[apoursh]

Change and mathematical scope

For every arity r>=3 and every odd modulus m>1, the fixed cubic-reader defect is a sum of at most three weighted cubes modulo m, using primitive integer source rows. The explicit source-frame construction uses integer determinant-one shears and exact inverse words, restoring both arbitrary dirty sources exactly while reproducing the eager output modulo m. At most 31 source shears and three nonzero query slots suffice; neither is a modular minimum claim.

This closes one possible obstruction route in Nick Petalas's #6653 Open next question 1: nonexistence in a finite odd quotient cannot prove an unrestricted four-query lower bound. Any finite collection of congruences is covered by one witness for their least common multiple. The result does not settle exact global dyadic feasibility at r=15 or rule out valuation arguments constraining one fixed global representation.

The proof combines a three-cube identity integral at 3 with the rational extension of #6215's pure-B identity integral away from 3. CRT patches a common five-shear word, producing actual integer source frames rather than assuming matrix lifting. An exact-versus-modular counterexample makes the scope boundary executable.

No new κ, physical cost saving or supplier admission. The proof records every retained child, preparation/restoration frame and query; physical prices and remaining caller overhead are unknown. Modular output equality does not meet the original exact-output contract.

Task, sources and overlap

Canonical upstream was pulled at entry and refreshed before publication, at e54523517d0f03e58352e647c403a5af3e525a57. Task joint-query-controller, scope key 58d8c9133832c5149bdc, explicitly changed scope to a bounded obstruction-method side result; the exact card and narrow family decision are in TASK.json.

Only research/cubic-local-solubility/ is added. Three exact live sources are pinned at the base: #6653's node lemma/open question, #6215's identity/restricted modulo-nine obstruction, and #6212's general-frame contract. Neither #7475 (exact r24 repair) nor #7478 (fixed-weight Pell classification) is a dependency.

Fresh all-state PR body/title searches found no duplicate. The dedicated overlap helper could not obtain remote corpora because gh is absent, and cached Discussion pages do not establish complete current claim coverage. Those limits are recorded; no global novelty guarantee is claimed.

Validation

CPython 3.12.14, macOS arm64, standard library only:

python3.12 -B scripts/research_package.py --package research/cubic-local-solubility
python3.12 -B scripts/check_research_commit.py --package research/cubic-local-solubility --check-modes --check-source-history
python3.12 -B scripts/export_research_package.py export --package research/cubic-local-solubility --ref HEAD --output /tmp/cubic-local.tar
python3.12 -B scripts/export_research_package.py check /tmp/cubic-local.tar

All four registered normal/optimized runs passed. Primary checks: two symbolic identities over Q[r], 3,682 literal integer-shear CRT certificates, seven algebra/program rejection controls, eight domain controls and three source-byte mutations. The independent checker imports neither primary nor predecessor code and instead uses primitive integer lifts and Bezout frames: rational expansions at r=3..128, 1,344 modular certificates, 5,376 dirty-input runs, three rejected faulty variants and a modulo-33 certificate that demonstrably fails exact output equality.

Committed bytes, executable modes and source history passed for all ten package/input files. Official export integrity passed. All four direct runs passed in a fresh read-only, Git-free export with empty PATH; all eleven files, including the receipt, retained their bytes and modes. The written proof establishes the infinite quantifiers; finite checks corroborate implementation. Separate Codex agents reviewed the proof and implementations; independent human/formal-kernel acceptance is not claimed.

Selected source-binding orientation passed; the selected scientific cold replay was not run because the selected construction is unchanged. Hosted CI is independently blocked at the base by test_current_ci_section_matches_selected_record: selected frontier7000 versus stale frontier6030 navigation. The same sole failure appears in #7475's planning job and #7478's planning job, before package replay. Existing #7276 covers that shared repair; this contribution does not duplicate it or weaken its test.

Credit and next questions

Prepared by apoursh with substantial OpenAI Codex and subagent assistance. Credit Nick Petalas (#6200 reader/defect, #6653 node lemma and question), apoursh (#6212 contract, #6215 pure-B identity and restriction), and their retained predecessor lineage. Classical CRT, localization and elementary matrix methods are not claimed as new. Apache-2.0.

Open next questions: analyze global valuation compatibility across one exact rational triple at r=15, or specify a restricted direction contract that can still support a modular obstruction. The blockers are global dyadic integrality and precise restrictions preserved by reduction/patching, respectively.