Indexed carry two-cycles and coordinate flags: conditional κ = 5.22712890914e-5

[chafreaky]

Change and mathematical scope

This composes anchored multi-split recursion, balanced coarse sums, exact retired/pending frame indexes, bounded carry two-cycles, PR74 node order, late-use output routing and coordinate flags into one frozen finite conditional witness:

κ = 261356445457/5000000000000000 = 5.22712890914e-5, with bit saving 408390793141/7812500000000000.

The exact value is about 0.8494742% above the reported pinned #74 construction (3f78d8967c009153f2df5c6a6577b6ab1d37e2ca) and 0.4583459% above the reported #82 value (3410b940aa26e5876202152dfa7c4f66451ee22c), observed 2026-10-08. These are historical comparisons. Complete pinned #74 and #79 recursive child lists are strictly excluded at the new accepted bit saving; #81/#82 comparisons use their reported values only. No foreign producer replay or current-global-frontier claim is implied.

A subsequent #74 revision at 428b5c167686b42a83f1abf7003c6d17aec561ea reports κ = 5.278214423139e-5, above this frozen witness. That later construction is outside this package and has not been replayed here.

The result retains the inherited ordered residual compiler, all-size recursion, finite scalar overhead, routing, prime selection, exact recovery, fixed-tape and analytic transfer interfaces. It does not assert an unconditional or fully formal multiplication theorem, global optimum, or measured practical speedup. This package is stacked on #79 and preserves its complete published construction source and finite evidence at 9a58e62cf9e8954a4dde3c9b1372d733c276a6d1.

Construction

Both axes use anchored point order, balanced coarse columns when i+j<ng-1 and rows otherwise, PR74 node order with natural region execution, pending next-use scoring, future unit completion, and up to three single-exchange and three two-cycle passes. h23 uses recursive split vector [4,1,1], baseline outgoing-use order and cyclic coordinate shift one. h25 uses [1,4,1], descending future-use deadlines and the pinned PR74 coordinate permutation. Code 4 splits first and middle eligible pairs with an explicit strict-contraction guard.

Quantity	This witness
h23 auxiliary roles	26,834
h25 auxiliary roles	35,348
Width W	132,466,108
Total recursive rank	76,166,165,200
Rank deficit	1,846,900
Largest child / parent dimension	529 / 575
Bit wire bits / row coefficient	27 / 843
Single exchanges, h23 / h25	1,492 / 1,554
Two-cycles, h23 / h25	103 / 119

The exact indexes preserve the prior candidate order and both pending-frame containment tests. Two-cycles preserve each affected output/carrier basis, future-use uniqueness and total cardinality; every accepted swap lowers a fixed integer heuristic score. Coordinate relabeling preserves the scalar map and physical role operations, while actual fixed-coordinate child profiles are freshly recomputed. All physical XORs, raises, copies, dirty inputs and cleanup remain charged.

Attribution

• Thomas DiFiore (@tomdif), #74: exact node order and coordinate relabeling, including the selected h25 permutation. Original source, license, notices and AI disclosures are preserved.
• Rohan Arun (@rohanarun), #78: coordinate-flag search idea; its published chosen permutations are not substituted into this selection. Also #65 and #67: exact refined assembly, schedules and profile-cost reclamation/oracle, with original AI disclosures retained.
• @eumemic, #69 and #57: balanced coarse sums, 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.
• Chafik Boukhalfa (@chafreaky), #60, #71, #79, and this composition: ranked reclamation, anchored splits, future completion, integer pricing, exact indexes, two-cycles, output routing, selected-coordinate search and certification, with OpenAI Codex assistance.

All earlier contributor, OpenAI, manuscript, license and AI-assistance notices remain attached. The 250-file published #79 construction closure, 56-file frozen R5 discovery bundle, and original #74/#78 contributor inputs remain in separate hash-checked archives.

Validation

The portable engine and PR74 node/relabel extracts are byte-identical to the audited frozen R5 source. Both parent physical words and both coordinate-relabeled words regenerate byte-for-byte. Construction 88ca39571907343a49e97f328971ec7bcd26fbfd has source freeze c5b314b4acff73f8fcd31802db2609f848ef6fcc2b63740fd985c353d994a612. The source/finite-input closure contains 580 pins, including the code, configurations, proof, tests, selected words, profiles and additive integration files.

• Full ordinary-input and arbitrary-dirty-basis replay in both orientations, actual literal frame reconstruction, fresh bounded-minor/CRT profiles with zero disagreements.
• Complete paid child-list reconstruction, two independent rational moment enclosures, next 10^-18 bit and κ grid rejection, all 47 strict inequalities and seven margins.
• 23 focused controls for physical mutations, omitted charges/centers, mass-preserving profile mutation, missing pins, configuration/provider restoration, optimized Python, stale bridge fields, and full coordinate round trips.
• Actual-source strategy audits: 2,250 basis completions and 34,141 literal XORs; 800 random plus five targeted single-exchange cases; 1,204 two-cycle cases; 23,572 retired-index mutations/11,799 ordered queries; 27,926 pending-index mutations/13,971 ordered queries, including actual frame raises and future-containment rejection.
• Independent final selection audit binds all selected/parent bytes, full physical replay, fresh exact CRT profiles, and all 7,569 complete paid pairings, with every other pairing strictly excluded. This is optimal within that enumerated frozen portfolio only.

The focused gate passed in 87.835 seconds. Independent portable-package review cleared all 580 working/staged/committed pins, all 250 original #79 archive files, unchanged R5/PR74/PR78 source bytes, both regenerated words and the complete exact arithmetic. The final 23 controls passed after two proof terminology corrections and preservation-only whitespace changes. The complete expanded repository run passed all 13 verification groups, 586 tests in 84 isolated modules, and 20 patch checks, with unchanged source freeze and a byte-clean tree (1815.025 seconds). All 42 construction CI checks passed. Receipt-only follow-up 424ef44609eda377250b2985a5210cb336ade19e records these results without changing the source freeze or numerical witness. All 42 final-head CI checks passed; this PR is ready for review and mergeable. The new indexed group retains every inherited verification group, formal package and workflow.

make indexed-cycle-verify
make verify

See research/indexed-cycle/PROOF.md for algebra and inherited scope, SOURCE.json for exact inputs, and certificates/indexed-cycle-validation.json for the focused receipt. 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: Retired/pending indices, atomic two-cycle changes and contracting multisplits.

Review and limits: Exact-head CI evidence; serial lifecycle and ordering obligations reviewed.

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: 424ef44609eda377250b2985a5210cb336ade19e. Publication: b84ba67156a01f9c2cb34fe1e3cb1dcdeab00d9c.