Anchored split frames with profile-cost live reclamation: κ = 5.1414646104039e-5

[chafreaky]

Change and mathematical scope

This adds a finite conditional witness with κ = 51414646104039 / 10^18 = 5.1414646104039e-5, a 0.7463402960% improvement over the complete pinned PR67 witness. The scalar graph uses split vector [1,1,2] and moves the lowest intact global pair to the front of each common-point permutation. The compiler combines PR67 profile-cost selection with PR68 pending live controls, then scores those controls against their known next-use frame.

Paid quantity	Pinned PR67	This witness
h23 auxiliary roles	27,918	27,455
h25 auxiliary roles	36,586	36,015
Physical width W	137,151,806	135,075,665
Recursive rank mass	78,860,441,550	77,666,660,475

The bit saving is 12854322426487/250000000000000000. Both words include every clearing XOR, role, endpoint copy and frame raise. The complete PR67 child profile is excluded at the new bit saving; the next bit-saving and κ grid points at denominator 10^18 are also rejected. The integer profile score is a discovery heuristic; the reported saving comes from the full paid profile and exact inequalities.

The new research/split-pair namespace preserves contributor originals, licenses and notices. Existing numerical certificates, physical words and maintainer-reviewed statements are retained. Only the three affected root-document pins and derived certificate digests are refreshed in the older PR60 certificate; its mathematical content is identical.

Contributor credits:

• @rohangar1 (Rohan Garg): split operation in #59.
• @ikeboy (Avi Eisenberg): interval pair graph in #62.
• @DominikScholz (Dominik Scholz): ranked composition in #63 and pending live controls in #68.
• @rohanarun (Rohan Arun): schedules/refined arithmetic in #65 and profile-cost selection in #67.
• @eumemic: reversible compiler and physical/profile checkers in #57.
• @chafreaky (Chafik Boukhalfa): ranked reclamation in #60 and this aligned split/next-use-score composition.

AI assistance: this composition was prepared with substantial OpenAI Codex assistance. Upstream AI-assistance disclosures remain in the preserved source notices.

PR #68's later 85781f662b57521743cce07cdd5ea194e6f3879d concurrently combines cost and live-control ideas; this package preserves the original 734c58e source and explicitly credits that contribution.

This remains a finite conditional witness. The all-size residual compiler, routing, prime selection, exact recovery, finite-alphabet/fixed-tape simulation and analytic transfer remain inherited dependencies. Arithmetic cutoffs are not full operational thresholds. No global optimality, fully formalized multiplication theorem, or measured practical speedup is claimed.

Validation

make -j1 split-pair-verify PASS: both portable words regenerated byte-for-byte, actual profiles and exact assembly matched, and all 23 focused test methods passed. Independent review checks the complete selected words, every input and dirty basis vector in both orientations, exact physical profiles, weighted recursive identities, pending-control eligibility, all tested complete pairings, the PR67 comparison, 47 strict assembly inequalities, seven margins, next-grid failures and source closure.

Expanded repository verification PASS across all eleven groups on 1bef94fd40a746452548c84a4a8f8834670a3113, including 539 isolated tests across 80 modules and 20 patch checks. The initial make -j1 verify completed five groups before Apple clang rejected an inherited OpenMP flag; the remaining six passed with a task-local adapter to installed libomp and existing hash-verified Boost headers. The working tree remained byte-clean, and both the source-manifest and exact-certificate hashes are unchanged.

All 36 construction CI jobs passed on that commit, including the three formal packages. The receipt-only final commit f59cb9f54d12a2a02b665afef24a9bb5061ae011 has also passed all 36 CI jobs. Independent completion review confirmed the unchanged sources and certificate, complete local coverage and exact published receipt. This PR is ready for review.

[DominikScholz]

Small arithmetic footnote for the construction at 1bef94fd40a746452548c84a4a8f8834670a3113: reducing the positive assembly backoff from 10^-12 to 10^-18 permits

κ = 25707323052097/500000000000000000 = 5.1414646104194e-5.

This changes κ by only 155/10^18 = 1.55e-16. The graph, compiler, physical words and bit saving 12854322426487/250000000000000000 are unchanged. This is a parameter footnote to your construction, not a separate construction contribution; I am withdrawing #73 and leaving the calculation here for reference.

Exact source/profile and arithmetic checks were reproduced without certificate drift, including the 47 assembly inequalities, seven margins and next-grid rejection. The common eventual arithmetic cutoff increases from 1.4e13 to 1.4e19; the smaller backoff trades a negligible exponent gain for a larger cutoff. Physical replay was not rerun for this parameter-only calculation, and the inherited conditional theorem assumptions remain unchanged.

Exact certificate · Reproduction checker · Proof and source scope.

Prepared with substantial OpenAI Codex assistance; construction credit remains with this PR and its cited predecessors.

[Mister-Meeseeks]

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

Contribution: Anchored splits with strict contraction and dirty-state obligations.

Review and limits: Retains the earlier fresh physical replay and 23 tests; exact-head CI supplements rather than replaces that scoped review.

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: f59cb9f54d12a2a02b665afef24a9bb5061ae011. Publication: b84ba67156a01f9c2cb34fe1e3cb1dcdeab00d9c.