[apoursh]

Change and mathematical scope

Answers the dyadic-coefficient question in Nick Petalas's #6200. For the fixed r-child program y += a^3; a += b, a source-restoring repair with two appended cube queries over Z[P^-1] exists exactly when
(2r-1)^2 - 6k^2 = 3 has a positive integer solution k and every prime divisor of k is inverted. The proof covers arbitrary nonmonic forms: rescaling cannot hide the forced reduced query-weight denominator k.

For dyadics, two queries are possible only at r=2. An integer construction attains the exact dyadic minimum of three for r>=5, r=2 mod 3. At r=14 it replaces #6200's non-dyadic weights +/-91/11 with three integer queries:

a -= 9*b
y -= 91*a^3
a -= b
y += 91*a^3
a -= 4*b
y -= 2730*b^3

The program starts at the actual drifted source a_entry+14b, retains all fourteen original readers, restores arbitrary source state, and preserves arbitrary incoming y. A separate integer construction gives four queries for every r>=3. Explicit dyadic three-query examples also cover r=3,4; the general three-versus-four classification is not claimed.

Scoped proof with exact finite corroboration; no new kappa or backend saving. Source operations, weighted cube queries, coefficient preparation, source access, private restoration, routing, precision, storage and dispatch remain separately paid. The complete logical event tariffs are written out. There is no admitted physical caller or favorable complete bill.

Task and overlap

Base main: a5bff3bf133e482541c09eba4e773884bdb2b1b7, priority revision 8. Published head: 9c159482e0495b6a97e17e629debe66a0ceb9466. Direct predecessor #6200: 8b705d15869ba12d55adcb9075c68a2c7d44602d.

This explicitly changes the joint-query-controller card to a bounded coefficient-feasibility side result answering #6200's first open question. It does not satisfy the card's full supplier gate. Canonical claim.

Bounded overlap covers the newest twenty PRs through #6211, all-state searches for “cubic dyadic”, “cubic repair” and “6200”, and the newest fifteen visible handoffs. The sparse local helper searched 30/8623 HEAD note paths and had no usable remote cache. This is not a novelty certificate. #6192/#6208 concern linear cyclotomic operators and #5084/#5132 a different norm-six feature factorization; no computations or savings are combined. Before submission, canonical main and #6200's head were unchanged, with no newer PR or later competing visible claim.

Only research/dyadic-cubic-repair/ is added. Two cited predecessor documents are retained byte-for-byte for the exact offline contract, with source URLs and SHA256 checks. The pending head is not asserted to be a publication ancestor; the source-history check covers the base provenance only. No predecessor executable is imported.

Validation

CPython 3.12.14, macOS arm64, standard library only:

python3.12 -B scripts/research_package.py --package research/dyadic-cubic-repair
python3.12 -B scripts/check_research_commit.py --package research/dyadic-cubic-repair --check-source-history --check-modes

• All four registered commands pass: both separately authored checkers under isolated, warning-strict normal and optimized Python.
• Primary: 255 universal integer programs, 85 congruence-family programs, arbitrary-input eager/deferred equality and literal inverses; seven Pell fixtures, 584 nonmonic normalizations and ten semantic controls.
• Independent: defect from binary concatenation of literal reader blocks and coefficients from exact elimination; 136 arities, eight Pell fixtures, very large arities, r=3,4 examples, four rejection controls.
• Actual altered copies of each retained source document and a changed source-head manifest all fail with the expected diagnostic.
• Exact committed bytes/modes and source history pass for all ten package files. Official export integrity passes. Both checkers pass normal/optimized from the read-only Git-free export with empty PATH; all eleven exported file bytes and all file/directory modes remain unchanged.

The written argument proves the all-r statement; finite scans do not. A prior binding-only orientation pass is reused after checking all 1227 covered paths unchanged. No selected-supplier cold replay, Lean/Docker run, independent human review, formal-kernel proof or scientific admission is claimed. Hosted CI status is separate from these local checks.

Credit

Prepared by apoursh with substantial OpenAI Codex assistance in derivation, proof, separately authored checkers, mathematical audit, documentation and publication. Nick Petalas retains #6200's defect, rational theorem, Pell criterion, schedule method and dyadic handoff. Sricharan Vagvala / theon199 retains #1532's quadratic fixed-reader interface; Adam Gall / #1205 and Hanson Sun / #1246 retain the linear lineage. Classical binary-form decomposition and localization retain their background credit. Apache-2.0; no historical-priority claim.

Open next questions

• Classify further arities admitting three dyadic queries, or prove that some require four. Begin with the three-form coefficient equations and odd-prime valuations, allowing nonmonic forms and requiring a legal restoring schedule.
• Bind an actual indivisible-reader caller and price the complete r=14 integer repair against eager restoration. The missing backend and favorable complete invoice block any paid-work or bound-improvement claim.

Detailed relevance, first steps and blockers are in README.md and TASK.json.