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Scoped negative result: all reps in PR #1347's fixed P112 ten-mask cover miss 10 diagonals (no new kappa) #1402

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Mister-Meeseeks merged 2 commits into from
Oct 14, 2026
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Scoped negative result: all reps in PR #1347's fixed P112 ten-mask cover miss 10 diagonals (no new kappa) #1402

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47c397f
Add exact diagonal obstruction for fixed P112 frame cover
1337isnot1337 Oct 14, 2026
e591725
Record seed52 fixed-cover result and PR handoff
1337isnot1337 Oct 14, 2026
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36 changes: 36 additions & 0 deletions research/p112-fixed-frame-diagonal-seed52/CLAIM.md
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# Seed 52 claim: alternate supports for a fixed P112 frame cover

- **Track / assignment seed:** Structural decoder, OS-random seed 52.
- **Base:** `bce54fb3f895cbe5cd4c0eb477c3afeed100190f`.
- **Predecessor:** PR #1347 (`64fd5f9e91961dedc16118fcc7ad71ee82efe28f`), especially its exact frame-only `COVERING.json` and bounded `DIAGONAL-SEARCH.json`.
- **Coordination claim:** [discussion #1163, comment 18911792](https://github.com/CrocSwap/integer-mult-bounds/discussions/1163#discussioncomment-18911792).
- **Recent overlap check:** refreshed open PRs and the pinned coordination discussion at 2026-10-14 00:05 UTC. PR #1347's refinement kept one first-enumerated representative per frame pattern; no current claim covers all representatives for its fixed ten patterns. Nearby claims #1316 (one odd free rectangle with a coordinate term), #1360 (rank-nine factorizations), and #1342 (one asymmetric-unit replay) concern other domains.

## Candidate selection

Three bounded candidates were compared. Seed `52 mod 3 = 1` selects candidate 2.

1. Jointly select ten maximum-hit frame patterns from the complete 5,376-pattern dictionary while imposing diagonal coverage. Deferred: this broadens PR #1347's explicitly open full-dictionary next question.
2. Keep the ten frame-hit masks in PR #1347's fixed `COVERING.json`; enumerate every maximum-hit rank-six support representative in each mask class, then decide whether one representative per class can cover all 112 live diagonals. **Selected.** This varies the omitted support representatives while holding the frame cover exactly fixed.
3. Extend the P112 two-coordinate free-rectangle audit to the first open 24-row support size. Deferred because it is adjacent to PR #1375 and active claim #1316.

## Hypothesis and exact scope

For the ten distinct frame-hit masks induced by the ten terms in PR #1347's `COVERING.json`, replacing each term by any other rank-six support triple with the same frame-hit mask preserves all 1,260 frame counts exactly. The question is whether one such triple per mask can make the union of the ten diagonal support masks equal the full 112-live-line set.

Enumerate the complete finite domain: 7,616 rank-seven tagged parent pairs, each with its 127 nonzero cut words (967,232 parent/cut cases). Retain precisely the cases with 1,120 frame hits and one of the fixed ten frame masks. Within each mask class, support triples whose diagonal mask is a subset of another triple's mask are safely dominated for this coverage question; keep the inclusion-maximal masks and exhaustively solve the one-per-class coverage problem with exact bit masks.

This is a support-only necessary-condition audit. A feasible tuple would not solve coefficient equations or produce a decoder. An infeasible result applies only to these ten fixed frame masks and says nothing about other ten-pattern covers, lower-hit terms, helper costs, or kappa.

- **Expected artifact:** source-pinned exact enumerator and verifier, complete candidate and Pareto counts, exact feasible tuple or exhaustive negative certificate, mutation controls, scope/assumption statement, and a concise handoff.
- **Reproducible entry point (planned):** `python3 -I -O -B research/p112-fixed-frame-diagonal-seed52/verify.py`.
- **Expected finish:** 2026-10-14 01:15 UTC; claim expires 02:05 UTC.

## Completed result

- **PR:** [#1402](https://github.com/CrocSwap/integer-mult-bounds/pull/1402).
- Exhaustive replay covered all 7,616 rank-seven tagged parent pairs and their 127 nonzero code words each (967,232 cases). The ten fixed frame-mask classes contain 40 exact rank-six support triples, two distinct diagonal masks per class.
- The union of all 40 masks has size 102/112 and misses live line IDs `18, 47, 53, 62, 88, 92, 96, 100, 112, 117`. Hence every one-per-class selection within this fixed mask family misses these same ten diagonals.
- The predecessor's frame counts, frame digest, and diagonal multiplicity histogram reproduce exactly. Zero-cut, cross-mask and omitted-diagonal controls pass.
- Local checks passed under CPython 3.14.7: exact verifier, package metadata/source pins, JSON syntax, and `git diff --check`. CPython 3.12 is not installed locally; the registered manifest is ready for the repository's Python 3.12 workflow.
- No coefficient feasibility, different-mask cover, physical realization or κ improvement is claimed. Exact next step beyond scope: include all 5,376 maximum-hit frame masks and all their representatives in an exact joint frame-plus-diagonal cover.
201 changes: 201 additions & 0 deletions research/p112-fixed-frame-diagonal-seed52/LICENSE
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21 changes: 21 additions & 0 deletions research/p112-fixed-frame-diagonal-seed52/NOTICE.md
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# Notice and attribution

Prepared by **Navinor** with help from opencode (research, proof development,
implementation, verification and documentation). Apache-2.0; new contributions are under
the repository's license.

Model and question: CrocSwap/integer-mult-bounds#717 (DreamingOfClouds), `research/format-limits-2/` at head
`4a97676294f9fa413b65986e9984fddeda71e4cc` (B14 decoder model, D1(a) masked reduction,
D1 mixed-beta open problem, D1(e) sampled covering bound; read for definitions only, not
vendored or executed). Prior decoder floors: CrocSwap/integer-mult-bounds#620 (Joel Pulikkan / GamingPuzzled),
CrocSwap/integer-mult-bounds#968 and CrocSwap/integer-mult-bounds#1077 (woklerx and the seed44/seed48 packages; both defer masked E-graph
identities), CrocSwap/integer-mult-bounds#845 (Mihir Srivastava; fixed-reader ninth total). The E8 line enumeration
is adapted from `research/complex-frame-pair-decoders/verify.py` (Apache-2.0, same
repository) with attribution; no predecessor file is modified or imported at runtime.

The selected construction chain (see `certificates/selected-result.json`) is read only:
Jonathan Capalbo, Chafik Boukhalfa, Jeremy Kothe, Hao Yan, DreamingOfClouds, eumemic and
the other credited authors of the T37/CrocSwap/integer-mult-bounds#589 chain. No selected-frontier change is proposed.

Third-party code: none. Standard library only. Retained source attribution and licenses of
all predecessor packages apply to their own files.
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