[petalas]

Summary

Scoped algebraic proof with exact finite corroboration: clean helpers, scalar copying and reset, discarded garbage, and changing live-register counts cannot reduce the binary-affine paid-call tariff of Gareth Davidson's #1333.

minimum calls = sum_j rank_F2([A_j-I | b_j])
delta(M) = rank(H(M) - epsilon(M) tensor I)
delta(UV) <= delta(U) + delta(V), including rectangular U,V

The paid alphabet remains I+([g]+1)uv^T, with binary rank-one idempotent coupling and homogeneous address displacement rank one. The changed free alphabet allows every rectangular binary scalar map. Only the designated output rows are required; helpers may be erased or discarded. Each free map has defect zero and each paid call has defect one. Output selection supplies the lower bound without invertible normalization. The upper construction is cited from #1333, PROOF.md Section 4, rather than replayed or claimed as new.

The proof and checker are in research/clean-affine-erasure-floor/. This contribution was produced by Nick Petalas. Credit Gareth Davidson for the original affine model, displacement-rank method and direct factorization, and its eumemic #288, Ali Romman #468, David Govea #720 and classical rank/reflection lineage. No general literature-priority claim, new kappa or selected-frontier change.

Evidence

Before, #1333 requires invertible free scalar maps and a complete restored-dirty boundary. After, the written rectangular augmentation-defect proof covers arbitrary clean allocation, singular scalar maps and retained outputs for every finite dimension and circuit length.

Local CPython 3.12.15 warning-strict normal and optimized checks pass. Each direct registered command took less than one second, below its 120-second limit. The checker enumerates 74,384 compatible small rectangular products, 682 free scalar maps, 315 paid factors, 48 new clean/erasing circuits, 2,304 faithful basis columns, 192 omitted-call controls and eight boundary controls. A separately enumerated row-span oracle checks the small-matrix ranks. Packed full formal multiplication and a set-based faithful regular-action interpreter check the new circuits. Finite cases corroborate the universal written proof; they do not enumerate all circuits.

The registered runner, metadata/source-pin checks, committed-byte/source-history preflight, official eight-file export with both declared inputs, and optimized read-only Git-free export replay pass at 5275d72e56afe7d22252f2a8bd7690e9a9344ded. The verifier emits a generated receipt bound to the six package source hashes. No predecessor executable or scientific census was replayed.

From the worktree root:

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

Substitute another CPython 3.12 executable elsewhere. Expected markers are PASS_CLEAN_AFFINE_ERASURE_FLOOR, PASS_REGISTERED_RESEARCH_PACKAGE and PASS_RESEARCH_COMMIT_BYTES. The standard library suffices. The source-binding orientation command also passed; it was not a selected scientific replay. Canonical main remained de5ec2e7a6074068452f45b1a191b5c72832df4a, priority revision 8, at submission refresh.

Limitations and unrun obligations

The circuit is finite, fixed and linear over the full formal group algebra. Free maps are scalar and address-independent. Nonlinear/adaptive operations, postselection and nonzero address-dependent advice are excluded. The homogeneous representation supplies necessary conditions only; it is not a formal endpoint oracle.

This is an abstract obstruction, not a physical price, supplier admission or complete multiplication theorem. Native address/scalar costs, movement, ownership, precision, tapes, stock, recursive profiles and outer use remain unproved. Independent human and formal proof-kernel review are unclaimed. Lean, Docker, native execution and the selected cold replay were not run. Both registered commands ran locally. No standalone linter is installed; warning-strict interpreter checks and AST inspection were used. The bounded CPython allocation deviation is documented in the README.

Open next questions

• Can an actual address-dependent native operation escape the defect bound at favorable complete cost? Start by computing one operation's defect and full input/output invoice. Physical legality and complete pricing are missing.
• For another specified representation, is the clean/erasing rank bound attained? Start with a cited direct construction and retained-output contract. A matching formal upper construction is required for an exact tariff.

Related open work

No open PR studying this binary-affine clean/erasing tariff was found in the live affine and #1333 searches. #2444 supplies general noncommuting exact-search tooling for a different KB2 candidate; it does not price this affine alphabet or establish this retained-output theorem. #5842 implements a physical runtime affine shear, a different operation and cost contract; no shared numerical quantity is compared. No extra computation was run to validate either contribution.

The completion refresh inspected this account's nine other open PRs created since the CLAIM at 2026-10-27T10:37:56Z, #5876–#5878 and #5880–#5885. None computes this binary-affine clean/erasing tariff. The closest boundary comparisons are:

• Prove sharp clean storage for reversible Gaussian overwrite (no κ) #5880 proves clean-bit capacity for reversible destructive Gaussian overwrite. It prices retained kernel information under an invertibility requirement; this PR grants scalar erasure and counts binary-affine calls. No common quantity or numerical disagreement is asserted.
• Prove exact additive cost for two anticommuting query types (no new κ) #5877 proves typed query-call costs for anticommuting involutions over characteristic different from two. Its algebra, paid instructions and shear endpoint differ from this characteristic-two affine group-algebra target; no common numerical cost is compared.

No additional mathematical computation was run to check those contributions.

The advance CLAIM, source pins and bounded overlap coverage are recorded in TASK.json. The cached card-wide search covered 426 open PRs and 5,331 Discussion records with complete remote pagination; its sparse local note coverage was incomplete and supplemented by canonical-main README searches. Search absence is not proof of novelty.

CI notes

Hosted status is plan failed; replay skipped on exact contribution head 5275d72e56afe7d22252f2a8bd7690e9a9344ded in run 40491123999. The plan job's only failed step is Onboarding helper tests. Its log ends FAILED (failures=8, errors=2, skipped=22), exit code 1. All ten failure/error entries belong to unchanged tests/test_research_current_assignment.py. The two errors are KeyError: 'follow_up_steps' and KeyError: 'catalog_first_step'; the eight assertion failures concern frozen-card resolution, missing current/historical references and sparse omission lists.

The ten failure/error headers and counts match PR #5877's plan log exactly. This is the reported shared main de5ec2e7a onboarding failure; maintainers will handle its repair. No shared tests or repository files outside this package were changed or rerun in response.

The replay job was skipped. Hosted replay has not certified the package; only local scientific checks have run. All registered commands and publication preflights listed above passed locally. This records publication/check status and does not weaken the scoped mathematical claim or claim a hosted pass.

Merge Danger

Door: two-way. Blast radius: isolated.

Only six files in one new opt-in research package are added. Shared implementations, frozen sources, predecessor credit and selected-frontier files are unchanged.