[petalas]

Summary

Scoped written mathematical proof with exact finite corroboration: every finite affine binary decoder for the cubic seven-cube, with each term costing at most rank 30, has total copied rank at least 200.

- Peak-30 total optimum in [140,240], PR1564
+ Peak-30 total optimum in [200,240]

The new support-sensitive factor-span lemma uses the complete coefficient identity. Both factor matrices must have rank at least five, so their rank-two product needs at least eight terms. At least four terms cost 30 by diagonal coverage, giving 20m+10k >= 200. The general lemma proves m >= 2 ceil(n/d) for affine parity decoding with linear supports at most d, allowing constants, repetitions and arbitrary finite term counts. Proof.

Hanson Sun / bbMisk's PR1564 supplies the exact low-rank classification, seven-cube model, earlier bounds and upper construction as cited premises. Its census and witness are not replayed. This contribution was produced by Nick Petalas. Credit Joel Pulikkan / GamingPuzzled, Ross Richardson, martinxxx9, icekylinx/eumemic and the retained decoder/frame lineage. Sylvester's inequality and affine counting are classical; no general literature-priority claim is made.

Evidence

Before, PR1564 explicitly left the peak-30 total optimum unresolved with lower bound 140. After, PROOF.md gives a universal lower bound of 200. The standalone checker corroborates the new span argument with 5,546 generated subspaces through seven variables, including 541 containing parity. A separate complete XOR-state BFS checks five small coefficient problems, allowing pure constants, with 132,112 total states and minima 2,4,2,6,4. It also checks 17,472 literal product entries, 4,752 independent affine-pair diagonal cases, 112 complementary-pair cases, three positive complete words, ten rejected contracts and the support/constant boundary fixtures.

Both registered CPython 3.12.15 commands pass locally, warning-strict, in 0.42 seconds each. Normal and optimized receipts agree. The registered runner prints PASS_REGISTERED_RESEARCH_PACKAGE; committed-byte/source-history checks pass for all six package files at 8eb92d3901231e7a172bbf99b780ec71bd4f3c6c. Official export integrity and declared-input coverage pass. Both exported modes also pass outside Git with an empty PATH and read-only files, with identical output and all seven export files unchanged. Receipts are generated from actual execution and bind all six package source hashes.

From the worktree root:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/seven-cube-peak30-floor/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -O -W error -B research/seven-cube-peak30-floor/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/seven-cube-peak30-floor
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/seven-cube-peak30-floor --check-source-history

Another CPython 3.12 path can replace the local one. The verifier uses only the standard library and writes nothing. The source classification and upper witness are pinned as historical provenance; there are no executable predecessor dependencies.

Canonical main remained 7c49fe32c55de02c2c873655c804b16e4c3fef0a, priority revision 8, selected PR6030. The selected binding example passed for orientation. Advance CLAIM. Live handoffs, recent account work, open and merged source/object/general-form searches and canonical-main notes were checked before claiming and committing. The broad cache covers 556 open PRs and 6,604 Discussion records with complete remote pagination; its sparse local note inventory is incomplete. Arbitrary historical diffs and unpublished work remain coverage limits. The card's change to this local algebraic side question is explicit in TASK.json. Research stopped at the first proved improvement.

Limitations and unrun obligations

The seven-cube application depends on PR1564's cited low-rank classification. The new universal bound rests on the written proof; finite checks are corroboration. Attainment of 200 and the exact optimum remain open. The upper bound of 240 is inherited, not a new construction or independently replayed result.

The predecessor's additive stage budget already excludes positive saving in that model. This tighter local tradeoff does not remove that obstruction. No physical producer/consumer, routing, source access, dirty restoration, hosting, precision, stock, tapes, compiler or outer admission, new kappa or selected-frontier change is supplied. All five selected framework assumptions remain. Independent human and formal proof-kernel review are unclaimed. Lean, Docker and the selected cold replay were not run. Both registered commands ran locally. The bounded managed-runtime allocation deviation is documented in the README.

Open next questions

• Determine the exact optimum in [200,240]. First test whether eight terms with four rank-30 terms can satisfy the full coefficient equations. The simultaneous equality conditions remain unresolved.
• Supply a complete two-stage retention interface changing the inherited additive charge. Begin with the predecessor's four-term decoder and price retained state, delivery and restoration. Physical legality and a favorable complete invoice are missing.

Related open work

No other open PR studying the affine seven-cube peak-30 total tradeoff was found in the live object search, the complete cached PR-body snapshot or the latest open-PR refresh. The direct predecessor PR1564 is merged. Recent projector-word, Gaussian-bank, selected-ribbon and two-capacity submissions use different objects and quantities; no numerical comparison or extra validation of them was performed.

The completion refresh inspected all 14 other open contributions from this account created since the advance CLAIM. None studies this seven-cube peak-30 total tradeoff, so no additional related-object line or numerical comparison is required. No extra computations were run to validate them.

CI notes

Both hosted checks passed for exact contribution head 8eb92d3901231e7a172bbf99b780ec71bd4f3c6c in run 41634840825: plan and package replay. The completed checks were read once; run metadata confirms the matching head, completed status and success. Hosted replay executes both registered warning-strict normal and optimized commands. Local metadata, committed bytes/source history, direct checks and read-only Git-free export replay also passed. This is execution evidence for the declared checks, not independent mathematical acceptance.

Merge Danger

Door: two-way. Blast radius: isolated.

Only six files in one new opt-in research package are added. Shared code, frozen predecessor packages, attribution and selected-frontier records are unchanged.