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Prove complete moment optimality of fixed eight-helper complex blocks - #1429
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100-PR batch review at Merged to Reviewed scope: All-order/all-length moment optimality proof for the twelve fixed eight-array late complex blocks, permitting repeated reads and dirty-coordinate changes. Exact interface and kernel-support premises pass for three source words; extra arrays, early outputs, descents and cross-block coding are excluded. 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. |
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The twelve late eight-helper blocks in the selected complex word cannot reduce the recursive moment through any finite rational refactorization with their retained input/output frames. The proof permits arbitrary order and length, repeated input reads, changed helper coordinates, and fusion with later work within each block. The current native words attain the complete lower bound, including helper completion to rank nine.
Review files: Proof and scope · Evidence · Source pins and credit.
This is a definitive dead end for that declared interface class. It makes no new kappa claim. Extra arrays, target scratch, earlier output access, frame descents, changed centres, and cross-block coding remain outside the class.
The two exact block minima, for every 0 < s < 1, are
24 + 8*2^s + 4*3^swhen the common frame has rank four and32 + 4*4^swhen it has rank five. Pointwise domination of the full complex supplier moment gives inclusion of the complete outer-recipe feasible sets. The copied-centre and bit costs are retained. The provider uses one canonicalword for both orientations: the backward program is the same-order
inverse-transpose dual, with identical frame climbs. Independent redesign of
a backward interface is outside the theorem.
The package binds the proof to the selected PR #1094 local word and both stronger pending PR #1319 providers. It extracts their input frames, late output entry frames, endpoint caps, and output matrices from the emitted words. Exact 16-by-16 prefix minors establish independent boundary inputs. Backward extraction and forward replay agree, using only eight original sources and the eight dirty helpers.
Reproduce with:
Expected:
PASS_FIXED_BLOCK_MOMENT_PREMISES blocks=36 subsets=9180 minors=36 controls=12.Validation: all 36 block interfaces, all 9180 nonempty source subsets, rational ranks via elimination and exact Gram minors, binary spans via two methods, 36 independent boundary minors, complete native helper profiles, and 12 targeted rejection controls. The
selected emitted orientation pair also passes equality of all 3486 climbs,
19493 exact non-centre scalar dual checks, and full profile equality. The package metadata and source pins pass. The worker completed direct Python 3.14.7 verification. The coordinator independently reproduced the direct verifier and the registered Python 3.12 package replay, checked the mathematical proof, and matched all four frozen input files to their exact source Git objects and the selected PR #1094 replay. No full construction cold replay or proof-assistant theorem is claimed.
Base: bce54fb3f895cbe5cd4c0eb477c3afeed100190f. PR #1319 source: 603defe58aac018c6a8457840481390a15846fe4. Exact file pins are in INPUTS.json. The selected frontier and frozen source packages are unchanged.
Credit Thomas DiFiore (@tomdif)'s comment in Discussion #1167 helper-basis screen, which left later coalescing open; Douglas Colkitt (@Mister-Meeseeks)'s PR #1240 budget screen; Rohan Arun (@rohanarun)'s PR #1094, PR #1297, PR #1319; Jeremy Kothe (@jeremykothe2)'s owner and helper-donor lineage; and all retained provider notices. Prepared by Ali Romman (@aliromman) with substantial OpenAI Codex assistance. Apache-2.0; inherited conditional realization assumptions remain.