Prove one-query endpoint and paid-scan bounds for the triangular carrier

[RossRichardson]

Summary

• For the fixed triangular carrier C_f=[[Q_f−I,N_f],[Q_f,0]], exclude an all-size one-query factorization over a fixed finite uniform-tensor physical-monomial endpoint dictionary.
• Allow growing nonuniform endpoint controllers and prove K_G·K_F≥4^f, hence at least 2f scalar bank ADDs. Growing paid O(f) ADD words remain permitted.
• Add only research/one-query-endpoint-scan/, including the actual 19 native 64×64 Gaussian base matrices and a restored-helper argument with its queried-width boundary.

Scope and limits

The helper extension retains its carrier-row vanishing premise and queried width ≤M. Changed cuts, multi-bank queries, wider queried helpers, nonmonomial endpoints and cross-cut sharing remain outside the theorem. Written all-parameter proofs are distinguished from finite base checks.

The repeated, varied-primal and primal/dual-route five-query searches are an archival appendix. The registered command hashes their receipts but does not rerun those searches; they do not establish a general five-query impossibility. No complete-caller stock floor, new κ, selected-frontier change, physical supplier admission or formal-kernel theorem is claimed.

See the proof, coverage and limits, and task, scope change and open next questions.

Test plan

• Fresh review at bf793d8739ad26f2aaba023de6bebca50fc67356: python3.12 -B scripts/check_research_commit.py --package research/one-query-endpoint-scan --check-source-history and python3.12 -B scripts/research_package.py --package research/one-query-endpoint-scan --check-only pass. These establish committed bytes, history, metadata and pins.
• Existing hosted CI run 39248202294 is completed/success at that exact head, including registered normal/optimized replay. Reproduce with python3.12 -B scripts/research_package.py --package research/one-query-endpoint-scan. This review reused that run; it did not rerun the package suite.
• Separate AI review inspected the tensor-ray, support, ADD and dirty-helper arguments and their actual native-source bindings; no actionable defect was found in that scope.

The earlier author-reported local and Git-free export checks are recorded in the canonical finding. AI review is not independent human or formal-kernel acceptance.

Sources and credit

Canonical base 20ecf2891d80b5acf18d980b2164fb44ea7faa90, priority revision 6. SOURCES, BINDINGS and RELATED-WORK retain exact ancestry and explain the changed task scope and nearest overlaps. Live Discussion/open-PR overlap coverage is partial, not a novelty certificate. Canonical publication.

Ross Richardson with substantial OpenAI Codex assistance; Apache-2.0. Preserve icekylinx/PR531 and Rohan Arun/kernel92 compiler notices and recorded ancestry; Chris R/Clear1nsight #3987 and ToikY #4002 signed-lift/phase credit; and Clear1nsight's distinct #3989/#4045 interface/volume work. Earlier #3630/#3632/#3645/#3649/#3655 and separate BIT #4021/#4023 are excluded from these new claims. No merge requested.

[RossRichardson]

Reviewed exact head bf793d8739ad26f2aaba023de6bebca50fc67356 against canonical main 20ecf2891d80b5acf18d980b2164fb44ea7faa90.

No actionable mathematical defect found in the assigned endpoint-proof scope. Separate AI review checked the finite-dictionary tensor-ray argument, sequential untouched-column vanishing, support uncertainty K_G K_F >= 4^f and the resulting >=2f scalar-ADD bound. The arbitrary-dirty-helper argument retains its carrier-row vanishing premise and queried width <=M. The actual native phase/tensor bindings and full-column Gaussian checker agree with the needed hypotheses. Growing paid controllers remain permitted.

The archived 24^5 / 26^5 / 26^5 five-query searches were not independently reexecuted or validated here; they remain archival evidence, not a general five-query impossibility. No complete-caller stock floor, physical supplier, formal-kernel result or new kappa is established.

Fresh committed-byte/source-history and registered metadata/pin checks pass. Existing CI run 39248202294 was read back completed/success at this exact head. The review used written-proof/source inspection, not a fresh package-suite run.

Size: 17,146 added lines across 41 files; 14,157 JSON lines and 2,105 Python lines, 400,558 bytes total. The description should use the required four-section format, remove the invalid -standalone commit suffix, and link the canonical finding. Prepared by Ross Richardson with substantial OpenAI Codex assistance; no independent human acceptance is claimed.