Aligned frame composition: κ=5.2789616935221e-5 with Lean arithmetic certificate

[chafreaky]

Change and mathematical scope

The aligned frame composition certifies the conditional saving κ = 52789616935221/10^18 = 5.2789616935221e-5 ≈ 2^(-14.2093862772), a 37.2695587404% increase over the original #36 κ=384569/10^10. Its exact bit saving is 10558480765231/200000000000000000. Exact rational values are authoritative. This is approximately 0.01415763596% above reported #74 at 428b5c167686b42a83f1abf7003c6d17aec561ea, and above the pinned #88 construction 3ffd9b1dc7ef2978fa1831fe3850cd4f534aea70. The reported external witness is accepted as a working baseline; no foreign producer replay is claimed.

The selected h23/h25 partitions are ['first6',1,1] / ['first9',1,1,2], with six/nine anchored pairs, depth-coarse word hhh, PR74 node order and its updated coordinate maps. The unchanged independently audited R10 compiler supplies future-compatible completion, exact retired/pending indexes and next-use prices, bounded single/two-cycles, h23 baseline output order, and h25 late output order with coordinate-aware pricing and three-cycle exchange. There are 26,387/34,772 auxiliary roles, width W=130,417,912, complete rank 74,988,452,500, and deficit 1,846,900. All recursive children are positive and at most 529<575; copied centers, endpoint copies, cleanup and data growth remain paid.

The proof in research/aligned-composition/PROOF.md establishes general partition/anchoring/coarse compatibility, source identity, and exact finite physical/profile/arithmetic claims. The conditional all-size conclusion still relies on the inherited framed/residual compiler, recursion, finite scalar overhead, routing, prime/precision selection, recovery, fixed-tape implementation and analytic transfer. Arithmetic cutoffs are not full operational thresholds; no practical benchmark or global optimum is asserted. The finite portfolio compares 59,045 complete paid pairs, with two equivalent winners differing only in the inactive h23 three-cycle setting.

The additive formal/lean/KappaCheck/AlignedFrameComposition.lean extends the inherited #26 arithmetic checks with 197 rational arithmetic theorems: supplied child counts, width/rank/deficit, rational accepted/rejected moment bounds, 47 strict slacks, seven strict margins and adjacent-grid rejection. Physical derivation of supplied arrays, real logarithm/exponential enclosure validity and all-size transfer remain external hypotheses. The module is byte-identical to its independent review and Lean/mathlib 4.21.0 compile receipt. Root imports, all 197 axiom entries and all 295 declaration/source rows are added without dropping earlier checks. The inherited axiom-output parser now trims comma-separated tokens so Lean's wrapped lists of allowed axioms are accepted; regressions still reject forbidden axioms, missing theorems and duplicate output.

Source credits: @eumemic #57 and #69, reversible compilation and coarse factoring; @rohangar1 #59, split recursion; @ikeboy #62, interval strips and core-aware assembly; @DominikScholz #63 and #68, ranked profiles and live pending controls; @rohanarun #65, #67, #78 and #82, exact refined assembly, profile-cost reclamation, coordinate search and coordinate-aware pricing; @alejandrozu #70, bounded carry exchange; Thomas DiFiore (@tomdif) #74, aligned firstK partitions, multi-pair anchoring, depth-coarse factoring, node order and selected coordinate maps; @chafreaky #60, #71, #79, #84, #88 and this composition, anchored splits, completion, indexed reclamation, next-use scores, two/three-cycle composition, bounded search and certification.

Chafik Boukhalfa prepared this package with OpenAI Codex assistance. All contributor originals, Apache-2.0, OpenAI/manuscript and AI-assistance notices remain attached, including Avi Eisenberg's Claude assistance and earlier OpenAI disclosures. The complete 1,290-file PR88 construction, updated 17-file PR74 source with fetch metadata, and all prior archives are preserved byte-for-byte.

Validation

Frozen source: research/aligned-composition/SOURCE.json, 2,720 pins, SHA256 7dcf54d1a136cc45fee258bdff80b191069c368a4a466b5862ad54d4623f2f57. Construction commit: 264f202b52edc04d0c72ca9cb3138282ca68b9ad.

make aligned-composition-verify passes: exact regeneration of both original and relabeled words; full ordinary/dirty basis replay in both orientations; fresh actual-frame CRT profiles; two independent rational moment enclosures; complete-profile comparisons; all 47 strict constraints and seven margins; neighboring 10^-18 rejection; eight actual-source strategy/Lean linkage audits; 33 physical/configuration/partition controls and four axiom-output parser regressions. Independent selected-witness clearance checks every physical/profile binding and all 59,045 paid pairs. Independent portable audit checks source, archive, proof, runtime, exact word/profile/arithmetic and additive formal integration bindings.

Full expanded sequential make -j1 verify passed all 15 verification groups on clean construction 264f202b52edc04d0c72ca9cb3138282ca68b9ad in 2110.262 seconds: 653 tests in 92 isolated modules and 20 patch-application checks, with regenerated files byte-clean and the source freeze unchanged. Construction CI passed all 48 checks. Final receipt-only commit 8d1a0838f3e906d372818df02b3564c8b36495cc adds this verification record and the exact completed #88 validation receipt from d5a5b936844104db18c5bdd871f89da2fbf99551, without changing the frozen witness, immutable PR88 construction archive or Lean module; all 48 final-head CI checks pass, including the integrated historical Lean package and both other inherited formal packages. Reproduce with make aligned-composition-verify, make verify, and make formal-historical-verify.

[Mister-Meeseeks]

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

Contribution: Aligned frames plus rational Lean endpoint arithmetic.

Review and limits: Finite numerical/physical CI and saved Lean source/axiom evidence retained. Generic formal(matrix) CI does not establish the new Lean module; no fresh full kernel rebuild claimed here.

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: 8d1a0838f3e906d372818df02b3564c8b36495cc. Publication: b84ba67156a01f9c2cb34fe1e3cb1dcdeab00d9c.