[petalas]

Summary

Extend Ross Richardson's PR4057 one-query obstruction for the triangular carrier C=[[Q-I,N],[Q,0]] to the full width range below the two carriers' combined dimension. Claim type: scoped written proof with exact finite corroboration. No new kappa or supplier admission.

 Under the explicit carrier-row vanishing / sparse-endpoint contract:
- excluded queried widths: w <= M
+ excluded queried widths: w < 2M

For unitary Q,N and endpoint carrier-block column supports s,t, every factorization satisfies w>=2M or s*t>=min(mu_Q^-2,mu_N^-2). The new step is 2M+K-w<=K: the invertible input endpoint must fit every untouched coordinate column into the K-dimensional helper space. Helpers are arbitrary independently dirty inputs and return exactly. Arbitrary active-coordinate positions are permitted.

An arbitrary query on all 2M carrier coordinates attains the threshold in the relaxed algebraic model. This is not a native implementation or a physical bank merger. Exact rational examples show that the support inequality must remain strict.

The result was produced by Nick Petalas. Credit RossRichardson / PR4057 at bf793d8739ad26f2aaba023de6bebca50fc67356 for the carrier, untouched-column equations, support method, native amplitudes and earlier width scope; icekylinx/PR531, Rohan Arun/kernel92, Clear1nsight/PR3987, ToikY/PR4002 and the retained native/compiler lineage. Rank-nullity and support uncertainty are classical. The predecessor's native evidence is cited without replay or audit. Its threshold 4^f and ADD coefficient 2f are inherited; their extended width domain is new here.

Advance CLAIM. Canonical main was refreshed before publication and remains 7c49fe32c55de02c2c873655c804b16e4c3fef0a, priority revision 8, selected PR6030.

Evidence

Before this change the stated variable-width result covers w<=M. The new written dimension proof covers M<w<2M as well, with arbitrary finite restored helper dimension.

Local CPython 3.12.15 on macOS 27.0.1 arm64 passes:

• Every one of 147,456 complete monomial endpoint pairs over F3 in dimension four requires all four active coordinates.
• All 50,304 permutation/active-set cases with M=2 and K=0,1,2 pass. The 33,720 excluded cases with w<4 include 15,696 at the newly covered width w=3. All 3,144 feasible active sets receive literal attaining queries.
• All 64 dense invertible operator pairs give 960 attaining queries and 6,400 checked arbitrary-input basis columns. Thirty-two one-sparse separation fixtures pass.
• Eighty rational-vector uncertainty checks, forty equality cases, exact two-sided carrier inverses, the width-2M attainer and strict-support/monomial boundary controls pass. Four malformed query descriptors reject.
• Warning-strict normal and optimized registered replays pass with byte-identical output. Each exact command was timed locally, at 0.375 s and 0.370 s respectively, below its 60-second timeout. These are timeout checks, not a performance comparison.
• Generated source-hash receipt creation and three receipt-path failure cases pass. Syntax/function-size inspection, staged whitespace, exact committed-byte comparison and declared source-history checks pass. Ruff was absent; no lint or static type-checker result is claimed.

Reproduce from the worktree root:

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

Another host may substitute its CPython 3.12 executable. Only the standard library is required. Expected markers are PASS_WIDE_ONE_QUERY_THRESHOLD, PASS_REGISTERED_RESEARCH_PACKAGE and PASS_RESEARCH_COMMIT_BYTES. Default verification writes nothing; optional receipts are generated and bound to all six package files.

Limitations and unrun obligations

The universal theorem is a written algebraic proof, with finite corroboration. It requires the stated carrier, coordinate query and endpoint support/separation contract. Dense active subspaces, multiple interleaved queries, discarded helpers, rounded execution and changed carrier maps are outside its scope. Rectangular helper dictionaries must actually satisfy the declared column bounds.

No native widened query, physical acquisition/restoration word, stock, hosting, precision/tape implementation, uniform compiler or complete paid supplier bill is supplied. No selected cold replay, Lean or Docker run is claimed. The selected frontier and five framework assumptions remain unchanged. Independent human and formal proof-kernel review are unclaimed. No local proof obligation is deferred solely to CI.

Open next questions

Can a physically legal query of width at least 2M improve a complete matched caller bill? Start with one actual widened query bank, its acquisition word, every arbitrary dirty input and exact restoration. Native realization and a favorable full invoice are the blockers. The relaxed attainer does not supply them. This contribution stops at the first width-domain improvement.

Related open work

• PR6400 excludes a single repair cut for a fixed five-call reflection border. This result concerns one arbitrary-width query for PR4057's triangular carrier and computes no repair-cut tariff.

• PR6596 removes an ownership cap for linear BIT coordinate-support obstructions. Its endpoint and quantity differ from this triangular carrier's one-query width threshold; no numerical comparison is made.

Completion refresh inspected all nine open account PRs created since the advance CLAIM, including this PR. None studies the same carrier-width threshold. No same-instance numerical disagreement was found, and no additional computation was run to check another contribution.

Targeted live open-PR searches found no other PR studying this carrier's query-width threshold. The cached card scan covered 540 open PRs and 6,569 Discussion records; its sparse local scan omitted 8,779 notes. Separate all-main object/formula searches and merged-PR searches covered relevant README text. These checks do not establish novelty against unpublished work or arbitrary historical diffs.

CI notes

Hosted Registered research packages run 41626477068 passed at exact head 063d1bdb9b1010a1cabf43a21d2f1860a0152ad9. Both plan and replay (research/wide-one-query-threshold) succeeded. The completed checks were read once with gh pr checks 6601 --repo CrocSwap/integer-mult-bounds. All registered commands also passed locally; no failed job or CI-only proof obligation remains.

Merge Danger

Door: two-way. Blast radius: package. This adds six files under research/wide-one-query-threshold/ and changes no shared implementation or selected record.