[petalas]

Summary

The A4 endpoint diag([a],[b],[aba^-1b^-1],[ab],1,...,1) still has exact minimum eight paid calls with any finite restored dirty workspace. The new result is a scoped written compression proof, with exact finite corroboration.

hypothetical six-call word on r>=4 roles
  both inherited rank-three floors are tight
  every paid output column lies in the first four roles
  restrict each rank-one idempotent factor to those roles
  obtain the four-role six-call word excluded by #2116/#4032

Before, #4032's exact minimum applied to four roles; its written floors already held at every role count. The new step closes the remaining workspace escape. The eight-call corollary retains Cinna Davis #2116's finite four-role exclusion and Sagar Thakrar #4032's radical-rank identities and determinant congruence as cited premises. Their searches and proofs were not audited or replayed for this contribution.

The six-file package is research/a4-dirty-workspace-minimum/. Sources are pinned at canonical main ec2780e8aeb1c88acaf5e9e470ef8c68e7fbbb6c, priority revision 7. The submission refresh found unchanged guidance and predecessor bytes. Advance CLAIM.

Evidence

Local CPython 3.12.15/macOS arm64 checks pass:

• All 29,791 nonzero binary column triples in five dimensions; 26,040 full-rank triples, with both support patterns checked by F4 elimination and a separate binary-span oracle. Exactly 336 triple/support pairs are compatible and confined to the required coordinates.
• All 3,600 admitted single-factor restrictions at role counts 4..7 and 256 additional deterministic words. The 3,856 fixtures check 287,616 complete regular-action basis columns. The eight-call upper-bound fixtures check another 264 columns.
• Nine controls cover seven malformed inputs, a rank-slack counterexample and a required nonzero dirty-input contribution. Two disposable optimized-script mutations, incorrect field inversion and omitted dirty reads, reject. The explicit pivot-inverse invariant was added after the initial rank-only fixtures did not detect that inverse-table mutation.
• Warning-strict normal and optimized registered/direct checks, metadata/pins, committed-byte/source-history preflight, Ruff 0.17.1 diagnostics and format, official export integrity, and normal/optimized read-only Git-free replay pass. All six exported package files remain unchanged. Each registered command took about 1.2 seconds locally, below its 60-second limit; these are single-run check timings, not performance benchmarks.

Exact reproduction from the repository root:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/a4-dirty-workspace-minimum/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -O -B research/a4-dirty-workspace-minimum/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/a4-dirty-workspace-minimum
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/a4-dirty-workspace-minimum --check-source-history

Any CPython 3.12 path may replace the local interpreter path. Expected markers include PASS_A4_DIRTY_WORKSPACE_MINIMUM, PASS_REGISTERED_RESEARCH_PACKAGE and PASS_RESEARCH_COMMIT_BYTES. Every verifier run generates a receipt bound to all six package files by SHA256. No receipt is handwritten.

Limitations and unrun obligations

Universal coverage comes from the written rank-saturation and restriction argument. The eight-call corollary depends on the predecessor's finite four-role certificate; a wholly written six-call exclusion is not supplied. The statement uses F2[A4], positive a/b calls and rank-one idempotent couplings. Nilpotent couplings, other fields, alphabets and targets are outside scope.

Native realization, frames, ownership, routing, hosting, precision/tape, stock, recursive costs, compiler and outer admission remain unproved and unpriced. No Lean, Docker, selected cold replay, independent human or formal proof-kernel acceptance is claimed. This is no new kappa or complete multiplication theorem.

Open next questions

Can the original four-role six-call exclusion be replaced by a wholly written contradiction? Start with compatibility of the two tight rank-three decompositions or the second radical layer. The missing compatibility argument is the blocker. This package does not reserve the question.

Related open work

• #5203 proves a four-call minimum for two nonidentity even S4 targets using every odd S4 paid label and arbitrary restored dirty workspace. This package instead uses positive a/b labels in A4 and four specified target roles. These are different alphabets and endpoints; no shared numerical value is compared.
• #3893 studies an adjacent S3 register rank floor with a different address group and endpoint; it supplies no competing value for this quantity.
• The completion refresh found nine other open account PRs created since the advance CLAIM: Prove the exact 60-call full signed ternary connector optimum (no κ) #5201, Prove the exact order-21 normal-subgroup call tariff (no new κ) #5202, Prove four-call optimum for all even S4 pairs with every odd label #5203, Prove exact seven-call minima for 360 Dic3 targets (no new κ) #5205, Prove the exact antipodal D16 spinor LP optimum (no new κ) #5206, Prove exact real Gaussian Schur fill signs for t <= 1/3 (no new κ) #5207, Prove the exact equal-nonunit dirty-fanout tariff (no new κ) #5208, Prove exact integer-transvection length λ(B^k)=2k (no new κ) #5209 and Prove a principal-ideal dirty-fanout obstruction over K[x,y] (no new κ) #5211. The remaining topics are a signed connector, order-21 and dicyclic targets, a spinor-code relaxation, Gaussian fill, dirty fanout and integer transvections. None studies this four-target A4 endpoint. No additional computations were run to compare other PRs. Sparse cached local notes and unpublished work remain outside exhaustive overlap coverage.

CI notes

Both hosted checks passed at contribution head 184dbcc0c5340f9b73f6abe988dbaf8b22da0b2b in run 39954056514: plan in 5m3s and the focused registered package replay in 1m13s. Run metadata confirms the exact head, completed status and conclusion success. The registered replay runs the warning-strict normal and optimized checks. Hosted execution is separate from scientific acceptance of the written proof and cited predecessor certificate.

Local registered/direct normal and optimized checks, metadata/pins, exact-head committed-byte/source-history preflight, lint/format, official export integrity and read-only Git-free replays also passed. All six exported package files remained unchanged. No hosted failure remains to report.

Merge danger

Two-way door. The change adds one isolated research package and touches no shared code, frozen predecessor or selected-frontier record. Archival merging is separate from scientific acceptance.

Credit

This contribution is produced by Nick Petalas. Sagar Thakrar #4032 supplies both radical-rank decompositions, the all-role floors and determinant congruence. Cinna Davis #2116 supplies the model, target, direct word and finite four-role exclusion. Retain Cinna Davis #2047/#2248/#2085, juper997 #1478, woklerx #1443, Gareth Davidson #1333, commdata2338 #1414, eumemic #288, Ali Romman #468 and the register-model lineage. Rank saturation and invariant-subspace restriction are classical linear algebra. Independent human and formal proof-kernel review are unclaimed. Apache-2.0.