast radius: one isolated opt-in research package. Existing shared code, predecessor evidence and the selected record are unchanged.