[petalas]

Summary

Classify every equality case in PR5146's sharp binary character remainder bound, answering its Open next question 2.

Z(c) = 19*3^(k-3)
iff c has exactly two arrows sharing a head or tail,
    with every other vertex isolated.
Exact labelled multiplicity: k(k-1)(k-2), all with positive sign.

Claim type: scoped mathematical proof with an exhaustive finite equality lemma, for every k>=3 and every zero-diagonal binary character matrix with at least two entries. The multiplicity is the coefficient in the unnormalized signed character power sum; Fourier and finite-GL normalizations remain separate. No new kappa or selected-frontier change.

The three-vertex equality lemma forces every induced outside field and constant to vanish. Single-vertex and two-vertex outside assignments then exclude every extra arrow. This adds the complete equality classification to the predecessor's bound and attaining example. PR5146 is a cited premise, not an audit deliverable. No predecessor executable or spectrum table is imported.

This contribution was produced by Nick Petalas. Credit Nick Petalas / PR5146 for the explicit equality question and sharp bound, PR4275 for the character law and PR4599 for the four-event antecedent; Ryanson Jonathan / pinterbanget PR4607 for the five-event predecessor and structural-spectrum motivation; and Cinna Davis, source527, commdata2338 and Jose Manuel for the retained request and normalizer lineage. Conditioning and character orthogonality are classical.

Evidence

Before: one sharp family, with complete equality classification explicitly open. After: all-dimension necessity, an attaining construction, and exact labelled signed multiplicity.

Local CPython 3.12.15/macOS arm64 checks pass:

• Every one of the 3648 three-vertex graph/field configurations, with exactly six equality rows.
• 4140 character matrices at k=3,4; all 16384 extensions of a fixed attaining core by two outside vertices; 756 distinct labelled attaining examples at k=3..8. Literal and positive-subset evaluations agree on all 57 eligible three-vertex characters.
• Eight invalid-domain rejections, six semantic checks, 63 nonzero-field rejections, and three actual code mutations rejected normally and under optimization.
• Normal and optimized receipts are identical and bind all six package files. Direct commands took 0.240 s and 0.241 s; both registered checks have 120-second limits. Metadata/pins, committed-byte/source-history preflight, official export/input integrity, and read-only Git-free normal/optimized exported replays pass. All seven exported files remain byte-identical.

From the worktree root, reproduce with these exact commands; another CPython 3.12 path can replace the local interpreter:

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

Expected markers are PASS_BINARY_CHARACTER_EXTREMIZERS, PASS_REGISTERED_RESEARCH_PACKAGE and PASS_RESEARCH_COMMIT_BYTES. The current selected-baseline orientation also passes, with metadata-only scope.

Limitations and unrun obligations

Zero diagonal, F2 and the uniform three-state sum are essential. The universal proof uses the finite equality lemma and PR5146's published reciprocal/matching bounds. Finite corroboration does not enumerate every dimension. No fourth-mode bound, full all-k probability inequality, negative-association extension, odd-field/weighted analogue, request producer, sampling/emission, native implementation, dirty restoration or complete physical invoice is supplied. All five selected framework assumptions remain. Selected cold replay, native builds, Lean and Docker were not run. Independent human and formal proof-kernel review are unclaimed.

Open next questions

• Determine the sharp fourth mode and its complete equality family, starting with the next conditioned local extrema and outside-field constraints. This could sharpen probability remainders; neither the value nor a global fourth-mode gap follows here.
• Bind one actual request producer and shared-header consumer to a complete bill for selection, emission, native alignment and restoration. Those physical costs are unknown.

Related open work

• PR6415 optimizes a complete literal full-guard header census. This contribution classifies character-sum equality, with no shared benchmark or new header-count comparison.
• PR6409 determines adaptive corank-one pivot-swap cost. The fixed uniform-header character sum here counts a different quantity and changes no pivot policy.

No open PR classifying this character-sum equality was found in the refreshed main, open/merged searches, claims and reservations. PR5146 and PR4607 are merged predecessors. PR5860's five-indicator probability theorem is separate and is not generalized. Sparse notes, unpublished work and arbitrary historical diffs remain overlap-coverage limits.

The completion refresh checked the twelve other account PRs still open and created after the advance CLAIM. They study different objects or quantities; none supplies a competing classification or common numerical instance for this character sum. No additional computations audited another contribution.

CI notes

Both hosted checks passed at exact head f41c74eb9c628c4998b8f7ed9813a7585d4bb301 in run 41543826856: plan in 3m52s and replay (research/binary-character-extremizers) in 1m17s. The completed checks were read once, and run metadata confirms completed success and the matching head. CI certifies execution of the registered finite checks and source bindings, not independent human or formal proof-kernel acceptance. All registered scientific obligations also ran locally; none was deferred to CI.

Merge Danger

Door: two-way.

Blast Radius: package. Only the new six-file research package is added. Scientific admission and selected-frontier changes remain separate decisions.