Complete far-retimed E8 replay and coupled bit frames; conditional kappa 0.000998634658090835

[ai-nanachi]

The retimed complex unit in #384 has a stronger nine-stage moment, but its description explicitly leaves the certificate-specific splice, dirty scalar/lifetime, new Clifford-frame and local finite-invoice checks unrerun. This change supplies those fresh checks in #377's complete physical construction, composes it with a fully invoiced zero-entrance bit word, and records the conditional result in the README and a checked candidate certificate.

Conditional κ = 998634658090834606116833 / 1000000000000000000000000000 = 0.000998634658090834606116833. It exceeds the pinned three-round #387 witness 998634657132636/10^18 (803b139) by 9.58198606116833e-13. This small headline gain applies applecrusher's #376 assembly refinement to the freshly replayed stronger complex supplier. It is not attributed to the new bit frames, and it is not a measured runtime improvement.

Change and mathematical scope

• Complete replay for Conditional κ = 9.98634656132003·10⁻⁴: #377's nine-stage E8 complex component with our #378 re-timed E8 unit, paired with our #382 bit word (+0.039 % over #382) #384's changed complex unit. Replays Sussman's reference/mirror checks, flipped-sign and source-span controls, and the rational operator on all 1,023 registers. Derives a separate κ = 9.98243154231047e-4: nine-stage complex E8 (≈2^-9.9683; complex +14.036%, bit side pending) #377 work tree following Rohan Arun's Retimed nine-stage complex + kernel bit: conditional kappa 0.000998390813251625 #385 procedure, with a digest for every certificate/pin/census/expected-price change. Fresh dirty scalar maps, lifetimes, exact Clifford recipes, data/cover phases, emitted global template, finite invoice and independent two-engine audit pass. No checker algorithm or control is removed.
• Connected bit frame coalescence. After Conditional κ=0.0011208649522582786: split centres with complete signed admission #376's R=822 helper reuse, a deterministic search coalesces connected one-, two- and three-gate groups. Seventy accepted moves change 98 distinct scalar-gate frames, preserving scalar order, operands, copied totals, endpoints and helper count. The bit coarse saving increases from 8128527708459/8000000000000000 to 63613071872999/62500000000000000. An exact rational Taylor/log bound proves the cost decrease, including fallback charges, for 0.998 <= p <= 0.9999; no all-exponent dominance is claimed.
• Fully checked unsplit bit B3. Uses κ = 9.95656966441037e-4: #370's E8 bit word in the nine-stage bridged word with #377's nine-stage complex E8 (above 2^-10; +2.94 % over #380) #381's generalized checkers at K=3, h=8, T=7: m=56, stock=10794, max child=34, least halving degree two, row reserve=6274<20161. Replays all 1,062 local and 1,542 global dirty columns, all 120 port geometries and rational routes, literal banks, and word/record/omitted-bridge controls. The fresh finite coefficient is 56346946940397<2^80, with power bound q^3137.
• Honest baseline and record update. The unchanged Conditional κ=0.0011208649522582786: split centres with complete signed admission #376 bit word also exceeds the new complex cap, so it reaches the same refined κ. This equality and the separate bit component gain are mandatory checks. Eight bootstrap steps, η=10^-24, β=10^-18 (the smaller positive backoff from Conditional kappa 0.000998634658090835: finite bootstrap and exact coarse precision #387), fresh coefficient-dependent cutoffs, all 47 exact outer inequalities, adjacent 10^-27 tick rejection complex-cap rejection, and zero-eta/zero-beta/unit-beta rejection are checked. README/candidate/contributor records distinguish this submitted candidate from the maintainer-reviewed selected result.

Validation

python3 -m pip install -r research/e8-coupled-bit-retiming/requirements.txt
make e8-coupled-verify

The command requires a fresh output directory and first runs the entire pinned #375 supplier suite, including its historical p=10 and E8 regressions. It then regenerates the selected bit word, runs the entire original eight-stage #377 suite, and runs the changed complex component's physical/Clifford/invoice checks. It compares the final reconstructed result with a frozen expected record; stored PASS receipts do not substitute for execution. Every imported byte has a source commit/path/SHA-256, and the verifier rechecks package integrity before and after execution.

• Fresh local κ = 9.5424356568133e-4: seven-stage E8 suppliers with the complex idle climbs at halving degree two (+2.40 % over #373) #375, κ = 9.98243154231047e-4: nine-stage complex E8 (≈2^-9.9683; complex +14.036%, bit side pending) #377, changed Conditional κ = 9.98634656132003·10⁻⁴: #377's nine-stage E8 complex component with our #378 re-timed E8 unit, paired with our #382 bit word (+0.039 % over #382) #384 component and both bit words: passed.
• Full packaged replay from a fresh output directory on Python 3.13: passed on the final head b3c45822ca847aec8c0c019bb04db30881fc2966, including both original supplier suites, the changed complex unit, the new bit word, both moment engines, all finite invoices, exact record equality and final manifest checks.
• make gen4-bank-verify: passed, all nine historical audited stages.
• Historical joint-dual full dirty replay/profile/assembly: passed; tests/test_joint_reclaim.py: 16 tests and 72 subtests passed. Three metadata files refresh the stale README/Makefile source pins and derived digests following Re-pin the root Makefile/README in research/joint-dual/SOURCE.json (fixes the failing 'Verify research artifacts' job on main and all PRs) #318; mathematical values are unchanged.
• make verify was attempted on Windows and stopped at the existing resource module dependency in the first historical group. The final head passed both dedicated Linux full-replay jobs on Python 3.13 and 3.14, including record checks and git diff --exit-code: workflow and downloadable verification artifacts. The repository's broader historical Linux matrix passed all 52 jobs on the same final head: complete matrix.
• git diff --check and candidate-record/source-manifest checks: passed.

The upstream workflows require a maintainer to authorize this first-time contributor's workflow runs; the dedicated upstream run is awaiting that authorization. The successful fork runs above execute the submitted workflow on the exact PR head. This PR is ready for review; numerical verification does not imply maintainer approval or acceptance of the retained mathematical interfaces.

Scope and attribution

Finite conditional witness. The upstream all-size affine/compiler/chart interfaces, recursive rank-cost realization, compressed Clifford windows and arbitrary dirty correlated banks, fixed tapes, precision/recovery, routing, restored-row and analytic transfer remain assumptions. No Lean proof of the complete multiplication theorem or astronomical cover enumeration is claimed. The proof note states the degree-two interface explicitly.

Credits: Dylan Crow / utcorvusvolat-dotcom (#370 bit); applecrusher (#376 reuse and exact refinement); DaysSky (#363/#373/#375 bridged chronology/full replay and #381 unsplit halving); Zhihao Chen / jacklightChen (#377 complete complex component); Chafik Boukhalfa / chafreaky (#378/#384 far-retimed unit); Rohan Arun (#380 common envelope, #385 derived physical replay and #387 smaller-beta witness); DreamingOfClouds (#315 retained interfaces); Jacob Sussman (E8 unit, certificates and checkers); Henry Grant, Evan McKinney, eumemic, Avi Eisenberg, icekylinx, RaD and the predecessor contributors retained in the original notices.

The included 25-trial campaign, five-family exploration and negative results explain the new bounded search without claiming worldwide priority. Prepared by ai-nanachi with substantial OpenAI Codex assistance. New code is Apache-2.0; source-specific licenses/notices are preserved.

[Mister-Meeseeks]

Thank you @ai-nanachi. This contribution is now retained on main in the scoped 100-PR review and contributor credits.

Contribution: Full far-retimed E8 replay and coupled bit frames.

Review and limits: Dedicated fork Linux replay is tied to its actual checkout and record checks. Preserve separate verification/packaging credit and the source's conditional boundaries.

The complete original source, notices and relevant execution evidence are retained through the pinned source archive. The review distinguishes fresh local runs, source-bound historical Linux CI and prior exact-head review; it does not claim 100 fresh full mathematical audits. Closing this historical PR as reconciled. This preserves parallel and cumulative credit without replacing the selected frontier or promoting priced research targets to completed constructions.

Reviewed head: b3c45822ca847aec8c0c019bb04db30881fc2966. Publication: 4040a0f46d67b41c36a472a7cd4af791fd4c36cc.