[petalas]

Summary

Scoped algebraic proof with finite corroboration: the formal Boolean-sector perturbation family needs exactly one owned workspace field even when arbitrary rational scalar overwrites and free ordinary linear mixing are allowed. Any finite number of arbitrary dirty borrowed fields may be used and restored. This settles the scalar-overwrite workspace question explicitly left open by Ali Romman's PR #5658.

ordinary matrix coefficients lie in Q[J_0,...,J_(s-1)], with J_0=Q^2
paired evaluations Q=+1 and Q=-1 agree on all ordinary coefficients
zero owned fields => det(M_+) = +/- det(M_-)
required r-block perturbation => det(M_+)/det(M_-) = 3^r
r>=1 => contradiction

The new contribution is the enlarged-alphabet lower bound, produced by Nick Petalas. PR #5658's one-field butterfly is cited for the matching upper bound, without replaying or auditing its implementation. Credit Ali Romman, Dugongue's T6/T7 and C8, PR #5626's sector interface and the retained Nick Petalas, Cao, DaysSky, Cinna Davis, Ryanson, zxv and Jacob Alan Sussman native/controller/compiler lineage. Determinant and character methods are classical. No new kappa or selected-frontier change.

Evidence

Before, the workspace proof excluded nonunit scales such as 3/2 because it used invertibility modulo three. After, a characteristic-zero paired-character invariant covers every rational scale and even arbitrary matrices over the ordinary involution algebra. Singular ordinary gates remain covered by the determinant equality.

Local CPython 3.12.15 verification passes at head 257e79abda93a7d586439727e9a00435a43ca4af:

• Warning-strict direct normal and optimized checks, locally timed at 0.388 and 0.378 seconds. These are check durations, not speedup claims.
• Both registered commands and metadata/source pins. Markers: PASS_SECTOR_SCALAR_WORKSPACE and PASS_REGISTERED_RESEARCH_PACKAGE.
• 63 literal projector characters through six axes; 2,016 target parity checks through sixteen blocks; every one of 4,096 two-by-two ordinary matrices over the declared eight-element rational alphabet, including 402 singular matrices, with an independent permutation-determinant oracle and both query locations.
• 400 seeded words through five fields, with the invariant checked at all 9,600 instruction cuts; eight scope/rank controls and three invalid-matrix controls.
• PASS_RESEARCH_COMMIT_BYTES for all eight package files, including full source-history preflight against the ancestral main base.
• Official export integrity with declared inputs checked, followed by warning-strict normal and optimized replay in a read-only Git-free export with empty PATH. All nine exported files, including the export manifest, retain their hashes. Generated stdout receipts bind every scientific package file; no receipt is handwritten.

Exact reproduction from the repository root:

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

Use the local CPython 3.12 path on another machine. Only the standard library is needed. The standalone checker has no predecessor executable dependency. The source README/proof and originating card are pinned as historical provenance.

Limitations and unrun obligations

The all-parameter result is a written proof for fixed finite B-linear words in the formal faithful-sector algebra. Ordinary matrices have coefficients in the involution algebra; Q-dependent ordinary overwrites, nonlinear/adaptive instructions and discarded borrowed outputs are outside the contract. Exact dirty restoration is essential. The finite checks corroborate the written proof; they do not enumerate all programs.

No extension of the predecessor's r+1 CLEAR-count lower bound is claimed. No physical native implementation, full caller, precision/tape/stock ledger, complete supplier, selected cold replay, Lean or Docker replay was run. Independent human and formal proof-kernel review are unclaimed. The selected frontier and five framework assumptions are unchanged.

Canonical main was fetched at entry and before submission and remains de5ec2e7a6074068452f45b1a191b5c72832df4a, priority revision 8. Live text overlap included all 459 open PR records and 5,402 Discussion records with complete reported pagination, this account's latest 200 PR bodies, current handoff comments and batch-independent source reservations. All-main README searches and merged PR searches found the source's explicit question, rather than a delivered enlarged-alphabet theorem. Sparse local helper coverage is incomplete; arbitrary diffs and unpublished work are outside that novelty check. The advance CLAIM records the exact question and scope.

Open next questions

• Determine the minimum CLEAR count under arbitrary rational scalar overwrites. Start with one block and an erasure invariant or a cheaper complete word.
• Bind a profitable physical caller to this sector-supported demand, including all input/return, ordinary shifts, precision and storage costs.

Related open work

No open PR on the same Boolean-sector scalar-overwrite workspace question was found in the inspected live metadata. Adjacent open work is distinguished explicitly:

• #5877 counts two anticommuting query types and supplies a one-field shear construction. This result proves a workspace obstruction for the commuting ordinary-sector perturbation family; it does not compare or alter those query minima.
• #5879 permits clean erasure in a binary-affine paid alphabet. Its characteristic-two address model differs from the rational group-algebra determinant argument here; no common numerical optimum is asserted.

The completion refresh inspected this account's seven other open PRs created since the advance CLAIM: #5909 and #5911 through #5916. Their questions concern target-partition fees, colored batching, rounded-shear history, carry-state information, cyclic quotient size, carry frequency and tensor phase recovery. None studies this Boolean-sector scalar-overwrite workspace quantity or computes a conflicting same-instance value. No additional related line or numerical comparison is required by that refresh.

CI notes

Hosted CI has not certified this package. At commit 257e79abda93a7d586439727e9a00435a43ca4af, run 40506014706 finished with plan failed; replay skipped.

The only failed step was Onboarding helper tests. All ten failure/error entries are in unchanged tests/test_research_current_assignment.py, ending FAILED (failures=8, errors=2, skipped=22). The two errors are KeyError: 'follow_up_steps' and KeyError: 'catalog_first_step'. The eight assertion failures concern frozen-card/current-command routing (0 != 16), missing current/historical references (ValueError not raised in five cases), and sparse omission lists (two cases).

All ten failure/error headers match PR #5911's plan log exactly; both logs were inspected. This confirms the reported shared onboarding failure on main de5ec2e7a6074068452f45b1a191b5c72832df4a. The failure is outside this package. Maintainers will address it; no shared source, workflow or onboarding test was changed.

The package replay was skipped. Hosted replay has not certified the package; only the local checks listed above have run. No package-specific failure was reported by this run.

Merge danger

Two-way door. Blast radius: isolated. Eight new files in one research package; shared implementations, frozen predecessors, source archives and selected records are unchanged. Apache-2.0.