[petalas]

Summary

Scoped proof with finite corroboration: solve the canonical-checkpoint question left in Ross Richardson's merged #1287. The fixed cyclic-equivariant child contract and its aligned recurrence are cited premises. No predecessor result is audited or replayed.

A <= B: b(k) = k A
A > B:  b(k) = k B + [0, A-B, min(2(A-B),A)][k mod 3]
canonical segments k_1,...,k_s: optimum = sum b(k_j)
sharp extra price of one cut = (A-B)+min(2(A-B),A), when A>B

An explicit schedule attains every prescribed-cut optimum. With A=3, B=1 and five children, a canonical cut after child 2 costs 8 boundary units, retaining the entire 15-to-8 saving. A cut after child 1 costs 9. Canonical batch frames do not imply completed source-only partial echoes.

Only research/cyclic-checkpoint-cost/ is added. This contribution was produced by Nick Petalas. Credit Ross Richardson #1287/#1145, Sagar Thakrar #1227, mightyOil/Philipp #1203, Hanson Sun #1488, Cinna Davis #1391 and the retained restoration/echo lineage.

Evidence

Before: #1287 leaves prescribed canonical checkpoints open and supplies a three-state recurrence. After: a quantified closed form, arbitrary-cut optimum, sharp surcharge and emitted attaining schedules, with a generated source-hash-bound receipt.

Local exact checks pass 16,320 cut-mask/price optima, 9,216 one-cut pairs, 72 rational-price cases, 756 literal fixtures and 10,962 arbitrary-dirty basis/rational probes. Fourteen input/word rejections and three formula controls pass. Warning-strict normal and optimized checks each take about 1.1 seconds, below the registered 120-second limit. Both registered commands pass on CPython 3.12.15 / macOS 27.0.1 arm64. Committed-byte/source-history preflight, official export integrity and an optimized Git-free replay of the final commit with empty PATH pass. All exported bytes are preserved. A relocated optimized replay passes; changed-source, changed-receipt and changed-formula copies reject and preserve every byte.

From the worktree root, exact reproduction commands:

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

Expected markers: PASS_CYCLIC_CHECKPOINT_COST, PASS_REGISTERED_RESEARCH_PACKAGE, and committed-byte/source-history success. Elsewhere substitute the local Python 3.12 path.

Limitations

The theorem concerns fixed indivisible order-three children, uniform equivariant readers, symmetric nonzero-power prices and read-only canonical batch cuts. It does not cover adaptive mutation, opened child bodies or completed external-source partial echoes at those cuts. The inherited child/alignment contract and six-ADD rotation upper bound are cited premises. Physical caller, routing, coefficient access, precision, native hosting, proof-kernel validation and complete multiplication obligations were not run or discharged. No admitted supplier, new kappa or selected-frontier change is claimed.

Open next questions

• Identify a literal read-only consumer of a canonical partial batch record. The missing consumer blocks applying zero-surcharge cuts.
• Specify and price the stronger completed-partial-echo observation contract, starting with three children and one cut. Charge every source injection and restoration.

Related open work

Refreshed 2026-10-22T22:28:57.032236+00:00. Read this account's ten open PRs created since the CLAIM at 2026-10-22T22:14:16Z. None studies this same order-three canonical-checkpoint optimum. The merged #1287 is the direct predecessor; merged #1488 and #1391 treat different transport groups. Search coverage does not include unpublished work or arbitrary diffs.

CI notes

Both hosted checks passed at e36ff5b6431d3014b9ca0a1a49a5274d82788252: plan and the registered package replay. Successful CI run. Local registered normal/optimized checks also pass. CI verifies these registered checks; it does not establish proof-kernel or complete multiplication-theorem acceptance.

The card's historical baseline command fails with KeyError: bit_lower against the selected record; the current selected-baseline source-binding positive control passes. This is an existing orientation mismatch, not a package check failure.

Merge danger

Door: two-way. Blast radius: isolated. Only the new six-file research package is added.