Classify every optimal width-two odd-cycle fitting matrix (no new κ)

research/odd-cycle-fit-rigidity/PROOF.md

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+# Rigidity of minimum-rank width-two odd-cycle fits
+
+produced by Nick Petalas. Apache-2.0.
+
+## Statement
+
+Let F be any field, n=2k+1 with k>=2, and let M be a 2n by 2n matrix
+partitioned into two by two blocks indexed by Z/n. Require
+
+    M_ii = I_2,
+    M_ij = 0 when i and j are distinct and nonadjacent on C_n.
+
+The two ordered blocks on each edge are arbitrary. They need not be
+symmetric or commute. JoyboyBrian's PR5023 gives the minimum rank n for this
+contract, as a special case of its unequal-width theorem. We use that rank
+value as a cited predecessor result.
+
+Define D on the two ports (i,left) and (i,right) of every vertex. Its diagonal
+blocks are I_2. Its only off-diagonal nonzero entries are
+
+    D[(i,right),(i+1,left)] = D[(i+1,left),(i,right)] = 1.
+
+**Theorem.** If rank(M)=n, there are invertible two by two matrices L_i over
+F such that, for L=diag(L_0,...,L_(n-1)),
+
+    M = L D L^-1.
+
+Conversely every such matrix fits the contract and has rank n. Thus all
+minimum-rank fits form one orbit under independent changes of basis at the
+vertices. This classifies the actual fitting matrices, not just the rank value
+or one attaining construction. Ordinary transpose, never conjugate transpose,
+is used in the argument.
+
+## Adjacent intersection lines
+
+Factor M=A B^T through F^n, with A and B of size 2n by n and full column
+rank. Let U_i and W_i be the row spaces of their two-row blocks A_i and B_i.
+The identity A_i B_i^T=I_2 makes both spaces two-dimensional and their mutual
+coordinate-bilinear pairing perfect. Forbidden blocks make U_i orthogonal to
+W_j whenever i and j are distinct nonneighbors.
+
+For nonadjacent i,j, U_i intersect U_j is zero. A vector in the intersection
+annihilates W_i by its membership in U_j, and perfect pairing on U_i forces
+it to vanish. The same statement holds for the W spaces.
+
+Fix an edge {i,i+1}. The k-1 vertices
+
+    J = {i+3,i+5,...,i+2k-1}, with indices modulo n,
+
+are independent and are nonadjacent to both endpoints. Their W_j form a direct
+sum of dimension 2(k-1). Indeed, pairing a relation with U_j kills every other
+summand and then perfect pairing kills its own summand. Both U_i and U_(i+1)
+annihilate this direct sum. Therefore
+
+    dim(U_i + U_(i+1)) <= n - 2(k-1) = 3,
+    dim(U_i intersect U_(i+1)) >= 1.
+
+Inside U_i the two intersections with its neighbors are disjoint, because
+i-1 and i+1 are nonadjacent. Both have dimension at least one and U_i has
+dimension two. Consequently, with H_i=U_i intersect U_(i+1),
+
+    dim H_i = 1,
+    U_i = H_(i-1) direct-sum H_i.                         (1)
+
+This extends the C5 equality argument of Nick Petalas's PR3248. The independent
+set J is the additional step for every longer odd cycle. Its use is also
+consistent with the rotated one-edge-plus-isolates inequalities of PR5023.
+
+## A dual edge basis and local similarity
+
+Choose nonzero h_i in H_i. The row h_i annihilates W_j unless j is i or i+1.
+For every other vertex j, at least one of those two endpoints is a nonneighbor
+of j, which proves this assertion from the ordered zero blocks.
+
+The n rows h_i are independent. In a relation sum c_i h_i=0, pairing with W_j
+leaves only c_(j-1)h_(j-1)+c_j h_j. This vector lies in U_j and annihilates
+W_j, so perfect pairing makes it zero. Equation (1) then makes both
+coefficients zero. This holds for every j.
+
+Let g_i be the dual row basis, h_i g_j^T=delta_ij. Each W_j annihilates all
+h_i except its two incident edges, and has dimension two. Hence
+
+    W_j = span(g_(j-1),g_j).
+
+For invertible matrices L_i and T_i we can therefore write
+
+    A_i = L_i [h_(i-1); h_i],
+    B_i = T_i [g_(i-1); g_i].
+
+The diagonal identity is L_i T_i^T=I_2, so T_i^T=L_i^-1. Pairing the displayed
+dual edge bases gives exactly the block D_ij. Thus
+
+    M_ij = L_i D_ij L_j^-1,
+
+which proves the claimed block-diagonal similarity over every field.
+
+## Spectral and local consequences
+
+A permutation of ports writes D as n disjoint copies of the all-ones two by
+two matrix J_2. Each copy has rank one over every field and J_2^2=2J_2.
+This proves the converse, and also proves
+
+    M^2 = 2 M,
+    characteristic polynomial = x^n (x-2)^n.
+
+If char(F) is not two, M is diagonalizable with eigenvalues zero and two, each
+of multiplicity n. If char(F)=2, M is square-zero with rank n, so its Jordan
+form consists of exactly n nilpotent blocks of size two. No eigenvalue or
+Jordan claim requires F to be algebraically closed because these forms split
+over F itself.
+
+Every ordered edge block has rank one. At each vertex the two return maps
+
+    P_i^- = M_(i,i-1) M_(i-1,i),
+    P_i^+ = M_(i,i+1) M_(i+1,i)
+
+are complementary rank-one idempotents. Products along two distinct consecutive
+edges vanish, in either orientation. These are necessary structural conditions
+on a proposed optimal fit. They do not impose symmetry in the original bases.
+
+The normal form is constructive. Choose a nonzero vector ell_i in im(P_i^-),
+and put r_i=M_(i,i+1) ell_(i+1). Then L_i=[ell_i,r_i] is invertible and gives
+the displayed similarity. The canonical form proves this assertion directly:
+ell_i is a nonzero multiple of its left basis vector, and r_i has exactly the
+matching scale at the other end of edge i. Each edge can be normalized
+independently; no cyclic scalar consistency condition remains.
+
+## Hypotheses and limits
+
+Odd length is essential for this rigidity. On any even cycle of length at
+least six, put an I_2 block on each ordered edge of a perfect matching, zero
+on all other edges and I_2 on the diagonal. Its rank is n, but its ordered
+edge ranks alternate between two and zero. Local similarity preserves these
+ranks, so it cannot be equivalent to D. The independent-set lower bound makes
+this a minimum-rank even-cycle example.
+
+The rank hypothesis is also essential. Replace one shared pair J_2 in D by
+I_2. The odd-cycle mask still fits, its rank is n+1, and its nonzero identity
+summands violate M^2=2M in every field. This is a boundary example, not an
+error in the predecessor minimum-rank theorem.
+
+The proof does not classify unequal widths, higher-rank fits or other masks.
+It gives no physical basis-change routine, complete-plane binary pairing
+tariff, native caller, dirty-state restoration or recursive bill. It changes
+neither kappa nor the selected frontier. Finite verification corroborates the
+written all-field argument; independent human and proof-kernel review remain
+unclaimed.

research/odd-cycle-fit-rigidity/README.md

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+# Every optimal width-two odd-cycle fit has one local normal form
+
+For every field and every odd n>=5, every rank-n fitting matrix on the cycle
+C_n with identity two by two diagonal blocks is block-diagonally similar to
+the canonical one-coordinate-per-edge construction. Thus every optimum obeys
+`M^2=2M`. In characteristic two its Jordan form is exactly n nilpotent blocks
+of size two. In other characteristics it is diagonalizable with eigenvalues
+zero and two, each n times.
+
+Claim type: scoped mathematical proof with exact finite corroboration. This
+contribution was produced by Nick Petalas. Independent human and formal
+proof-kernel review are unclaimed.
+
+JoyboyBrian's [PR5023](https://github.com/CrocSwap/integer-mult-bounds/pull/5023)
+already proves the rank value n as part of its unequal-payload theorem. This
+package classifies **all attaining matrices**. It uses that rank value as a
+cited premise and does not replay the predecessor's capacity computations.
+The [proof](PROOF.md) extends Nick Petalas's older
+[PR3248](https://github.com/CrocSwap/integer-mult-bounds/pull/3248) C5 edge-line
+argument to all longer odd cycles, then obtains the local similarity and the
+spectral identity from a dual edge basis. It permits arbitrary directed edge
+blocks over any field, without symmetry, positivity or commuting-block
+assumptions.
+
+## Concrete contract and consequence
+
+Write the two ports at vertex i as left and right. The canonical matrix D
+has diagonal I_2 and one coefficient in each direction between right at i
+and left at i+1. All other off-diagonal coefficients are zero. The theorem is
+
+    rank(M)=n  implies  M=L D L^-1,  L=diag(L_0,...,L_(n-1)), L_i invertible.
+
+A port permutation makes D a direct sum of n all-ones two by two matrices.
+Every ordered edge block of an optimum has rank one; the two edge return maps
+at each vertex are complementary rank-one projectors, and every directed
+product along two distinct consecutive edges vanishes. These conditions can
+reject a proposed optimum before a decoder construction is priced.
+
+The width-two and odd-length hypotheses remain explicit. Even-cycle optima
+with alternating rank-two and zero edge blocks defeat the normal form.
+Rank-(n+1) odd-cycle fits need not obey the quadratic identity. Both boundaries
+are executable controls.
+
+## Reproduction and exact evidence
+
+From the repository root, use CPython 3.12 with its standard library:
+
+```sh
+/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/odd-cycle-fit-rigidity/verify.py
+/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -O -W error -B research/odd-cycle-fit-rigidity/verify.py
+/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/odd-cycle-fit-rigidity
+/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/odd-cycle-fit-rigidity --check-source-history
+```
+
+Any CPython 3.12 interpreter can replace that local path. Each direct verifier
+prints a hash-bound evidence summary and `PASS_ODD_CYCLE_FIT_RIGIDITY`, and
+writes no files. [RESULT.json](RESULT.json) is generated by the verifier and
+binds every package source and declared predecessor input. The separate
+`--write-result PATH` option writes a fresh receipt outside the package.
+
+The bounded replay checks:
+
+- 200 exact fixtures over Q, GF(2), GF(3), GF(5) and GF(101), at cycle lengths
+  5, 7, 9, 11 and 13, including rational nonintegral basis changes.
+- All 7,776 local GL2(GF(2)) basis choices on C5. Their matrices are distinct;
+  direct reconstruction finds the canonical form and independent column-bit
+  elimination checks their rank.
+- Every binary assignment to two ordered edge blocks of the canonical C5 or
+  C7, for every choice of the two slots. Among 11,520 assignments on C5,
+  55 have rank five; among 23,296 assignments on C7, 105 have rank seven.
+  Every such optimum passes reconstruction and the local/spectral identities.
+  The counts enumerate assignments, including repeated matrices. They are
+  exhaustive only for this specified two-block completion domain.
+- 22 rejection controls, covering rank slack in five fields, even-cycle
+  boundaries, forbidden and nonidentity blocks, malformed coefficients,
+  matrix capacities and an unsupported modulus.
+
+The all-field, all-odd-length result rests on the written proof. The finite
+checks corroborate it and exercise the constructive normalizer; they do not
+form an exhaustive census of all fitting matrices.
+
+## Scope, credit and family decision
+
+This changes the `complex-rank-shape-decoder` card to a structural side result
+on an exact read-mask interface. It adds no new query-capacity value, physical
+decoder tariff or kappa. It does not extend or recheck the recent bipartite
+capacity computation. The selected frontier is unchanged. Canonical base is
+`a5df48c55ea806d7952b65acc9c76d878ff09b13`, priority revision 7.
+The [advance claim](https://github.com/CrocSwap/integer-mult-bounds/discussions/1166#discussioncomment-19144836)
+records the target. Initial cached overlap covered 331 open PRs and 4,230
+Discussion records; 6,756 of 6,918 tracked notes were omitted by the sparse
+local search. Relevant predecessor notes and main-tree/merged searches were
+read separately. This coverage does not establish novelty for unpublished work.
+
+Credit JoyboyBrian #5023 for the all-field unequal-cycle rank theorem and the
+rotated subspace inequalities; Nick Petalas #3248 for the C5 equality/dual-basis
+argument; Tekalign #3362 and Sagar Thakrar #2484/#3232 for the cycle-mask and
+decoder lineage. Their existing notices and frozen sources are unchanged.
+New proof, constructive normalizer and finite checks are produced by Nick
+Petalas. Apache-2.0. No literature priority is asserted for standard matrix
+factorization, duality, change of basis or Jordan-form facts.
+
+Family decision: finish this classification and stop. Physical acquisition,
+copies, numerical conditioning of the local bases, dirty restoration, binary
+complete-plane tariffs, ownership, hosting, preparation, stock/precision,
+tapes, compiler and outer recovery remain unpriced. A linear similarity is
+not a free physical operation. No supplier or complete multiplication theorem
+is supplied.
+
+## Open next questions
+
+Can an explicit minimum-rank unequal-width odd-cycle fit fail to be locally
+equivalent to an integral edge/private-channel construction? Start with the
+smallest width vectors for which the half-total bound has slack, inspect the
+neighbor-intersection dimensions, and compare local similarity invariants.
+The present dimension-two equality argument does not classify those cases.
+Do not assume that finding a normal form pays for its physical realization.
+
+## Verification implementation limits
+
+The executable is intentionally bounded to matrices of at most 26 by 26 and
+the five stated coefficient fields; the proof has no such finite restriction.
+Input coefficients have numerator and denominator bit-length at most 256.
+Python exact arithmetic allocates dynamically, a power-of-ten rule 3
+deviation. Live matrices have fixed bounded dimensions; the only retained
+census set has 7,776 encodings of 100 bytes. The three source inputs are
+fixed hash-bound documents. No recursion, solver, subprocess or background
+work is used. Negative-control callbacks have three fixed call targets.
+Normal and optimized registered runs retain every check.
+
+The isolated script adds only its fixed package directory before importing
+its sibling module. The single E402 annotation documents that required import
+order; the remaining selected Ruff diagnostics are enabled without exclusions.

research/odd-cycle-fit-rigidity/RESULT.json

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+{
+  "claim_type": "scoped proof with exact finite corroboration",
+  "evidence": {
+    "binary_gauge_matrices": 7776,
+    "completion_digest": "a8915e04186cc3644afe59f54f428acf5e6ee8627567beae6d8ec651b9604097",
+    "field_fixtures": 200,
+    "negative_controls": 22,
+    "two_block_completions": [
+      {
+        "minimum_rank": 55,
+        "n": 5,
+        "tested": 11520
+      },
+      {
+        "minimum_rank": 105,
+        "n": 7,
+        "tested": 23296
+      }
+    ]
+  },
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research/odd-cycle-fit-rigidity/SOURCES.json

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