[petalas]

Summary

Prove the exact appended cube-query cost of restoring a fixed cubic-reader batch over rational source frames. For every integer r>=2, retain all r children y += a^3; a += b and reproduce the eager endpoint on arbitrary incoming y,a,b.

q = (2r^2 - 2r - 1)/12
rational repair queries = 2 if q is a rational square, otherwise 3
real repair queries = 2

Claim type: scoped mathematical proof with exact finite corroboration. At r=3, two real queries suffice but three rational queries are necessary. At r=14, an explicit two-query rational program restores the actual dirty source. It saves eleven source-shift events while adding two cube queries. No backend cost saving, supplier admission or new kappa is claimed.

The new step is the arithmetic-sensitive cubic criterion and its restoring schedules. Credit Sricharan Vagvala / theon199's PR1532 for the quadratic fixed-reader/paid-repair interface, Adam Gall / PR1205 and Hanson Sun / PR1246 for the linear lineage, and classical binary-form power-decomposition results. No predecessor computation is rechecked as this contribution and no general Waring-rank theorem is claimed.

Evidence

• Before allowing only rational coefficients, the real relaxation predicts two queries at every arity. After imposing the rational interface, coefficient comparison proves the exact square criterion and the three-query floor at r=3.
• CPython 3.12.15 passes 63 symbolic arities with eager/repaired/inverse endpoints, 1,798 dirty-input executions with inverse checks, 3,933 rational slope-pair coefficient cases, seven semantic controls and seven invalid-input controls.
• The exhaustive square-criterion scan covers every integer 2<=r<=10,000 and returns exactly 2,14,134,1322 as two-query arities. The all-r theorem rests on the written proof, not this scan.
• Normal and optimized direct checks take 0.26 and 0.28 seconds locally. Both registered checks pass, each below one second and its 120-second bound. Metadata/pins and committed-byte/source-history checks pass for all six package files at 8b705d15869ba12d55adcb9075c68a2c7d44602d.
• Official committed export and input-coverage checks pass. Normal and optimized direct replay pass from a read-only Git-free export with empty PATH in 0.273 and 0.271 seconds. All seven exported files retain their hashes.
• Every direct run generates a stdout JSON receipt bound to SHA256 hashes of the package files. No receipt is hand-written.

Exact reproduction from the repository root:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/cubic-reader-rational-repair/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -O -B research/cubic-reader-rational-repair/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/cubic-reader-rational-repair
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/cubic-reader-rational-repair --check-source-history

Another machine may substitute its CPython 3.12 executable. Direct success is PASS_CUBIC_READER_RATIONAL_REPAIR; registered replay ends with PASS_REGISTERED_RESEARCH_PACKAGE.

Limitations and unrun obligations

The original calls are indivisible. Source maps are linear and independent of the output; there is no intermediate observation. Cube-query count excludes additional source transformations and coefficient preparation. PROOF.md gives the complete symbolic event tariff, with every original call and repair shift charged. Opening the child interiors permits a direct endpoint computation outside this comparison.

No supported supplier with these cubic readers is identified. Physical native realization, precision/storage, routing, compiler closure, actual backend pricing and supplier admission remain unproved. No Lean, Docker or selected scientific replay was run. Independent human and formal proof-kernel review are unclaimed.

Canonical base a5bff3bf133e482541c09eba4e773884bdb2b1b7, priority revision 8, was refreshed before committing. The selected record resolves to PR6030 at .003125804 and is unchanged. The card changes explicitly to this nonlinear-interface side result. Live handoffs, reservations, account contributions since October 22, full main README searches and targeted merged/open PR searches found no matching result; sparse local-note and unpublished-work coverage remains incomplete.

Related open work

The completion refresh inspected this account's twelve other open PRs created since the advance CLAIM, #6192 through #6199 and #6201 through #6204. None studies fixed cubic-reader restoration or computes this rational cube-query cost. No extra computations were run to check their results. PR1532 is the merged quadratic predecessor and supplies the credited interface.

Open next questions

1. Which dyadic source and query coefficients attain the same two-query count? Start with prime-denominator tests on the normalized slopes and weights forced by the proof. Rational existence does not yet give a dyadic implementation.
2. Can an admitted indivisible cubic-reader backend make the r=14 tariff profitable? Pin one return map and price every displayed cube accumulation, source shift and physical coefficient operation. No such backend is supplied here.

CI notes

Hosted CI passed at 8b705d15869ba12d55adcb9075c68a2c7d44602d: plan and replay (research/cubic-reader-rational-repair) both succeeded in run 40911961999. The checks were confirmed once with gh pr checks 6200 --repo CrocSwap/integer-mult-bounds. Local normal/optimized replay, metadata, publication bytes/source ancestry, export integrity and read-only Git-free replay also pass. CI certifies the registered replay at this head; independent human and formal proof-kernel review remain unclaimed.

Merge Danger

Door: two-way.

Blast Radius: isolated. Only research/cubic-reader-rational-repair/ is added; no shared runner, selected record or frozen predecessor changes.