[petalas]

Summary

Prove the exact minimum of 68 complete executions for extracting arbitrary diagonal blocks on 66 nonzero grades by equally weighted sign conjugations. Yiyang's merged #5037 uses 128 complete root executions at this grade count.

128-run Walsh extraction
  -> 68-run Paley sign extraction
  -> exact accumulator division by 4 and 17
  -> the same dyadic block-diagonal output

Claim type: scoped algebraic proof with an exact finite sign certificate. Rank and three-column divisibility give the lower bound; the classical Paley order-68 Hadamard matrix attains it. The application and exact-grid wrapper are the new step. Paley's construction and character averaging are credited methods, not claimed inventions. The reduction saves 60 complete executions; copying, signs, accumulation and exact division are additional work. No wall-clock gain, native invoice change or new κ is claimed.

Only six files under research/grade-sign-extraction/ are added. The proof applies at arbitrary nonzero grade dimensions, with arbitrary off-diagonal blocks, and includes exact integer divisibility by 17. The wrapper is applied once at the root; repeated input copies earn no productive credit.

Evidence

• Before: Construct a bounded-lane border core without an exact boundary provider #5037's source-bound extraction note specifies K=2^ceil(log2(H+1)), hence 128 at 66 grades. After: the exact sign design uses 68 rows, and the matching scoped lower bound rules out 66 and 67.
• At head eb0c12811253174a6a7834c118e2cf827cc9b706, normal and optimized warning-strict CPython 3.12.15 replay pass. The final Git-free invocations took about 0.35 seconds each, well below the 120-second registered limits. These are check durations, not physical performance measurements.
• Exact evidence includes 2,346 full Gram entries, 28 dense/triangular unequal-grade fixtures with 1,712 output components, a six-coordinate variable-denominator fixture recovered through the integer divider, 8,201 signed exact divisions and 11 rejection controls. Three disposable semantic corruptions reject. Removing a redundant remainder guard preserves behavior because the quotient postcondition still enforces exactness; it is not counted as a rejected semantic corruption.
• Registered replay, AST/function-bound checks, six-file committed-byte/full-source-history preflight and official export integrity pass. Both normal and optimized standalone checks also pass in a read-only Git-free export with PATH=/nonexistent; all seven exported files remain byte-identical. The verifier writes no files and imports no predecessor executable.

From the worktree root, reproduce with the actual local interpreter:

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

Expected direct output includes PASS_GRADE_SIGN_EXTRACTION, runs: 68, literal_cases: 28 and signed_exact_divisions: 8201. On another host substitute its CPython 3.12 path. The dimension-uniform claim rests on the written proof, not extrapolation from fixtures. Independent human and formal proof-kernel review are unclaimed.

Limitations

The optimality contract requires equal positive weights and grade-constant ±1 conjugations that work for every permitted off-diagonal block. Unequal weights, complex phases, independent input/output maps and additional sparsity are outside it. The exact divider needs the proved output-grid/divisibility premise; arbitrary division by 68 is not a free dyadic operation.

Actual native shears, recursive source/exit matching, rounded arithmetic, physical ownership/movement/restoration, stock, BIT transfer and outer admission remain open. No full physical tariff, selected cold replay, native/Lean/Docker suite or complete multiplication theorem is claimed. The selected frontier is unchanged. Python's bounded managed-runtime allocation and diagnostic limitations are documented in README.

Related open work

• A coherent rank gauge still misses the actual two-full fresh-D endpoint #5106 studies the actual two-full fresh-D endpoint and gives a coherent-gauge obstruction. This wrapper assumes a completed same-grade program and does not discharge that endpoint or re-check its counts.
• No other open PR on the 66-grade equal-weight sign-extraction minimum was found in the inspected coverage. The refreshed snapshot covers 333 open PRs and 4,251 Discussion records with complete reported pagination; sparse local notes, arbitrary unseen diffs and unpublished work remain outside complete coverage.

Open next questions

• Bind one literal native caller from Construct a bounded-lane border core without an exact boundary provider #5037 to the completed graded extraction contract, preserving every source, exit and complement. Coherent shears and whole-job recursive matching remain blockers.
• Compare the complete physical tariff of the 68-run and 128-run wrappers, including signed copies, workspace reuse, accumulation and exact division by 17, once such a caller exists.

Provenance and credit

Scientific base and submission refresh: canonical main 81e9bd69e4a5b00ba9c713a1620e36d576d5522e, priority revision 7. TASK records the changed scope from complex-rank-shape-decoder; exact predecessor README/extraction-note, catalogue and selected-record blobs are retained as provenance. Their scientific verifiers were not replayed. The cheap selected-source metadata orientation passed, separately from operator admission.

Advance CLAIM. The delivered scope matches it. Current claims, this account's recent contributions, open PR bodies, merged-main distinctive terms and merged/all-state PR searches were checked before claiming and again before committing. No competing result was found within that bounded coverage.

This contribution was produced by Nick Petalas. Credit Yiyang / Lyy-iiis #5037 for the bounded-lane/root-only extraction setting, Dillon #4623 for border recovery, Cao Yifan #4610 for native context, Jacob Alan Sussman and the finite-precompiler contributors, and the retained native/helper/precision/outer lineage. Classical Paley/Hadamard methods retain their credit. Predecessor files and notices are unchanged. Apache-2.0.

CI notes

CI blocked upstream. Run 39933925464, for contribution head eb0c12811253174a6a7834c118e2cf827cc9b706, failed in plan / Onboarding helper tests; replay was skipped. The log reports 408 tests, eight failures and 17 skips. No hosted package replay or mathematical acceptance is claimed.

The checkout log identifies tested merge 5b6a029383b29e00c5684eeba983eba23b896f0c, combining this contribution with main 20a1e31bc6009b9b3f4d084bbaadc6350e34fcba. This PR changes only the six files under research/grade-sign-extraction/; it changes no shared scripts, tests, workflow or historical fixture.

Two upstream fixture problems account for the eight failures:

• Seven failing subcases in tests/test_research_descendant_source_roles.py stop at working source drift: scripts/check_research_commit.py, before their expected helper/history-test diagnostic or acceptance. The historical research/isolated-descendant-bytecode/SOURCES.json still declares that script as an input with SHA256 ecf206a82bf1f4e748efb6cc639919582050d173b3537ec12f7265a4f7b55ead, while the tested main version hashes to 28fed1e156fe8601633ed453f682a158cb15e94981e6c89ed96ae72ffc9d0c6f. The expected diagnostics include source blob provenance mismatch: scripts/research_package.py, source SHA256 provenance mismatch: scripts/research_package.py and the corresponding tests/test_research_source_history.py messages. The unchanged fixture copies the current shared script before authenticating it against the historical live-input pin.
• tests/test_research_onboarding.py::OverlapTests::test_report_survives_a_cp1252_stdout copies only scripts/find_overlap.py into its temporary repository. The current helper imports git_source_env from research_package, but that module is absent in the fixture, giving ModuleNotFoundError: No module named 'research_package'.

Upstream commit 862ac2a0f changes both shared scripts and is an ancestor of the tested main base. Its diff and the base's actual bytes confirm the new import and historical-pin mismatch. No shared-code fix is included here.

The same eight failures, with the same diagnostics, are independently visible in PR #5116, “Implement complete owned query streams on selected signed records”, head ee4bebf4bb1d441bbbcd43ff4bc2bcb2734a679e, run 39933826517. This comparison is limited to the failed onboarding-job logs; that contribution's mathematics was not reviewed or replayed.

Both registered package commands and the committed/source-history/export checks passed locally at the unchanged contribution head, as recorded above. Upstream onboarding repair and a successful hosted replay remain outstanding. No DONE has been posted.

Merge Danger

Door: two-way.

Blast Radius: research.

Isolated new package only; no shared code, frozen source or selected-frontier change.