Certify ordered-frame composition at κ = 5.2514124198891e-5

[chafreaky]

Change and mathematical scope

This composes rank-preserving envelope ties, coordinate-aware oracle prices and bounded three-donor carrier exchanges with the complete indexed construction from #84:

κ = 52514124198891/1000000000000000000 = 5.2514124198891e-5, with bit saving 52516882077117/1000000000000000000.

The exact value is 0.4645669003% above the pinned #84 construction 88ca39571907343a49e97f328971ec7bcd26fbfd. Complete #84 and reported pinned #74 paid recursive profiles are strictly excluded at the new accepted bit saving. This also exceeds the reported #85 value 5.2272192668365e-5 (fb1f2c57681cd7c0b86a192de9367a7af87c9cca) and #86 value 5.2275423755251e-5 (d1b2a3a7caec9a101363e5c534a4680a1d2086ba), observed 2026-10-08. These are historical reported-head comparisons; the actual pricing source is #82. No foreign producer replay or current-global-frontier claim is implied. A later #74 update (428b5c167686b42a83f1abf7003c6d17aec561ea) reports κ=5.278214423139e-5, above this frozen witness; it does not change the historical comparison or numerical certificate here.

The result remains a finite conditional witness under the inherited ordered residual compiler, all-size recursion, finite scalar overhead, routing, prime selection, exact recovery, fixed-tape and analytic transfer interfaces. It is not an unconditional or fully formal multiplication theorem, a global optimum or a practical speed benchmark. The complete original #84 construction and all earlier source and finite evidence remain preserved.

Construction

Selected choice	h23	h25
Recursive split vector	[4,4,4]	[4,4,1]
Rank-first envelope ties	descending cover, then core	original PR74
Output routing	baseline	descending future-use deadline
Oracle coordinate pricing	identity	final selected coordinates
Three-donor passes / accepted cycles	0 / 0	at most 3 / 2
Auxiliary roles	26,581	35,133

Both axes retain anchored point order, balanced half-plane coarse sums, future-compatible unit completion, exact retired/pending indexes, pending next-use scoring and bounded single/two-cycle exchanges. Both use explicit coordinate maps selected by our bounded R6 search. The full width is 131,503,443; recursive rank is 75,612,632,825; rank deficit is 1,846,900; largest child/parent dimension is 529/575. Bit wire bits and row coefficient remain 27/843.

Distinct scalar envelopes strictly increase cover-size minus core-size along an edge, so rank-first ordering preserves dependencies while retaining original order inside each envelope. Pricing changes only serialized heuristic masks, leaving legality checks and in-memory frames untouched. Three-donor cycles independently check each final local basis and preserve distinct future-use ownership and cardinality while strictly decreasing a fixed integer score. Final profiles are freshly recomputed; all XORs, frame raises, copies, centers, dirty inputs, endpoints and cleanup remain charged.

Attribution

• Thomas DiFiore (@tomdif), #74: original rank-first node order and exact coordinate relabeling. Original source, license, notices and AI disclosures are preserved.
• Rohan Arun (@rohanarun), #82: coordinate-aware oracle pricing, active in the selected h25 compiler. Also #78, coordinate-flag search; #65 and #67, exact refined assembly, schedules and profile-cost reclamation/oracle. Original AI disclosures remain attached.
• @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, #84 and this composition: ranked reclamation, anchored splits, future completion, integer pricing, exact indexes, two-cycles, output routing, rank-preserving tie orders, three-donor cycles, selected-coordinate search and certification, with OpenAI Codex assistance.

All earlier contributor, OpenAI, manuscript, license and AI-assistance notices remain attached. The 590-file original #84 construction closure, immutable 90-pin final discovery runtime and original #82 inputs remain in separate hash-checked archives.

Validation

Both portable engine and node helper are byte-identical to the independently audited final runtime. The original and coordinate-relabeled physical words regenerate byte-for-byte. The package source and finite-input closure contains 1,280 pins, checked against working, staged and committed Git bytes.

• Independent dense global triple-input support checks for every scalar node, exact envelope, disjoint addition and target output.
• Full ordinary-input and arbitrary-dirty-basis replay in both orientations; 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 rejection, all 47 strict inequalities and seven margins.
• Actual-source audits for basis completion, both frame indexes, single/two-cycle exchanges, 1,604 three-cycle cases and 400 coordinate-pricing writer cases. The whole engine AST differs from the preserved original only by the pricing writer and audited three-cycle helper/call.
• Physical, source, configuration, coordinate, provider-restoration and stale-bridge negative controls; seven order/dense-check controls.
• Independent selected-word/profile/arithmetic clearance for all 51,525 complete paid pairings: two equivalent best pairs and 51,523 strict exclusions. They differ only by an inactive h23 cycle setting; the freeze selects zero passes. This is a finite-portfolio statement only.

Focused make final-frame-verify passed in 111.367 seconds, including all seven source audits and 30 controls. An independent portable review cleared all 1,280 source pins, the complete 590-file original #84 archive, physical words, paid profiles and arithmetic. Construction commit a24042385d515cafe1e09248664925e49a8477a1 is preserved. Full expanded sequential make -j1 verify passed all fourteen groups on clean tested commit 3ffd9b1dc7ef2978fa1831fe3850cd4f534aea70 in 2007.95 seconds: 616 isolated tests across 87 modules and 20 patch-application checks, with byte-clean regenerated artifacts. Its only change from the construction is a normal merge of the completed #84 validation receipt. Final published commit d5a5b936844104db18c5bdd871f89da2fbf99551 adds only this package's complete repository-validation receipt; all 45 required CI checks pass on that exact head, including all three inherited formal packages. The frozen source and finite witness remain unchanged. All 13 inherited verification groups and all three formal packages remain; the new final group adds the selected witness.

make final-frame-verify
make verify

See research/final-frame/PROOF.md, SOURCE.json, and certificates/final-frame-validation.json. 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: Whole-construction selection from a bounded portfolio.

Review and limits: Exact-head CI evidence; 51,525-candidate search scope retained, not a global ceiling.

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: d5a5b936844104db18c5bdd871f89da2fbf99551. Publication: b84ba67156a01f9c2cb34fe1e3cb1dcdeab00d9c.