Shrunk-frame birth and terminal composition: conditional kappa = 1.12802185097556e-4

[chafreaky]

Change and mathematical scope

This separate candidate composes shrunk frames, dead-slot birth reuse and terminal elimination. Under the inherited OpenAI #109 all-size, analytic and fixed-tape interfaces, it certifies

T(n) = O(n (log n)^(1-kappa)), with kappa = 112802185097556/10^18 = 0.000112802185097556 and conservative dyadic corollary kappa = 2^-14.

The selected word retains 2,108 birth reuses and removes 381 distinct-target terminal roles, leaving 26,216 physical auxiliary roles. Complete reconstruction gives width 114,315,520, rank 65,843,877,440, largest child 574 and conservative scalar charge 3,116,113,870,848. Exact rational replay checks all 2,024 source and all 26,216 dirty columns in both orientations. The joint choice is checked against the actual complete paid histogram, including source and target fronts, auxiliary exteriors, copied-center losses and final endpoint correction.

The saving is approximately 193.32105577297% above PR36's 384569/10^10 and 113.68252252328% above our PR91's 52789616935221/10^18. The comparison with the pinned #126 value 111192082577/10^15 is approximately +1.44803702138%. The final fully paginated refresh found newer, larger reported exponents: #138 at head 877da1c6e000cfaa72d8163a88acda31ed8057d6 states the exact body fraction 206798323483/500000000000000 = 0.000413596646966, while its title displays the different value 0.000413596659368; the exact body fraction is used here. #139 at head 52ca13342bd40027029feefe23e67dcd53ba01f6 reports 0.00045019844704 as a finite arithmetic candidate and explicitly leaves its combined physical execution theorem unproved. Those are recorded as reported working inputs with their stated scopes. This PR is a separate verified finite composition checkpoint and makes no current-frontier or global-best claim.

The final Gaussian/layer margin controls the next 10^-18 grid point in this fixed-profile balanced family. Two rational log/exp enclosure methods, three simultaneous row stocks and the completed-child bridge support all 47 strict inequalities and seven assembly margins. The 240-theorem Lean companion proves scoped rational/count statements. The inherited all-size realization, semantic/common-grid, analytic precision/recovery and fixed-tape obligations remain explicit in PROOF.md; this is neither a global-optimality nor a practical-speed claim.

The portable package is under research/shrunk-birth-terminal/. It includes the selected word, compressed original inputs with stored/decoded hashes, complete source closure, proof, terminal packet argument, transfer certificate, reconstruction commands, independent-audit evidence and the unchanged generated Lean module. Sources reconstruct the selected witness from the pinned graph and recorded moves. The descriptive scalar-count correction is documented separately and changes no word, histogram, scalar charge, numerical bound or Lean theorem.

Chafik Boukhalfa (@chafreaky), with OpenAI Codex assistance, prepared this composition and its verification. Retained contributions are credited to:

• @GamingPuzzled (Joel Pulikkan), shrunk frames: Shrunk lifted frames with saturated deferral: conditional κ = 1.10591581647e-4 #125.
• @jamesyc (James Chang), birth reuse and balanced-layout lineage: Birth-read slot recycling: conditional kappa 1.1038332094346e-4 #124 and Reversed two-stage basis and balanced assembly for conditional κ = 1.639226629e-5 #34.
• @SovereignSteak, terminal identity and majorization: Terminal accumulator elimination with exact majorization audit #122.
• @eumemic, scalar DAG and earlier compiler: Deferred bit word and a 28,705-role complex producer under stopped-product accounting: conditional κ = 1.044939e-4 #117.
• @DanieleCorso, endpoint-frame search and marginal placement costs: Physical operation frames with deferred readouts: conditional κ = 1.11192082577e-4 #126.
• @rohanarun (Rohan Arun), signed/deferred balanced transfer and unit readouts: Balanced transfer of deferred signed networks: conditional κ = 6.3978919675787e-5 #100, Saturated readouts with complete signed/balanced integration: conditional κ = 6.398282083297e-5 #103 and Deferred readouts on PR117’s complex producer: conditional κ = 1.085428e-4 #118.
• @ikeboy (Avi Eisenberg), deferred physical compilation: Deferred readouts on both stopped networks: conditional κ = 8.58478e-5 #110 and earlier work.
• @icekylinx, stopped product ring, rational centers and ordinary wrapper: Stopped product-ring interchange and rational centers for conditional kappa = 7.79476e-5 #104.
• @Swapnil-jain (Swapnil Jain), the round-seven bit word: integer-mult-kappa.
• @jacklightChen (Zhihao Chen), controlled-basis and semantic interfaces: Semantic precision and bulk resampling for a conditional 1099/10^8 > 2^-17 bound #23 and Two-stage controlled corners for a conditional 15536/10^9 > 2^-16 bound #29.
• @hipotures and the RaD project, paid positional, routing and semantic machinery: Alternating pair-block networks for conditional κ = 4.034168124 × 10⁻⁵ > 2⁻¹⁵ #41 and retained sources.

Original authorship, licenses, notices and Anthropic Claude/other assistance disclosures remain intact. references/CONTRIBUTOR-PINS.json records the verified GitHub attribution metadata. These credits identify dependencies and do not imply review or endorsement of the new composition.

Validation

• One successful complete make -j1 verify run passed on clean construction 7b3b6f2113c0d7e61836cab409b6228e4c9b3a4d in 1377.869 seconds in a fresh source-only checkout: all 12 groups, 525 tests in 79 isolated modules, and 20 patch-application checks. The first attempt stopped at the hash gate after 1193.323 seconds because the inherited producer restored numeric histogram-key ordering; parsed values were unchanged, and the repair retained those stable bytes. A second attempt stopped after 939.309 seconds on leftover ignored scratch required to be absent by an inherited producer. The fresh clone kept the exact commit, tree and 281 source/input pins while removing that runtime-state dependency. Both failed attempts are retained in the evidence. The final commit updates only the derived repository-validation receipt.
• make verify-shrunk-terminal passed in 62.215 seconds. It reconstructs the geometry and complete word, compares every paid bin and scalar bill, runs 18 adverse arithmetic controls and audits the Lean linkage. Nine focused repository controls also pass.
• Independent package and final source/cache audits passed. An independent Lean-to-word audit recalculated 233,458 frame transitions, 71,928 Gram matrices and all 34 paid-width bins. Fresh integrated Lean compilation and axiom checks are included in the formal CI jobs.
• All 39 published-head checks pass at 3ddc4dcf7312ec5ab3bff843fd3e8c10cce041dc: 12 verification groups on Python 3.11, 3.13 and 3.14, plus the historical, Gaussian and matrix formal jobs.

The local compiler environment, commands and exact log hashes are recorded in certificates/shrunk-birth-terminal-repository-validation.json; REPRODUCE.md gives portable commands and separates arithmetic checks from fresh formal compilation. Foreign producers were accepted as reported working inputs and were not replayed solely as an admission gate. The final frontier refresh read every page of open PRs and retained exact heads and reported certificate comparisons.

[Mister-Meeseeks]

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

Contribution: Shrunk-birth terminal elision with exact adjoint and redirects.

Review and limits: Exact-head physical CI evidence and source-linked saved 240-theorem arithmetic evidence. No fresh Lean kernel rebuild or analytic formalization claimed.

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: 3ddc4dcf7312ec5ab3bff843fd3e8c10cce041dc. Publication: b84ba67156a01f9c2cb34fe1e3cb1dcdeab00d9c.