Classify all extremizers of the third binary character mode (no κ)

research/binary-character-extremizers/PROOF.md

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+# Exact equality in the third binary character mode
+
+This contribution was produced by Nick Petalas. Apache-2.0.
+
+## Contract and result
+
+Let k >= 3 and let c be a zero-diagonal k-by-k matrix over F2, with at
+least two nonzero entries. Interpret c_ij=1 as an arrow i -> j. Define
+
+```text
+Z(c) = sum over s in {0,1,2}^k of
+       (-1)^(sum over i,j of c_ij [s_i=1][s_j=2]).
+```
+
+Nick Petalas's [PR5146](https://github.com/CrocSwap/integer-mult-bounds/pull/5146)
+proves 0 < Z(c) <= 19*3^(k-3). Its second open question asks which graphs
+attain the upper bound. The following is the new classification.
+
+**Theorem.** Equality holds if and only if c has exactly two arrows, and
+they have a common tail or a common head. Every other vertex is isolated.
+There are exactly k(k-1)(k-2) labelled equality matrices. Each has even
+entry weight, hence positive sign in the signed character expansion.
+
+The proof uses an exhaustive new local equality lemma. It does not infer
+the universal statement by testing a finite list of dimensions.
+
+## The local equality lemma
+
+For a zero-diagonal three-by-three binary matrix a with at least two
+arrows and a binary field f=(u_0,v_0,u_1,v_1,u_2,v_2), put
+
+```text
+L(a,f) = sum over s in {0,1,2}^3 of
+         (-1)^(sum a_ij [s_i=1][s_j=2]
+                  + sum_i (u_i[s_i=1]+v_i[s_i=2])).
+```
+
+**Finite lemma.** |L(a,f)| <= 19. Equality in absolute value holds exactly
+when a consists of two arrows with a common head or tail and f=0.
+In those six cases L(a,0)=+19. There are no cases with L(a,f)=-19.
+
+`verify.py:local_equality` proves this finite lemma by enumerating every
+one of the (64-1-6)*64=3648 graph/field configurations and all 27 states
+for each. The graph bits, in order, are
+
+```text
+(0,1), (0,2), (1,0), (1,2), (2,0), (2,1).
+```
+
+The complete equality rows are below. The field mask is zero in each row.
+
+| Graph mask | Arrows | L |
+|---:|---|---:|
+| 3 | 0->1, 0->2 | 19 |
+| 10 | 0->2, 1->2 | 19 |
+| 12 | 1->0, 1->2 | 19 |
+| 20 | 1->0, 2->0 | 19 |
+| 33 | 0->1, 2->1 | 19 |
+| 48 | 2->0, 2->1 | 19 |
+
+The checker derives its expected equality condition from endpoint
+incidence, rather than using this table to compute the sums. It retains
+the complete table in the generated receipt. Every test uses explicit
+exceptions and also runs under optimized Python.
+
+## Reducing global equality to three vertices
+
+We use PR5146's reciprocal-pair and directed-matching bounds as cited
+premises. A reciprocal pair gives Z(c) <= 5*3^(k-2), strictly below the
+target. If the arrows form a directed matching with r >= 2 arrows, then
+
+```text
+Z(c)/3^k = (7/9)^r <= 49/81 < 19/27.
+```
+
+Otherwise some two arrows share a vertex. Their endpoints form a set H
+of three vertices with at least two induced arrows. For each fixed
+outside assignment t, separate the character exponent into the induced
+quadratic part on H, its induced unary field f(t), and its outside
+constant q(t). Thus
+
+```text
+Z(c) = sum over t in {0,1,2}^(k-3) of (-1)^q(t) L(c_H,f(t)).
+```
+
+The finite lemma bounds each real summand above by 19. If the sum equals
+19*3^(k-3), every summand must equal +19. At t=0, the field and constant
+are both zero. The equality lemma forces c_H to be a two-arrow star.
+For every other t it then forces f(t)=0 and q(t)=0, since its only
+absolute equality value is L(c_H,0)=+19.
+
+These requirements remove every possible outside arrow. For an outside
+vertex v, set all other outside states to zero. The choice t_v=1 gives
+
+```text
+coefficient of [s_i=2] in f(t) = c_vi for each i in H.
+```
+
+The choice t_v=2 gives the coefficients c_iv of [s_i=1]. Thus f(t)=0
+for these two assignments implies that no arrow joins v to H, in either
+direction. This holds for each outside vertex.
+
+If an arrow v->w existed entirely outside H, assign t_v=1, t_w=2 and
+all other outside states zero. The only possible active outside term is
+c_vw=1, even if the reverse arrow exists. Hence q(t)=1, contradicting
+q(t)=0. Therefore the outside graph is empty. This proves necessity,
+including k=3 where the outside set is empty.
+
+## Attainment, count and signed multiplicity
+
+For a common-tail star, condition on its center. States 0 and 2 each
+contribute 9; state 1 contributes (1+1-1)^2=1. The sum is 19. Swapping
+states 1 and 2 at every vertex transposes the graph and gives the same
+value for a common-head star. Every isolated vertex contributes a factor
+3, proving sufficiency.
+
+Choose the center, an unordered pair of other vertices, and one of the
+two orientations. The center and orientation are unique in every such
+graph, so the exact labelled count is
+
+```text
+2*k*binomial(k-1,2) = k(k-1)(k-2).
+```
+
+Each matrix has exactly two entries. In PR4275's signed character law
+its sign (-1)^|c| is therefore +1. Write its unnormalized power sum as
+
+```text
+S_n = sum over zero-diagonal c of (-1)^|c| Z(c)^n.
+```
+
+The coefficient of (19*3^(k-3))^n in S_n is exactly k(k-1)(k-2).
+The walk count has the additional Fourier factor 2^(-k(k-1)); header
+probabilities have their further finite-GL denominator. None of these
+normalizations is included in the stated spectral multiplicity. This
+is a direct corollary of the classification, not a calculation of the
+rest of the spectrum or a new all-dimension probability inequality.
+
+## Dependencies and limits
+
+PR5146 supplies the character interpretation, sharp bound and explicit
+equality question. The cited reciprocal and matching bounds are used as
+published results, without auditing their predecessors. PR4275 supplies
+the signed character law for the multiplicity corollary. Ryanson Jonathan
+and PR4607 supply the five-event spectral context. The retained request
+and normalizer lineage is credited in README.md.
+
+Zero diagonal, F2 and the exact uniform three-state sum are essential.
+The proof supplies no odd-field or weighted analogue, fourth-mode bound,
+complete negative-association theorem, header producer, sampling/emission
+procedure, native implementation, dirty restoration, complete physical
+invoice, new kappa or complete multiplication theorem. The five selected
+framework assumptions remain. Independent human and formal proof-kernel
+review are unclaimed.

research/binary-character-extremizers/README.md

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+# Exact extremizers of the third binary character mode
+
+This contribution was produced by Nick Petalas. Apache-2.0.
+
+Claim type: scoped mathematical proof with an exhaustive finite equality
+lemma. For every k >= 3, the zero-diagonal binary character matrices with
+at least two entries attaining Z(c)=19*3^(k-3) are exactly two arrows
+with a common head or tail, plus isolated vertices. Their exact labelled
+count is k(k-1)(k-2). Every one has positive sign in the unnormalized
+signed full-guard character sum, so this is its exact third-mode
+coefficient. The Fourier and finite-GL normalization factors are separate.
+
+This answers Open next question 2 of
+[PR5146](https://github.com/CrocSwap/integer-mult-bounds/pull/5146).
+That predecessor proves the sharp bound and supplies one attaining
+family. The new result is the complete equality classification and exact
+multiplicity. Its published bound is a cited premise. No predecessor
+program, census or spectrum table is imported or audited.
+
+[PROOF.md](PROOF.md) gives the universal argument. A three-vertex equality
+lemma forces every outside-induced unary field and constant to vanish.
+One-vertex and two-vertex outside assignments then rule out every extra
+arrow. Labelled stars give the matching construction and count. No new
+kappa or selected-frontier change is claimed.
+
+## Reproduce
+
+From the repository root, using the Python 3.12 standard library:
+
+```sh
+/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
+```
+
+Any CPython 3.12 interpreter can replace the displayed local path. The
+direct commands write nothing and emit `PASS_BINARY_CHARACTER_EXTREMIZERS`
+inside a newly generated JSON receipt binding all six package files by
+SHA256. Neither a receipt nor mathematical output is hand-written.
+Each registered command has a 120-second timeout. The standalone checker
+requires neither Git nor predecessor files.
+
+## Evidence
+
+The checker enumerates every three-vertex graph with at least two arrows
+and every unary field. Its literal ternary-state sum establishes the new
+local equality lemma. A separate positive-subset sum checks global
+classification at small dimensions and every extension of a fixed
+attaining core by two outside vertices. It also constructs all labelled
+stars for k=3 through 8 and checks their distinctness, values and count.
+The generated receipt records exact executed counts and the six complete
+local equality rows. Invalid domains and false equality rules reject,
+including a directed path, reciprocal arrows, a three-arrow star,
+nonzero unary fields and an outside edge with a changing constant sign.
+
+Executed coverage is 3648 local graph/field cases, 4140 character matrices
+at k=3,4, all 16384 two-outside-vertex extensions of one fixed core, and
+756 distinct labelled attaining examples at k=3..8. The local sums and
+the positive-subset oracle agree on all 57 eligible k=3 characters.
+There are eight invalid-domain rejections, six semantic checks and 63
+nonzero-field rejections. The local equality table has exactly six rows.
+Three actual code mutations, omitting fields, changing star incidence and
+ignoring outside edges, reject in both normal and optimized local runs.
+
+The finite checks establish the local finite lemma and corroborate the
+universal argument. They are not a census at every dimension. Normal and
+optimized modes run the same explicit checks. Source pins are historical
+provenance only, with no external executable inputs. The current baseline
+orientation reports `PASS_RESEARCH_BASELINE_BINDINGS_ONLY` for selected
+PR6030; that is metadata orientation rather than a scientific replay.
+
+## Scope and limits
+
+The independent-moment card is explicitly changed to this original
+spectral side question. Binary arithmetic, zero diagonal and the uniform
+three-state weights are essential. The result does not give a fourth
+spectral mode, a complete all-k probability inequality, negative
+association, an odd-field or weighted analogue, a request producer,
+sampling/emission procedure, native realization, dirty restoration or
+complete physical invoice. All five selected framework assumptions remain.
+No complete multiplication theorem is claimed. Selected cold replay,
+Lean, Docker and native builds were not run. Independent human and formal
+proof-kernel review are unclaimed.
+
+All verifier loops have explicit finite input bounds, with no recursion,
+subprocesses or dynamic execution. Matrices have at most twelve rows;
+literal sums allow at most six, and executed local sums use three.
+The extension loop has exactly 16384 cases and handles one matrix at a
+time. Each function fits within sixty physical lines. Explicit exceptions
+retain contracts under optimization; registered commands treat warnings
+as errors. Python and standard-library runtime dispatch and allocation
+are managed-runtime deviations from fixed allocation and static calls.
+The declared dimension/file bounds and one-at-a-time enumeration bound
+live work. No real-time or fixed-memory runtime guarantee is claimed.
+A larger census must revise and verify the capacities first. Syntax and
+function-length inspection pass. External Python analyzers were unavailable;
+warning-strict execution is the recurring registered diagnostic.
+
+## Open next questions
+
+1. What is the sharp fourth character value and its complete equality
+   family? Begin with the next local conditioned extrema and outside-field
+   rigidity. This proof supplies neither that value nor a fourth-mode gap.
+   The answer could sharpen full-guard moment remainders.
+2. Can an actual request producer exploit a shared header at a favorable
+   complete price? Start with one source-bound producer/consumer and
+   account for sampling, selection, emission, native alignment and dirty
+   restoration. Those physical costs are unknown here.
+
+[TASK.json](TASK.json) records structured relevance, first steps and blockers.
+Research stops at this first verified improvement; no follow-on is reserved.
+
+## Credit and coordination
+
+Nick Petalas / PR5146 supplies the sharp remainder bound and explicit
+equality question, and PR4275 supplies the signed character law. Ryanson
+Jonathan / pinterbanget PR4607 supplies the five-event predecessor and
+structural-spectrum motivation. Nick Petalas / PR4599 supplies the
+four-event antecedent. Cinna Davis, source527, commdata2338 and Jose Manuel
+retain the request and normalizer lineage credited by those sources.
+Conditioning, finite character sums and the triangle inequality are
+classical. The scoped classification and its checker are the new work.
+Frozen sources and predecessor attribution are unchanged.
+
+Base: canonical main `7c49fe32c55de02c2c873655c804b16e4c3fef0a`, priority
+revision 8, selected PR6030. The
+[advance claim](https://github.com/CrocSwap/integer-mult-bounds/discussions/1166#discussioncomment-19362336)
+records the target. Live claims, local reservations, current main README
+searches, merged/open PR searches and recent account work were checked.
+The cached overlap snapshot has 387 open PRs and 6319 Discussion records
+with complete remote pagination, but searches only 114 of 8917 local
+notes. Separate Git searches cover main README terms. Unpublished work,
+sparse notes and arbitrary historical diffs remain coverage limits.
+PR5860's five-indicator probability comparisons and PR6415's literal
+header-census optimization are separate quantities and are not extended.

research/binary-character-extremizers/SOURCES.json

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