Split-pair dual-suffix frames: conditional κ = 4.83109396e-5

[chafreaky]

Change and mathematical scope

This composes PR #59's split-pair group choices with PR #55's dual-suffix layout and the unchanged ranked joint compiler from PR #60. It improves PR #60's conditional exponent saving by 1.3974275711%:

$$ T(n)=O\left(n(\log n)^{1-\kappa}\right),\qquad \kappa=\frac{120777349}{2500000000000}=4.83109396\times10^{-5}. $$

The result remains conditional on the inherited base theorem and all-size analytic, residual-compiler, routing, recovery and fixed finite-alphabet multitape interfaces. This comparison is to #60. Main now records the stronger reviewed saving 5.1016920170078e-5; this contribution preserves a separate, weaker conditional construction and does not claim to improve that current frontier. No global optimization or practical runtime claim is made.

Construction

Use split-group vectors [1,3,2] at h=23 and [1,2,2] at h=25. Both axes use PR #55's native suffix layout: input order at the top, the inherited deeper support orders, and the original total fold orders. The exact PR #60 ranked reclamation engine is unchanged. No paid clones or recorded PR #59 permutation overrides are used.

Quantity	PR #60	This composition
Auxiliary roles, h=23	30,688	30,118
Auxiliary roles, h=25	40,338	39,663
Physical width W	150,167,598	147,661,173

Every selected frame transition, clearing XOR, output copy and cleanup is included. The exact bit saving is 2415663683/50000000000000, full rank mass is 84903327575, and rank deficit is 1846900. The fixed I+J data geometry and complex branch are inherited unchanged.

The new package is separate in research/split-dual/, with dedicated compiler, verifier and certificates. It privately loads the graph sources and restores temporary provider/module substitutions, including exceptional exits. Original PR #59 files and a complete immutable PR #60 comparison certificate remain pinned. Adding the target and notices refreshes only the old package's root-file source bindings and derived metadata; its code, words, profiles, numerical result and existing verifier remain intact.

Attribution

• Rohan Garg (@rohangar1), PR #59: split-pair recursion and group choices, with substantial OpenAI Codex assistance.
• Rohan Gupta (@gupt1156), PR #55: dual-suffix layout, with Anthropic Claude assistance.
• @eumemic, PR #57: joint reversible frame compiler and independent word/profile machinery, with substantial OpenAI Codex assistance.
• Chafik Boukhalfa (@chafreaky), PR #60 and this PR: ranked reclamation, explicit graph composition, isolation, comparison and integration, with OpenAI Codex assistance.

The inherited construction retains Avi Eisenberg (@ikeboy) (#53), Rohan Arun (@rohanarun) (#44, #49), RaD (@hipotures) (#41, #51), @icekylinx (#36), Dominik Scholz (@DominikScholz) (#35), James Chang (@jamesyc) (#34), Zhihao Chen (@jacklightChen) (#7), and Aurel Prosz (@Paureel) (#1).

Earlier contributions and retained checks also credit @Bortlesboat (#2), @dleen (#4), Ryan S (@princezuda) (#26), Alejandro Zarzuelo Urdiales (@alejandrozu) (#45), and Douglas Colkitt (@Mister-Meeseeks) (maintainer integration). Swapnil Jain (@Swapnil-jain) retains the linked two-stage credit recorded in #29. OpenAI, David Harvey, Joris van der Hoeven and every earlier contributor retain their manuscript/source attribution. All original notices, licenses and contributor-specific AI-assistance disclosures remain attached to their sources.

Validation

All 36 checks now pass at final receipt-only head 701b1246e2def8b022a14309b5171138f2b12a52, including all three inherited Lean packages. The preceding receipt head hit an external Lean 4.31.0 download HTTP 500 before the matrix proof ran; the accurate metadata-only retry preserves every mathematical source, pin, word, profile and numerical certificate.

The verified construction is b07443b7fb73200eeca4e681a2bccc226ef938b1, with source freeze 43368c897bf61e6cab9d7275451241d57083f8381af9871cf4bb8cc1d67e129f. The final repository receipt is a metadata-only follow-up; it does not change any source pin, word, profile or numerical certificate.

• Fresh word regeneration matches both selected words byte-for-byte. Independent complete source/target/dirty-basis replay in both orientations and actual CRT profile regeneration pass.
• Exact rational moments exclude the complete PR Ranked slot reclamation: conditional κ = 4.764513337e-5 #60 child list at the new bit saving. All 47 strict assembly inequalities, seven margins, eventual cutoffs and next-grid rejections pass.
• 14 focused split-dual controls pass, including physical mutations, omitted charges, provider restoration on failure and assertion-disabled Python rejection. Independent review additionally rejects seven source/manifest/artifact mutations.
• Fresh make -j1 verify-joint passes at the verified construction in 309.334 seconds, with a byte-clean checkout. This includes the predecessor witnesses and independent PR60/61 arithmetic/source binding.
• All eleven verification groups complete locally on the merged source tree, including 530 isolated tests across 79 modules and 20 patch checks. The initial run on 29a504f completed five groups before Apple clang rejected an inherited OpenMP flag; the remaining six completed with a task-local adapter to installed libomp and existing hash-verified Boost. Exact attempt and continuation records are retained. Canonical ordering of two derived receipts was restored without changing their parsed JSON values, then rechecked clean at b07443b.
• All 36 unique CI checks pass at b07443b: eleven groups on Python 3.11, 3.13 and 3.14 plus all three inherited Lean packages. Both workflow invocations are green, 72 successful check runs total.
• The original ac0bc93 full run also passed: 443 integrated tests, all focused checks and 20 patches in 1096.909 seconds. That historical record retains its own tested source freeze.

Main fc74109 is integrated additively: all upstream additions and the complete current CI architecture remain. Independent reconciliation review confirms unchanged physical/numerical witnesses and only three root-file pin updates. The maintainer-reviewed headline remains 5.1016920170078e-5; this PR's separate conditional construction is weaker than that current frontier.

Reproduce with:

make split-dual-verify
make verify

certificates/split-dual-repository-validation.json records exact commits, source hashes, checks and log digests. See research/split-dual/PROOF.md for the local algebra and inherited all-size residual-compiler, routing, recovery and fixed-tape obligations. The inherited Lean checks do not constitute a formal proof of this complete multiplication theorem.

[Mister-Meeseeks]

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

Contribution: Joint split/dual source combinations with restored import state.

Review and limits: Exact-head CI evidence; retains the independent word checker and complete profile rather than assuming additive gains.

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: 701b1246e2def8b022a14309b5171138f2b12a52. Publication: b84ba67156a01f9c2cb34fe1e3cb1dcdeab00d9c.