[apoursh]

Change and mathematical scope

This answers Open next question 2 of Nick Petalas's #6653: do its equal-weight cubic repairs at arities 7 and 97 extend to an infinite family?

For every integer r>2, the fixed defect representation

sum_{i<r}(A+iB)^3 - r*A^3
  = -r*A^3 + (r/2)*(A+dB)^3 + (r/2)*(A+eB)^3

holds with rational directions exactly when r^2-3*n^2=1 for a positive integer n, with {d,e}={(r-1-n)/2,(r-1+n)/2}. The directions are automatically integers. An elementary descent proves that the recurrence (r,n)->(2r+3n,r+2n) from (1,0) exhausts the solutions.

An infinite subsequence lies outside all earlier infinite classes listed by #6653: 7,97,18817,262087,50843527,.... The first two remain Petalas's finite examples. Visiting directions e,d,0 gives three queries and three integer shears, restores arbitrary dirty sources, retains every original child and produces the eager output. With #6212's cited dyadic lower bound, the query minimum is exactly three at these arities. No shear-optimality claim.

Scoped written proof with exact checks; no new κ or supplier admission. This classifies the fixed weights, not arbitrary three-query repairs. Physical query/shear prices, coefficient preparation, routing, precision/storage and the complete caller invoice remain unpaid.

Task, sources and overlap

Canonical main was pulled at entry and fetched again before publication; both resolve to e54523517d0f03e58352e647c403a5af3e525a57. The full joint-query-controller card is retained, explicitly marked changed scope to this algebraic side question. Revision 8's PR5599 assessment differs from selected PR7000; the selected bound remains 363005817/100000000000 with its five assumptions.

Direct predecessor #6653 is merged, unchanged head 3ea9965e13457fdafc70b784305d026ebfa8faa2. Four exact source notes are declared live replay inputs and authenticated by SHA256 and Git blob ID. No predecessor executable is imported and no frozen file is edited.

Live all-state PR searches, current predecessor metadata and the full HEAD filename inventory found no later solution. #6212's Pell equation concerns two-query feasibility; #5209 concerns integer-transvection length. The dedicated source-overlap helper could not populate remote corpora because gh is absent; live connector searches supplemented it. Public Discussion retrieval did not provide complete November 12 comment coverage. These limitations are recorded, not treated as proof of novelty. An advance claim could not be posted through the connector (403); no reservation is claimed.

Validation

CPython 3.12.14 on macOS arm64; standard library only:

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

All four registered isolated, warning-strict normal/optimized runs passed. The primary engine checks the exact symbolic off-Pell residual, 96 recurrence terms, 32 terms outside the prior classes, arities 3–10,000, 12 algebra/domain controls and four in-memory source-byte mutations. The independent engine executes seven literal polynomial programs through arity 18,817, 40 dirty rational examples, 1,063,536 bounded direction pairs, 528 descent steps, nine program/algebra mutants and five domain controls. Its defects are accumulated from the children without the primary power-sum formulas.

Commit bytes, file modes and source history passed for seven package files plus four inputs. Official export integrity passed. All four direct runs also passed from a read-only, Git-free export with empty PATH; all 12 files (including the export receipt), directory modes and byte hashes remained unchanged. Selected source-binding orientation passed on 221 paths; the selected scientific cold replay was not run.

A separate Codex agent audited the proof and both checkers and found no blocker. This is not independent human or formal-kernel acceptance. Finite checks support the implementation; the all-arity argument is in PROOF.md.

The base still has the previously reproduced CI navigation mismatch: the selected record names make frontier7000-verify, while docs/ci-verification.md still names #6030. Pending #7276 covers that shared repair. This package leaves it unchanged; hosted CI success is not claimed.

Open next questions

• For non-Pell arities such as r=15, investigate unequal weights or general dyadic frames using Attain three dyadic cubic-repair queries at 17 new arities (no κ) #6653's node condition and weight valuations. This proof is not a four-query obstruction there.
• Price the three-shear schedule at r=7 or 97 under one pinned backend and Classify dyadic cubic repairs on slopes 0,d,2d (no new kappa) #6250's complete comparison before downstream expansion.

Credit

Prepared by apoursh with substantial OpenAI Codex and subagent assistance. Credit Nick Petalas (#6653 question/examples; #6200 fixed reader), apoursh (#6212 interface/lower bound; #6215), AAAmirinu (#6246), CubeJerry (#6250), and their retained predecessor lineage. Pell equations, descent and power sums are classical; no literature-priority claim. Apache-2.0.