Cost-aware retired-slot selection: conditional κ = 5.1033760584215e-5

[rohanarun]

Conditional improvement

κ = 10206752116843/(2×10^17) = 5.1033760584215×10⁻⁵, 0.00935843% above #65, under the inherited analytic, all-size compiler and fixed finite-alphabet multitape hypotheses. Bit saving: 51036365162009/10^18.

Mechanism

Start from #63's interval-strip pair graph and compiler, keeping the original region ordering in both dimensions. Instead of accepting the first eligible dependent retired slot, inspect eligible dependence witnesses across all frame ranks. Select using an integer score derived from actual fixed-basis transition profiles and rational logarithm enclosures.

The score is a discovery heuristic, not the acceptance criterion. Every selected dependence is realized by literal XORs; all operand raises, final cleanup, copied centers, and endpoint corrections remain paid. The complete serialized physical word and the resulting recursive moment determine acceptance.

Both axes retain 27,918 and 36,586 roles, with W=137,151,806. Clearing XORs are 1,415 and 1,950. No graph/matching optimum is claimed.

Checks and review status

Focused checks pass: full input and dirty-basis replay in both orientations, independent word replay, physical transition reconstruction and fixed-basis profiles with CRT checks, the complete paid rank identity, exact moment enclosures, all 47 strict constraints and seven margins, and next-grid rejection. An independent rational logarithm/exponential calculation confirms the arithmetic.

Full make verify PASSED on b3745601e947a94316bf25c2c6263d93c06e3364 in 25m38s (467 core tests and 20 historical patch checks). Log SHA256: d5b7816bf25b70f99d4f840df0e9cd4a65a62f171ca873a0161c4f8399d7ee47. Receipt: research/slot-cost-rank-pair/validation.json.

Upstream was integrated afterward with unchanged consumed source and physical witness bytes. Focused source/profile checks, eight arithmetic tests and seven physical controls passed during integration; the complete newly imported make verify-joint also passed, including ten controls. Its root-document provenance and derived metadata were refreshed. Newly imported upstream groups were not part of the earlier full local run; all 45 expanded GitHub checks passed on integrated head 46991a21d13f37faf2350591de7d794cfaa4455e.

Run make slot-cost-rank-pair-verify. The proof note, portable generator, source pins and exact certificates are in research/slot-cost-rank-pair/. The branch retains the previous schedule witness separately for reproducibility.

Credit

Dominik Scholz (#63), Avi Eisenberg (#62), Chafik Boukhalfa (#60), eumemic (#57), Alejandro Zarzuelo Urdiales (#61, parameter refinement), RaD/hipotures, icekylinx, James Chang, Zhihao Chen, Aurel Prosz/Paureel, Swapnil Jain, Douglas Colkitt, OpenAI and all inherited contributors. Source notices and licenses are retained.

Prepared by Rohan Arun with OpenAI Codex assistance. These finite checks are not a formal proof of the full multiplication theorem or a practical speed benchmark.

[Mister-Meeseeks]

Thank you @rohanarun. This contribution is now recorded on main in the scoped reconciliation review and contributor credits.

Contribution: Optimize actual profile cost with stable candidate evaluation.

Review and limits: Exact-head CI evidence; heuristic objective does not certify global optimality.

The complete original source, proof/research notes, notices and relevant replay/CI evidence are retained through the pinned source archive. The review distinguishes fresh runs, exact-head historical CI and prior integration evidence. Closing this historical PR as reconciled; its obsolete headline does not replace the current selected bound, and draft/retracted claims retain their explicit qualifications.

Reviewed head: 46991a21d13f37faf2350591de7d794cfaa4455e. Publication: b84ba67156a01f9c2cb34fe1e3cb1dcdeab00d9c.