Research note: sharp off-target proper-star puncture threshold

[lukemansoor]

PR669 proves the sharp puncture threshold for target-local stars and explicitly leaves punctured decoders using arbitrary off-target stars open. This research note / scoped negative result closes that larger scalar class: for every power-of-two k >= 2 and p >= k, decoding one retained target requires exactly 2(p-k)+1 deleted ports. No new kappa or paid supplier is claimed.

The written elementary proof establishes the stronger identity d(C_{p,k}) = 2(p-k+1) for the global degree-<k star evaluation code, including the empty star, for every k >= 2. It uses generalized-cube parity, an even-face lemma and restriction induction. At a power-of-two k, permitted target overlap-class corrections add no functions to that code. The attaining witness uses one nonempty (k-1)-star.

A ten-star p=5,k=4 example reaches the same threshold while all 16 target intersection cells survive. Thus PR669's minimum threshold extends globally, while its local cell feasibility criterion does not. Exact elimination checks local infeasibility for this example.

Changes are limited to research/global-star-puncture/ and one scoped workflow. The accepted conditional kappa remains 51110230636731/31250000000000000; selected-result, existing suppliers, generators and upstream/ are unchanged.

Reproduce

python3 -B research/global-star-puncture/verify.py

Expected: PASS_GLOBAL_STAR_PUNCTURE: sharp deletion thresholds 3/5; 2 independent F2 engines; 960 correction rows; 12 negative controls.

The complete deterministic receipt is compared on every run. A packed row-space quotient and a separately written list-RREF/nullspace implementation agree on 80/240-port domains, ranks 65/155 and distances 4/6. All 28,680 pair syndromes in the second domain are distinct. Coverage also includes all 38,344 codewords in seven small domains, exhaustive even-face support checks in dimensions 1–4, valid generalized-face cubes, and a non-power-of-two positive escape control. Finite replay supports the written proof; it is not a formal proof of the multiplication theorem.

Validation and limits

• Python 3.13.5 on Windows: normal and isolated optimized exact replays pass.
• Four harness tests pass: relocated minimal replay/export, corrupted receipt rejection with output preserved, changed contextual source rejection, and changed source-pin rejection.
• Separate AI proof/code review and an independent enumeration of all 131,072 local coefficient assignments corroborate the off-target example.
• Selected-record/source integrity and staged whitespace checks pass; the final checkout is clean.
• Scoped hosted Linux verification passed on Python 3.11 and 3.13 at the submitted head: fork run 38348453202. Both jobs passed the exact receipt, four failure-detection tests, selected-result integrity and clean-checkout check. Upstream PR-triggered workflows await maintainer approval (action_required).
• Full make verify and frontier626-verify were not repeated for this isolated additive research package.

The theorem concerns one specified retained target, arbitrary global proper-star coefficients and fixed coefficients within each overlap class. Simultaneous all-target delivery, paid total preparation, dirty restoration, legal reader frames, hosting, recursive costs and analytic transfer remain separate obligations. Degree-k stars, source-dependent overlap corrections, nonlinear mechanisms and other geometries are outside scope.

Base: 66849e1475c930fd7732b1a0067ffee7e1e95f60. Submitted head: 80431a74c31e807581f5c2258c87abe2084caacd. OS-random assignment seed 43, structural decoder research; the narrow local claim and result link are in CLAIM.json. No pinned coordination issue existed. Final overlap metadata through #721 and fresh #669/#715/#717 checks show different scopes; PR669 remains at 308f1e6ec3bb0b6ec0584cf7bfe219addc219b1e.

Credits: Sam Sung / the-missionary-company (#669), Joel Pulikkan / GamingPuzzled (#231), icekylinx (#144), eumemic and the retained paired-cube lineage. Selected-construction context preserves Chafik Boukhalfa, Jeremy Kothe and eumemic's credits. Exact source pins and notices are included.

Prepared by Luke Mansoor (@lukemansoor) with substantial OpenAI Codex assistance in research, proof development, implementation, independent AI verification/review and writing. Apache-2.0. No independent human review, proof-assistant verification or worldwide priority claim.

[Mister-Meeseeks]

100-PR batch review at 80431a74c31e807581f5c2258c87abe2084caacd.

This remains open after scoped triage. Next review step: Uniform global-star proof and package-path collision with #725 require coordinated review.

This is a specific continuation point, not a rejection or a request to redo already recorded checks. Completed local evidence is retained separately from outstanding whole-head obligations.

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