Research: two clean checkpoints reduce the 50-factor leaf bill 1728 to 1248 (no new kappa)

research/two-clean-checkpoints/CLAIM.md

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+# Sprint claim
+
+- Track: recursive interfaces; OS-random assignment seed: **50**.
+- Base: `66849e1475c930fd7732b1a0067ffee7e1e95f60`.
+- Start: 2026-10-13 05:45 UTC; planned finish: 06:30 UTC. Final quarter reserved for verification and handoff.
+- No pinned coordination issue exists; claim recorded locally.
+- Object: ordered binary product-shear trees with two initially clean, reusable arrays, arbitrary remaining dirt, and reserved ancestor connectors.
+- Method: exact interval/resource recurrence, literal forward/inverse words, independent exhaustive tree enumeration, and exact noncommutative/rounded execution.
+- Seed-selected fixture: 50 ordered factors, six total scratch arrays, two initially clean versus one. Weighted small fixtures provide separate controls.
+- Deliverable: one research note, executable construction, exact evidence, full cost recurrence and explicit clean-provisioning obligation; no new kappa claim.
+
+## Candidates considered
+
+1. Borrow idle ancestor connectors in all-dirty echoes. Harder lifetime proof; pending #730/#758 already cover adjacent static scheduling classes.
+2. Two reusable clean checkpoints at fixed total scratch (selected). #714 has one clean root; #788 permits one clean array to move between sequential subtrees. Neither allows a second clean connector while the first remains reserved.
+3. Optimize #788 with dispatch-aware prices. Useful but less directly addresses restoration work and depends on an actual cost model.
+
+The selected search changes the explicit initial-space contract, not just computation or the random seed. Domain: ordered binary trees, positive integer leaf prices, fixed total scratch and clean counts. Dynamic programming exhausts every split and legal connector type; it does not search arbitrary reversible programs.
+
+Overlap reviewed: #628, #663, #673, #689, #714, #730, #744, #758, #784, #788; initial open PR metadata through #794; full 618-open-PR overlap refresh through #799. The ledger ends at #626; it records no maintainer review of these pending interface PRs. The selected result remains #626/#543 at kappa 0.001635527380375392.
+
+## Result
+
+The two-clean recurrence attains 1,248 complete leaf calls at 50 factors and six scratch arrays, versus the independently checked one-clean minimum 1,728. Full-capacity unit cost is exactly `4^(s-c)3^c`. Complete dispatch/provisioning/precision costs and inherited obligations are stated. Exhaustive, symbolic, actual rounded-leaf and corruption checks passed; see VALIDATION.md.
+
+Publication overlap refresh: 638 open PRs through #825; #788 remains the sole reusable one-clean comparator and its pinned head is unchanged. No pinned sprint issue appeared. Source pins and credit distinguish this changed two-clean contract from pending all-dirty work.
+
+Concurrent #805 (VictorLiwentao, `ef34bd5b32fcf2d5640dae90b5b8f28488738d5a`) supplies a different in-place real cyclic leaf plus dirty scalar. Its contract and execution prices differ from the additive leaves optimized here; no source is consumed and no cross-contract runtime comparison is claimed.
+
+Result branch: https://github.com/Better-Call-Paul/integer-mult-bounds/tree/research/two-clean-checkpoints-seed50/research/two-clean-checkpoints
+
+Verification extended beyond the planned finish to resolve local dependency
+configuration and complete the selected cold replay. Remaining unrelated
+historical searches were stopped after `verify-certificates` and
+`verify-ternary` passed; no full-suite pass is claimed.
+
+Immediate pre-submission review also read #822 (@shhxera, `8c5e1b557f59bca29139c4c2dcc65edfb808372b`): it proposes reuse of restored sibling connectors with entirely dirty scratch. That changes connector lifetimes; this contribution keeps ancestor connectors reserved and changes the number of clean inputs. Neither minimum is asserted over the other class. No #822 source is imported.