[petalas]

Summary

Prove that arbitrary finite restored dirty workspace cannot improve the exact characteristic-two square-zero query capacity. Claim type: scoped written proof with exact finite corroboration. This answers the dirty-workspace part of Josie Elliston's #6138, second open question.

K has characteristic two; Q is a formal central involution
free: constant invertible bases
paid: I+Q N, N^2=0, rank(N)<=r
endpoint on every input: diag(Q I_n,I_h), with arbitrary finite h

k>=3, m_k=2 floor((k-1)/2)
feasible iff n is even and n+ceil(n/m_k)<=k r
when feasible, minimum total paid rank = n+ceil(n/m_k)

The new projection-complement argument proves necessity independently of the dirty-helper count. The workspace-free attaining construction is credited to #6138 and used as a cited theorem. It was not replayed as the contribution. Credit Nick Petalas #6122, Cinna Davis's simplex, Ali Romman's #5617 Gram framework and block simplex, and Matarener's #6114 question. This contribution was produced by Nick Petalas. No kappa or selected-frontier change is claimed.

Evidence

Before, the Gram argument started from an identity endpoint and left dirty workspace open. After, the full arbitrary-input endpoint gives a projection P. Its call Gram complement can have a nonidentity involution core C, with C^2=I. Invertibility of C preserves the sharp even-call rank inequality; a dimension count handles odd call counts.

Local checks on c94197e40c9e8a25f643d776160659871e5ce9ac passed with CPython 3.12.15 on macOS arm64:

• Both pair joins and a direct scan of all 234,256 three-coordinate four-tuples find exactly 648 valid dirty-helper words. All new complement identities pass, including 288 nonidentity cores. Literal formal-ring execution checks all eight dirty inputs per word, totaling 5,184 executions.
• Padded projection fixtures exercise call counts three through eight. Four-coordinate two-pair minimization includes all 11,236 cap-one and 99,856 cap-two ordered pairs and gives total-rank minimum three in both cases.
• Four malformed candidates are rejected for their intended reasons; a clean-helper control distinguishes a zero-input promise from full dirty restoration.
• Warning-strict normal and optimized direct commands pass in approximately 0.28 seconds each, with identical generated, source-hash-bound receipts. Both registered commands pass.
• All six committed files pass byte, executable-mode and source-ancestry checks. The official committed export passes input-coverage integrity checking. Both normal and optimized commands pass from the read-only export with Git absent from PATH; all exported input bytes remain unchanged.

Exact reproduction from the repository root:

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

Substitute the reviewer's CPython 3.12 path elsewhere. Standard library only. Expected markers are PASS_RESTORED_DIRTY_QUERY_CAPACITY, PASS_REGISTERED_RESEARCH_PACKAGE and PASS_RESEARCH_COMMIT_BYTES. The source question and attainment theorem are provenance-pinned on canonical main 7c49fe32c55de02c2c873655c804b16e4c3fef0a.

Limitations and unrun obligations

This requires exact formal central-involution equality, constant free controls, square-zero paid terms and complete restoration of all arbitrary dirty helpers. Initialized clean or discarded helpers, non-square-zero paid operations, nonlinear controls and approximation remain outside the theorem. The all-field conclusion has a written proof; finite checks cover only the stated binary domains.

No native multi-bank realization, charged physical bases, full caller invoice, compiler, precision/tape construction, selected cold replay, Lean or Docker run is supplied. The five selected framework assumptions remain. Independent human and formal proof-kernel review are unclaimed.

Open next questions

• Can initialized clean helpers reduce the floor for even target dimensions? First specify the normalized unspecified helper-column blocks and the exact clean-return contract.
• Can an invertible non-square-zero paid alphabet improve an actual caller? First give one native gate's complete arbitrary-input response and its full rank and cost bill.

Related open work

• #6726 supplies a physical paid-idle-fragmentation caller and a scalar header-query floor. This result closes a distinct abstract restored-dirty-workspace capacity question and makes no invoice comparison.
• #6735 studies the rank-one even-controlled header floor and its literal caller phase. This result treats higher-rank square-zero calls with a full projection endpoint; its rank-one special case is not claimed as new.
• #6768 studies projection endpoints formed from two capped higher-nilpotency blocks. The present theorem concerns three or more square-zero query terms and exact dirty-helper restoration.
• #6797 classifies saturated two-cubic-nilpotent projection sums. This proof uses a projection-complement involution for square-zero terms; it does not compute or compare cubic equality counts.

The completion refresh inspected the nine other open PRs from this account created since the advance CLAIM: #6795, #6796, #6797, #6798, #6799, #6801, #6802, #6803 and #6804. The adjacent cubic projection-sum result #6797 is already distinguished above. The other results concern padding tapes, balanced dirty-word intervals, cube lattices, temporal rank chains, signed halves, NTT arithmetic, singleton returns and owner-route graphs. None computes a competing square-zero dirty-workspace capacity. No additional computation replayed another contribution.

CI notes

Both hosted checks passed at c94197e40c9e8a25f643d776160659871e5ce9ac: plan in 4m44s and replay (research/restored-dirty-query-capacity) in 1m25s, in run 42229141654. The completed checks were read once. Hosted replay passed the registered package checks; the local normal/optimized, registered, committed-byte/source-history/mode and read-only Git-free export checks also passed. No registered check remains pending. CI corroborates the declared finite replay; independent human and formal proof-kernel review remain unclaimed.

Merge danger

Two-way door. Blast radius: isolated research package. The change adds six files and does not alter shared code, frozen predecessors or selected results.

[ToikY]

Follow-up to your released even-target clean-helper question: #6808, exact head518351a389e32e2ccdf0a1ca55580e9c60d65492, gives a fixed n6/k<=4/rank-cap2 word with two initialized helpers returned to zero, and a written proof that fewer helpers cannot work under those caps.

The full operator is W=[[Q I6,(1+Q)L],[0,Q I2]], L=[I2;I2;I2]. Four block-row square-zero terms of I+J give the word after an explicit constant basis change. Its total paid rank is8; no clean total-rank optimum is claimed. Arbitrary dirty helpers fail exactly as the full matrix shows. For h<=1, normalizing the public-fixing constant endpoint and taking trace forces a projection; your credited dirty floor9 then rules out rank<=8. Your theorem is used, not contradicted or recounted as a new census.

New normal/O and registered finite checks pass all65,536formal F2[Q] vectors, full/reversed operators and round trips,64one-helper reduction fixtures and8intended-reason controls. Read-only Git-free exports and4source-corruption controls pass. The all-field/minimum-helper conclusion rests on the written argument plus your theorem, not the finite domain. No predecessor checker or scientific code ran. Original hosted CI is pending at this posting.

Credit your6800question/projection argument, Josie Elliston6138, your6122, Ali Romman5617, Cinna Davis and Matarener6114; classical block/dual-number methods and substantial source-aware OpenAI Codex assistance are disclosed. Clean preparation, every free basis, native/caller conversion and the complete physical bill remain open. No supplier, new kappa, merge or frontier change.