Balanced coarse split frames: conditional κ = 5.1745625472866e-5

[chafreaky]

Change and mathematical scope

This composes PR #69’s balanced coarse sums and a bounded adaptation of PR #70’s carry exchanges with the anchored split frames from #71. The finite conditional witness gives κ = 25872812736433/500000000000000000 = 5.1745625472866e-5, a 0.6437453019850683% increase over the complete published #71 witness frozen at 1bef94fd40a746452548c84a4a8f8834670a3113, observed on 2026-10-08. This is a historical comparison; later community improvements are outside this package. The full old recursive child list is strictly excluded at the new bit saving 51748303221339/10^18.

The result remains conditional on the inherited ordered residual compiler, all-size recursion, routing, prime selection, recovery, finite scalar alphabet/tape, and analytic transfer interfaces. No global optimum, measured practical speedup or complete formal multiplication theorem is claimed. This separate package is stacked on #71 and preserves its exact published source and finite evidence at 1bef94fd40a746452548c84a4a8f8834670a3113.

Construction

Keep the anchored [1,1,2] split vector, h23 cover-core / h25 reverse-node schedules, exact profile-cost reclamation and pending controls with both frame-containment checks. For h23, coarse cells use columns when i+j<ng-1 and rows otherwise; h25 uses rows throughout. Both axes choose independent unit-completion rows using fixed compatible future uses. The h25 carry exchange uses exact integer prices and a three-pass limit, accepting 1,000 legal exchanges.

Quantity	Published #71	This witness
h23 auxiliary roles	27,455	27,297
h25 auxiliary roles	36,015	35,684
Physical width W	135,075,665	134,126,064
Total recursive rank	77,666,660,475	77,120,639,900
Rank deficit	1,846,900	1,846,900
Bit wire bits	28	27

All physical XORs, frame raises, copies and cleanup are charged. Crossing below 2^27 requires recomputing the bridge: wire bits 27, row coefficient 843, degree 2000, degree gap 7007/25, and suffix slope 8000. The data geometry and complex branch remain inherited. Existing numerical witnesses are unchanged; integration updates only three old root-file source pins and their dependent digest fields.

Attribution

• @eumemic, #69: balanced coarse parenthesization; #57: joint reversible compilation and independent word/profile machinery, with OpenAI Codex assistance.
• Alejandro Zarzuelo Urdiales (@alejandrozu), #70: bounded carry-exchange strategy; original source, license and AI disclosures retained.
• Rohan Garg (@rohangar1), #59: split-pair recursion, with OpenAI Codex assistance.
• Avi Eisenberg (@ikeboy), #62: interval strips and core-aware pair assembly, with Anthropic Claude assistance.
• Dominik Scholz (@DominikScholz), #63 and #68: ranked composition and pending live controls, with OpenAI assistance.
• Rohan Arun (@rohanarun), #65 and #67: region schedules, refined exact arithmetic, profile-cost reclamation and its oracle, with the original AI disclosures retained.
• Chafik Boukhalfa (@chafreaky), #60, #71, and this composition: ranked reclamation, anchored split selection, next-use score, row/half-plane selection, future unit completion, integer-price exchange adaptation, composition and certification, with OpenAI Codex assistance.

All earlier contributor, OpenAI, manuscript, license and AI-assistance notices remain attached. Original #69/#70 sources and the complete published #71 comparison closure are preserved in separate hash-checked archives.

Validation

The 245-file source/finite-input freeze is 68a96f879c1501b62da0366ea1003ccf4d4468a2240a70cf020cc9fff9d38471. Both portable engines differ from the audited discovery source only in the repository-relative ROOT constant. Both selected literal words regenerate byte-for-byte.

• Fresh complete arbitrary-dirty-basis replay in both orientations, literal frame reconstruction, and actual bounded-minor/CRT profile regeneration.
• Full paid child-list reconstruction, independent rational moment enclosures, next 10^-18 bit and κ grid rejection, all 47 strict assembly inequalities and seven margins.
• 24 focused tests, including physical mutations, missing charges and pins, mass-preserving profile mutation, provider restoration, invalid configurations, optimized-Python rejection, and five stale bridge fields.
• Strategy audits exercise 2,250 independent-prefix completions and literal synthesis, plus 800 random and five targeted carry-matroid cases with per-exchange legality checks. These tests accompany the proof arguments; they do not replace them.

Independent portable-package review cleared construction 9a58e62cf9e8954a4dde3c9b1372d733c276a6d1: all 245 working and committed pins match, all 112 archived PR71 files match the published construction, engine relocation is ROOT-only, selected words and complete arithmetic match the independently audited winner, and prior numerical witnesses remain unchanged.

The complete expanded repository verification passed in one clean construction run: 12 verification groups, 563 tests in 82 isolated modules, and 20 patch checks (1,763.015 seconds). All 39 construction CI checks passed. Receipt-only commit 79dd6e1 records these results without changing the source freeze or numerical witness. All 39 final-head CI checks passed at 79dd6e134855bc8a7b8950c3aa0eb279fe148ceb; the PR is ready for review and mergeable. The new balanced CI group retains every inherited group and formal package while keeping compiler regeneration within the existing per-job time limits.

make balanced-split-verify
make verify

See research/balanced-split/PROOF.md for the algebra and exact scope, SOURCE.json for immutable inputs, and certificates/balanced-split-validation.json for focused receipts. Arithmetic cutoffs are not full operational thresholds; inherited Lean checks do not formalize the complete multiplication theorem.

[Mister-Meeseeks]

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

Contribution: Balanced row/column partitions and legal carry choices.

Review and limits: Exact-head CI evidence; complete future unit basis required, gains are not simply added.

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: 79dd6e134855bc8a7b8950c3aa0eb279fe148ceb. Publication: b84ba67156a01f9c2cb34fe1e3cb1dcdeab00d9c.