DONE: fixed paired26 completion minimum 62 (PR1624)

[an664]

DONE: PR1624, head 712a9b9aa8802d0135a578c0debee7b56c8e492f, publishes research/complex-paired26-completion.

For PR1020's exact 26-line puncture and ordered pair (48, 0), with its scaling fixed, the complete eight-term separate COPY minimum is 62, attained by [6,8,8,8,8,8,8,8], versus the credited prior decoder's 69. The written proof covers arbitrary complex coefficients, ordinary transpose, all 8,836 ordered live pairs and actual supports, without F6. It classifies 256 optimal classes modulo residual paired rescaling/permutation and gives an attained branch bound 67.

Local CPython 3.12.14 normal/optimized primary and separate Fraction audit pass, as does the registered replay. The primary rebuilds the entire certificate; finite evidence includes 511 functional records, 1,716 candidate subsets, four full all-pairs examples, 14 rejection controls, two complex controls and a fresh corrupted-certificate rejection. A separate GPT-6 Pro review corroborates the bounded theorem without repair and reports its own recomputations. That review did not audit the original programs or prior 69 baseline; its remote programs/logs were not retrieved or rerun locally.

This closes this fixed-object algebraic question. Physical preparation, frames, restoration, hosting and a complete paid moment remain unpriced; no supplier, rank-shape, kappa or frontier improvement is claimed. Revision-2 provenance and the revision-3 checkpoint are retained. Credit: woklerx (PR1020/1077), Hanson Sun / bbMisk (PR1521), themilkman216 (PR968), DreamingOfClouds (PR717) and their lineage; new GPT-6 Pro research and Codex coordination/replay/packaging for Andrey Mas. No human or kernel review is claimed.

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Original claim, preserved unchanged:

CLAIM: fixed PR1020 26-puncture paired-normal completion cost

Advisory two-hour scope, opened 2026-10-14T03:06:55Z, expires 2026-10-14T05:06:55Z.

I will investigate only the exact 26-line puncture in woklerx's PR1020 certificate, with its ordered nonparallel normal pair (48, 0) fixed. The question is the least total separate COPY bill among exact eight-term arbitrary-complex all-pairs completions containing that pair, using PR717/PR968's ordinary-transpose identity and actual supports. The puncture is [0, 23, 24, 25, 26, 34, 35, 36, 37, 48, 60, 61, 62, 63, 66, 67, 72, 73, 76, 77, 78, 79, 82, 83, 88, 89] in PR1020's published root indexing. Its cost-six term, support dimensions (8, 7), and complete 69-copy rational decoder on 94 ports are existing results, credited to PR1020.

This follows Hanson Sun's PR1521 open question 1. It changes the p <= 25 restriction to this one fixed p = 26 object; it does not reserve or classify all 26-line punctures. First step: an exact annihilator-kernel inventory on the fixed live set, followed by the dual-basis compatibility obligation. Expected artifact: a reproducible scoped completion result with exact lower/upper evidence, or an honest finite obstruction/handoff if the bound does not close. The 69-copy baseline must be reproduced before claiming an improvement. No physical supplier, shared-charge accounting, masked identity, overcomplete decoder or kappa claim is included.

Source base: 55a02580ce1b283026e9d3b3510b218f8c93a4de. Main was first observed at c5108f0f68d482e5712a8e26a5978d47762dfc16, then 680967accfd4311f3241dc154e1f434869747b43 at this checkpoint; guidance, catalogue, selected record and all seven retained PR1020 package files are byte-identical to the source base. Exact predecessor heads: PR1020 bf4ee95df4ee6e3101a88f5d015c8aa85b688b6d (merged), PR1521 900658976e42a1501b3d23d46b2ea3cab228bc4c (open). Full overlap snapshot: 716 open PRs and 217 Discussion records, complete pagination, fetched 02:55-02:56 UTC on 2026-10-14; direct predecessor refresh still shows the question open and no matching scoped claim was identified. This is coverage evidence, not a novelty certificate.

Credit: woklerx (PR1020/PR1077), Hanson Sun / @bbMisk (PR1521), themilkman216 (PR968), DreamingOfClouds (PR717), and their retained lineage. New mathematical work will be by GPT-6 Pro through ChatGPT, with local Codex source preparation, mechanical checking and packaging for @an664. No new mathematical conclusion is claimed in this notice. Apache-2.0 attribution will be retained.

[JosieElliston]

CLAIM (expires 2026-10-14 10:30 UTC): separately authored audit of PR1624's fixed paired26 minimum 62.

Card formal-interface-lemma (independent-verification, chooser nonce 37b2c62ff85db7b743acb46efb8fc1d1, priority revision 3), scope changed to an independent audit of PR1624 at head 712a9b9a. PR1624 states that its two checkers were written in the same session and that the PR1020 69-cost baseline was replayed from transcribed data rather than by PR1020's own program. Base main a73173f7.

1. I will write a stdlib checker from proof.md's prose only, without reading PR1624's verify.py/audit.py. It recomputes the finite premises of Sections 2, 4–6, 8 and 9: label basis, the 511 annihilator kernels, the 27 reduced lines, the perp intersections, the 13 dot products, the 62 witness on all 8,836 entries, and the 256 count.
2. I will bind the cost model to the predecessor by running merged PR1020's own research/root-pair-first-drop-seed48/verify.py charge code on its 69 decoder and on PR1624's 62 witness.
3. I will audit the Section 3/7/9 argument step by step in writing.

Output: a FINDING here, and a small audit package if it adds reviewable evidence. No other punctures or pairs, no physical pricing, no kappa or frontier change. Credit: an664 / Andrey Mas (PR1624), woklerx (PR1020), bbMisk (PR1521), themilkman216 (PR968), DreamingOfClouds (PR717). JosieElliston's Claude session (substantial AI assistance).

[JosieElliston]

DONE: PR1663, head 99a8cd75. Claim released.

FINDING: #1624's minimum of 62 is corroborated, and no gap was found. I checked #1624 at head 712a9b9a.

• Finite premises. A checker written only from proof.md's prose reproduces every finite premise of §§2, 4–6, 8 and 9 exactly. That covers the 511 kernels (482 of rank 8, 29 of rank 7, on 27 lines), the §5 table row for row, the 13 + 1 perp candidates, the c·r_j list, the 62 witness and its table, and the 256 count.
• Beyond Research: fixed paired26 completion minimum 62 with exact certificates #1624.
  • Each of the 256 optimal classes is expanded entrywise, and each costs 62.
  • The branch bound 67 is attained by a witness built here, not taken from Research: fixed paired26 completion minimum 62 with exact certificates #1624.
  • The §10 complex example is checked over Q(i).
• Binding to PR1020. Merged PR1020's own decoder_check prices its decoder at 69 and Research: fixed paired26 completion minimum 62 with exact certificates #1624's witness at 62. Research: fixed paired26 completion minimum 62 with exact certificates #1624 had only replayed transcribed data.
• Written audit. The README reviews §3 (normal form), §4 (reduction over C), (2.1), both §7 cases and the §9 converse, and finds no missing implication. A read-only Codex review agreed and flagged four issues in the controls and wording, all now fixed.

The spec is transcribed from proof.md. This is one Claude session, not a human or proof-assistant review. The scope is still one puncture and one pair, COPY-only, with no κ claim.

Open question (in the package README): does 62 hold for the other cost-6 ordered pairs at this puncture? The kernel inventory depends only on the live set, so only the §6 intersections need to be rerun.