Conditional κ = 9.67261003170394e-4: seven-stage E8 bit with nine-stage complex (+1.32149% over #378)

[rohanarun]

Pairing #377's nine-stage complex component with #373/#375's already checked seven-stage E8 bit supplier gives conditional κ = 483630501585197 / 500000000000000000 = 0.000967261003170394, 1.32149% above #378's published bound at c248962c3cf641d4f6226d63d86843f2c096a7d0.

#377 evaluated its new complex component with the older frozen #361 bit supplier. This composition uses the stronger existing E8 bit supplier. Both circuits and constructions remain unchanged. The complex saving is 999240641319551/10^18; bit saving is 968197503705007/10^18, and the bit side now binds. This is not the hypothetical larger bit saving used in #377's title projection. The resulting κ remains below 2^-10.

Compatibility and scope

The full nine-stage invoice fits the same retained envelope used by the bit supplier and outer assembly: G = 2^30000, E = 2^100000, C0 = 2^210000, row allowance 20161. Its actual physical-row coefficient is 15445, required-C0 bit length 200012, and induction gap 49B. Both upstream packages use byte-identical outer.py. With the inherited three bootstrap steps, eta = 10^-12 and beta = 10^-9, all 47 strict constraints pass and the adjacent 10^-18 κ grid point fails.

The result remains conditional on both supplier interfaces. In particular, #377's compressed-window, exact Clifford/compiler, arbitrary correlated-bank composition, common-grid, row and analytic recurrence interfaces remain obligations; #373/#375's B3, nine bit totals at dimension eight, geometry, tape/recursion and finite-bill interfaces remain inherited. No new Lean proof, new circuit, astronomical-cover enumeration or globally optimal exponent is claimed. Executable compatibility checks do not discharge the retained theorem interfaces.

Validation

• PASS: all eight stages of κ = 9.98243154231047e-4: nine-stage complex E8 (≈2^-9.9683; complex +14.036%, bit side pending) #377's standalone component reconstruction from its pinned sources, including exact Clifford phases, local dirty admission, banks, cover corrections, global template, complete invoice, two rational moment engines and negative controls.
• PASS: the unchanged seven-stage bit supplier's full replay in the previously completed Conditional κ = 9.54395132518163e-4: coupled complex retiming after #375 (+0.01588%) #374 package; its dedicated Python 3.11/3.14 replays and all 58 repository checks passed.
• PASS: exact combined pricing with the nine-stage complex saving, bit binding, all 47 constraints and adjacent-grid rejection.
• PASS locally: the full combined package replay at 6e95ce5fb4112e5e98382886d7904aee29e31c56, including both complete supplier replays, the shared-envelope checks, all 47 rational slacks, adjacent-grid rejection and the unchanged bit finite bill.
• PASS on GitHub: both Python 3.11 and 3.14 full replays. Remaining repository checks are still running. Draft submitted first as requested.

python -m pip install -r research/e8-seven-bit-nine-complex/requirements.txt
python -B research/e8-seven-bit-nine-complex/verify.py /path/to/fresh/output

The package vendors #375 and #377 unchanged, preserves every license and notice, runs both original verifiers in full, checks their shared envelope, reprices the freshly reconstructed bit word with the freshly certified complex bound, and compares the unchanged bit finite bill. All 47 rational slacks are saved in assembly.json. RESULT.json is written only after all checks pass. Expected receipts never replace computation.

Credit

Zhihao Chen with OpenAI Codex supplied #377; Jacob Sussman supplied the E8 complex unit and proof/checker framework; Dylan Crow / utcorvusvolat-dotcom with Claude supplied the E8 bit word; DaysSky with Claude supplied B_K and #373/#375. Earlier pricing and assembly authors are credited in the preserved notices. This composition and verifier were prepared by Rohan with substantial OpenAI Codex assistance. New work is Apache-2.0.

An isolated prerequisite commit reuses #318's unchanged three-file joint-dual source-pin repair, as in #366/#368/#374. It makes no mathematical change.

[ahrazzle]

Independent replay on third hardware — PASS

Ran this PR's verify.py --boost-include /opt/homebrew/include end-to-end on Apple M1 Max, Python 3.14.6 (numpy 2.5.4/sympy 1.14.0/mpmath 1.3.0, GCC 16.2): PASS_SEVEN_STAGE_BIT_NINE_STAGE_COMPLEX_COMPOSITION, κ = 483630501585197/5·10¹⁷ = 9.67261003170394e-4, exit 0, gain_percent = 1.32149152825611 (matches the stated +1.32149%).

Replayed receipts: full seven-stage bit baseline (1,585 formal columns), full eight-stage #377 complex component (row coefficient 15,445), combined exact pricing with bit binding (c = 9.68197503705007e-4 < b = 9.99240641319551e-4), all 47 strict outer constraints, adjacent-grid rejection, unchanged bit finite bill.

I also re-derived your assembly arithmetic independently (three bootstrap steps from a₀ = 384599/10¹⁰, η = 10⁻¹², β = 10⁻⁹): my implementation reproduces this κ exactly, and reproduces #377's projected 998243154231047/10¹⁸ from the projected bit saving — the projection arithmetic in #377 is sound. The gap this PR sits inside is 3.21% (bit 9.6820e-4 → cap (1−β)·b = 9.99241e-4); per my headroom comment on #352, closing it needs a new bit construction — which #382 (retimed + shared-donor kernel E8 bit, unsplit B₃, c = 1.006907e-3) has now done, hitting exactly the projected ceiling. This composition is superseded in value by that line but remains a correct incremental step; it also correctly documents that the split-degree-1 layout leaves the complex bound unused.

No code changes — independent replay only.

Automated posting by agentic team with human oversight.

[Mister-Meeseeks]

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

Contribution: Seven-stage bit with nine-stage complex supplier.

Review and limits: Full composition wrapper checks both pinned suppliers and the changed layout; retain the uniform compiler and analytic assumptions.

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: 6e95ce5fb4112e5e98382886d7904aee29e31c56. Publication: 4040a0f46d67b41c36a472a7cd4af791fd4c36cc.