[petalas]

Summary

Scoped mathematical proof with exact finite corroboration: every minimum-rank fitting matrix on an odd cycle of length n>=5 with two coordinates per vertex is block-diagonally similar to the canonical edge-sharing matrix.

identity 2x2 diagonal + ordered cycle zero mask + rank n
    -> local similarity to n disjoint all-ones 2x2 blocks
    -> M^2 = 2M

In characteristic two, the Jordan form consists of n nilpotent blocks of size two. Otherwise the eigenvalues are zero and two, each n times. Every directed edge block has rank one, with complementary return projectors and vanishing products along two distinct consecutive edges.

JoyboyBrian's #5023 supplies the rank optimum. This contribution classifies all attaining matrices; it does not recheck that optimum. The proof extends the C5 equality argument of Nick Petalas's older #3248. Credit Tekalign #3362, Sagar Thakrar #2484/#3232 and the retained cycle-mask/native lineage. The contribution is produced by Nick Petalas. Independent human and formal proof-kernel review are unclaimed.

Evidence

Before: the longer-cycle rank value and an attaining edge/private-channel construction were available. After: all width-two optima lie in one local-similarity orbit over every field, with an explicit basis-recovery algorithm. Odd length and minimum rank have executable boundary counterexamples.

Local CPython 3.12.15, macOS arm64, standard library only:

• Direct normal and optimized commands pass in 4.515 and 4.751 seconds. Registered normal/optimized replay and metadata/pin checks pass.
• 200 exact fixtures over Q and GF(2), GF(3), GF(5), GF(101), at lengths 5,7,9,11,13; all 7,776 distinct C5 binary local-basis matrices; 22 rejection controls.
• All 11,520 C5 and 23,296 C7 two-ordered-block binary assignments are enumerated, including repeated matrices. Their 55 and 105 rank-optimal assignments pass reconstruction and the spectral/local identities. This is exhaustive only for that explicitly bounded completion domain.
• Committed-byte/source-history preflight passes for all 11 package/input files. Official export integrity and read-only Git-free normal/optimized replay pass with unchanged bytes. Ruff E,F,B,I,UP,C4,SIM and formatting pass.

From the repository root:

/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 can replace the local interpreter path. Expected direct marker: PASS_ODD_CYCLE_FIT_RIGIDITY. RESULT.json is a generated receipt binding the package sources and three pinned predecessor documents. The written proof supplies universal coverage; finite tests are corroboration.

Limitations and unrun obligations

The executable is bounded to 26-by-26 matrices and the stated five fields. The written theorem covers every field and odd length at least five under the exact width-two/minimum-rank mask contract. Unequal widths, higher ranks and other masks remain outside the classification.

Physical basis changes, acquisition/copies, conditioning, dirty restoration, complete-plane tariffs, owners/hosting, preparation, stock/precision, tapes, compiler and outer recovery remain unpriced. No physical supplier, complete multiplication theorem, new kappa or selected-frontier change is claimed. No Lean, Docker, full selected cold replay or independent human/kernel proof review ran.

The historical card orientation command fails with KeyError: 'complex_profile' against the changed selected schema; it is not used as evidence. The current selected source-binding orientation passes. No shared script or predecessor was changed.

Open next questions

Can a minimum-rank unequal-width odd-cycle fit fail to be locally equivalent to integral edge/private channels? Start with the smallest width vectors having slack in the half-total bound; compare neighbor-intersection dimensions and local similarity invariants. An explicit fit outside the present equality argument is needed before physical construction work.

Related open work

No open PR studying width-two odd-cycle fitting-matrix rigidity was found in the refreshed live PR-body search. Merged #5023 is the rank-capacity predecessor; merged #3248 supplies the older C5 equality argument. The recent bipartite-capacity result studies a different mask/rank-value question, rather than this classification. Unpublished work remains outside coverage.

CI notes

CI pending for head 3e1645cdee74aa5efb722f7d218f7465d219e66c. All registered commands ran locally; no hosted result is claimed. The exact local checks, generated receipt and source-history/export evidence are recorded above.

Merge Danger

Door: two-way. One new isolated research package.

Blast Radius: research. Shared code, frozen predecessor inputs and the selected frontier are unchanged.

Advance CLAIM; publication base a5df48c55ea806d7952b65acc9c76d878ff09b13, priority revision 7.