[JoyboyBrian]

Exact query-dimension contract

For every odd cycle C_(2k+1), k>=2, arbitrary nonnegative payload widths and every coefficient field, the minimum fitting rank is exactly max(maximum independent-set payload weight, ceil(total/2)). Permitted directed rectangular blocks may be nonsymmetric and noncommuting. A compressed edge/private-channel constructor attains the bound with O(n log(max(d)+2)) integer operations and O(n) compressed storage; it does not expand payload coordinates.

The all-field proof sums the rotated one-edge-plus-isolates subspace bounds. Classical fractional vertex-cover duality and half-integral cycle matching give attainment, and a concave closing-edge search implements it without a solver. Seven width-two payloads need seven query coordinates, versus eight for two scalar solutions; (2,1,2,1,2,1,1) instead needs six because of its heavier independent set.

This answers a distinct longer-cycle interface question alongside Tekalign #3362's C5 theorem and Nick Petalas #4909's bipartite theorem. It does not claim literature priority for classical odd-cycle coding or matching. Family gate: expand the mask class once, close this dimension contract, then stop before any unfunded physical construction. Canonical investigation/claim: Discussion #1587. Base canonical main bee95ce88f9f19802f0613f92838895cd0e14a1f, priority revision 7; selected conditional kappa 0.00302015 unchanged. Initial overlap covered 310 open PRs, 4096 Discussion records and all 6690 tracked notes, with complete reported inventories; arbitrary diffs and unpublished work remain limits.

Verification

• Registered Python 3.12 normal/optimized replays pass: 3723 complete bounded width vectors; independent edge-sharing and independent-set enumeration; 11169 attaining ranks over GF(2), GF(3), GF(101); every 17408 directed binary scalar C5/C7 fitting matrix.
• Separate column elimination checks all proof inequalities on 288 rectangular-block matrices and a sharp C7 attainer whose allowed 2-by-2 blocks do not commute. A 4097-bit-width example remains compressed. Malformed dimensions, underpaid/broken recipes and a forbidden ordered block reject.
• Separate read-only peer review confirms the uniform proof and actual constructor. A separately written edge DP checked 384 weighted C11/C13/C15 cases, all closing-edge values and concavity. The review found and resolved quadratic witness copying/list membership; final reconstruction and validation are linear. This is AI peer evidence, not human or proof-kernel acceptance.
• Exact committed bytes/source-history preflight passes at 1b069fc3d6bb463ab212f3ebc644abe838efa550 (13 files). Official export integrity passes; Git-free normal/optimized cold replays pass in 0.584/0.604 seconds. Source corruption and a bool-for-integer receipt mutation reject.

Reproduce: python3.12 -B scripts/research_package.py --package research/odd-cycle-query-capacity.

Remaining obligations and credit

This counts intermediate linear field-coordinate dimension. Physical reads/copies, dirty compensation/restoration, ownership, frames, hosting, coefficient preparation, stock/precision, fixed tapes, compiler/outer recovery and an actual P112/caller embedding remain unknown. No selected cold replay, Lean, Docker, full multiplier theorem, native supplier admission or new kappa is claimed.

Credit Tekalign #3362; Sagar Thakrar #2484; ywz65 #2471; Nick Petalas #4909; Cinna Davis, Adam Gall, ctoth, Philippe Van Velzen and the retained P112/native lineage. New scoped proof, constructor and checks: JoyboyBrian, with substantial OpenAI Codex assistance and separately assigned peer review. Apache-2.0.

Submission refresh (2026-10-23 06:22–06:24 UTC): fetched canonical f259dabc7d70e0db8d6d36bfda888f008bbfecc1 preserves guidance, selected record and all declared input bytes from scientific base bee95ce88f9f19802f0613f92838895cd0e14a1f. New package paths are disjoint from upstream changes. Refreshed remote coverage304PR/4114Discussion, local6690base notes complete; no novelty guarantee.

Supplemental AI peer review is local-only evidence; the committed written proof, source bindings and registered verifiers are the public reproduction artifacts. Peer-only observations do not constitute independent human or scientific acceptance.