Research: P112 rank-six star trims have sharp frame-demand bill 60

[1337isnot1337]

Result and scope

Research note / scoped structural proof. No masked decoder, physical supplier, or new kappa.

For every arbitrary rank-six support restriction of an original cost-seven P112 C/N root-star atom, at most 1,120 of the 1,260 live orthogonal frames can be met on both factors. A rank-factorization lemma reduces all joint cuts to maximal one-axis codeword cuts. The complete census covers 7,616 tagged parents and 967,232 maximal slices; two separately implemented engines agree on every maximum, attaining cut, minimizer count, full codeword/hit digest, and certificate field.

The sharp per-parent joint maxima are 1,080 for 5,824 tagged parents and 1,120 for 1,792. The uniform averaged frame bound is 54. The simultaneous integer frame-demand relaxation has sharp copy bill 60: the inherited nine-root obstruction excludes nine terms, while the exact certificate supplies ten rank-six terms meeting every frame at least eight times.

The recorded cover leaves 21 diagonals uncovered, so it cannot satisfy the masked identity for any coefficient choice. This is a concrete negative control, not a proposed decoder. A single 120-second refinement requiring diagonal coverage returned no feasible incumbent; it did not exhaust the search or establish infeasibility. That refinement retained one support representative per frame-hit pattern and excludes other representatives even with identical frame hits.

Changed hypothesis and overlap

Base: bce54fb3f895cbe5cd4c0eb477c3afeed100190f. Fresh OS seed 20, complex supplier track. Sprint discussion1330.

PR1236 proves the cost64 floor for exact C/N supports; arbitrary further subsets are excluded there. PR1323, pinned at c1144855242a56ffc1090be2f075f977d123d947, establishes the first support-erasure threshold. This package measures all rank-six descendant frame capacities and solves the resulting frame-only relaxation. It does not repeat the minimum-erasure census or reopen the original exact-support theorem.

Only research/p112-rank6-frame-coverage/ is added. Original rank-eight/nine parents, descendants of rank five or lower, mixtures with other cost classes, different punctures and shared-copy prices are outside the optimization. Complete helper, source, frame, restoration, hosting and recursive charges are optimistically zero for this necessary-condition screen. No coefficient identity or complete supplier invoice is supplied.

Overlap coverage: 821 open PR bodies, 73 discussion/comment records, 60 recent all-state PRs, and local notes/ledger; source heads and latest claims refreshed before publication. No competing coverage census found. The report records coverage limits and nearby distinct P112 work.

Reproduce and validation

python3 -I -O -B research/p112-rank6-frame-coverage/verify.py

Expected:

PASS_P112_RANK6_FRAME_COVERAGE atoms=7616 slices=967232 max_hits=1120 average_floor=54 cover_bill=60

The independent command is python3 -I -O -B research/p112-rank6-frame-coverage/independent.py.

Full registered replay passes on Python3.12.15:

python3.12 scripts/research_package.py --package research/p112-rank6-frame-coverage

All eight tests pass on Python3.12.15 and 3.14.7. Mathematical regressions cover all 4,096 binary3-by-4 matrices and 524,160 joint restrictions. Failure controls reject invalid copy/frame witnesses, changed archives, wrong roots, a false coverage table and receipt overwriting; both engines pass after optimized relocation. Four frozen archives were checked against their exact source Git blobs. Runtime hashes and scope are in VALIDATION.json.

No native supplier admission, selected cold replay, full multiplication formalization or remote CI pass is claimed.

Credit and next question

Navinor/PR1226 supplies the model and star census; David Arias/PR1236 supplies the original cost64 theorem and nine-root obstruction; Hanson Sun/PR1242 independently corroborates it and identifies further trims as open. Cinna Davis/PR1323 supplies the first-erasure result. DreamingOfClouds/PR717 and retained PR968 geometry keep their credit in the frozen notice. Exact dependency heads, Git blobs, archive hashes and Apache-2.0 licensing are retained.

New work by Cinna Davis (@1337isnot1337) with substantial OpenAI Codex assistance. Both implementations belong to this contribution; independent human review is not claimed.

Next step: a full-support-dictionary diagonal/per-frame rank relaxation, followed by exact coefficients and a complete physical invoice if a favorable candidate exists. README.md and TASK.json state the blockers.

[Mister-Meeseeks]

100-PR batch review at 64fd5f9e91961dedc16118fcc7ad71ee82efe28f.

Merged to main, preserving the original contributor commits and package attribution.

Reviewed scope: Complete rank-six P112 descendant capacity census and rank-factorization domination argument, with sharp ten-term frame-demand relaxation of COPY60. The exhibited cover misses diagonals; no exact decoder or supplier is claimed.

The pinned scoped command passed; the batch ledger contains its command, input hashes, environment and log. This incorporates the research/tooling contribution and does not change the selected κ or discharge the inherited multiplication-theorem assumptions.

Batch review, attribution and evidence. Thank you for the contribution.