P112 rank-57 cover with scoped rational decoder cost floors

[flyingviper]

The original PR2041 cover has strict scalar fitting rank 99. This contribution changes the cover of the same 112 selected labels and certifies minimum scalar fitting rank 57, with explicit factors and a matching identity minor. It then proves why minimizing scalar rank does not produce an affordable decoder under the full port-space tariff:

• Exactly 57 rational rank-one terms cost at least 334, over every fitting matrix on this mask. The saved construction attains 334, so this floor is sharp.
• Every fit with at most 58 nonzero rational terms costs at least 297, including asymmetric factors, exterior directions that cancel, and fitting matrices of rank 58. This lower bound is not claimed sharp.
• The sharp minimum is 334 at any factor count when the fitted matrix has rank 57 and both factor families remain in its column and row spaces. With m=57+s nonzero terms, C >= 334+2s. This supersedes the earlier valid but weaker 308 lower bound; adding terms inside both spans cannot improve the saved construction.

All three floors exceed the favorable positive-saving allowance 2672/9 (integer charge at most 296). A successful scalar decoder on this mask therefore needs at least 59 terms, and must also escape the rank-57/both-spans restriction. Those cases, changed covers and operator contracts remain open. No cheaper decoder, native supplier, selected-result change or exponent improvement is claimed.

The minimum-length proof uses the forced identity minor, Schur complement and duality. The sharp any-length/both-spans extension uses the same complete 40-space closure and 272 cheap rectangles. Retaining the identical pure-J count in the full57 and restricted54 acyclic bounds gives C >=2m-3P+229 and C >=2m+P+217; their one-quarter/three-quarter convex combination is C >=2m+220. The new 58-term proof handles rank 57 through its one-sided exterior normal form, and rank 58 through a rank-one Schur complement whose support lies in a forest. Exact censuses cover 1,750 physical-triad spans, 2,318 affected-set unions and all 314 nonempty Schur support rectangles. Singular paired maps, the exceptional 4/24 branch and a possible zero retained source column are handled explicitly. Rational coefficients are not reduced modulo two; binary linear algebra prices group geometry.

See the sharp canonical-span proof, rank-tight proof and exterior-factor proof for the written arguments. Validation on CPython 3.12.14, standard library only:

python3.12 scripts/research_package.py --package research/p112-rational-decoder-gate

All eight registered commands pass under -I -O -B. Separate implementations check the exact endpoints, whole-group tariffs, span closures and Schur census; semantic controls reject endpoint corruption, underpricing, incomplete closures and an added forest cycle. Three separate AI agents split constructive research and independent proof/checker review. These checks are not independent human review or proof-assistant verification. Bounded coordinate-motif searches found no cheaper construction and are not universal impossibility claims.

Task coverage-law-change, explicitly changed scope, retains its family budget and open questions in TASK.json and README.md. The construction's scientific source pins are unchanged. The sharp canonical-span refinement was checked against refreshed canonical main 28f54620ff770dd051d5df35882bd84303d13c6f; guidance, selected record and consumed inputs were unchanged. The latest 100 updated open PR bodies were checked at 22:38 UTC on 2026-10-15. New PR2798 addresses a different fixed complex root-completion/COPY contract; PR2782 addresses positive half-integral Gram decoders, whereas this theorem permits arbitrary rational asymmetric factors in the stated full-group spans. Prior discussion access returned HTTP 406; cached discussions and fresh local notes were used without retrying the blocked route, so overlap coverage is bounded.

Credit is retained for Cinna Davis (#2041/#2126), Adam Gall (#1889), Sagar Thakrar (#2532 and different-mask rank certificate #2484), ywz65 (#2471), ctoth and collaborators, Philippe Van Velzen, Jeremy Diviney (#2203), icekylinx, woklerx, and the selected PR1094 source lineage. The general projection/excess-rank framework is prior work, including Hanson Sun's decoder-excess-rank package following joaqcoo/#1224; the new result is the cost consequence for this full-group mask. Exact distinctions and pins are in the package.

Prepared by flyingviper with substantial OpenAI Codex assistance in construction, proof, implementation and independent-agent checking. Apache-2.0.

Published head 18da668: all 18 package files were fetched back and matched the tested local files byte-for-byte.