Compressed nine-stage E8 windows and whole-block banking for a conditional κ = 9.98243154231047e-4 > 2^-10 bound

[jacklightChen]

Change and mathematical scope

This combines the local E8 bit word from #370 with the nine-stage complex component from #377 and supplies a new compressed nine-stage bit construction. The resulting conditional bound is

κ = 998243154231047 / 10^18 = 9.98243154231047e-4 ≈ 2^-9.9683211064 > 2^-10.

The excess over 1/1024 is exactly 21680654231047 / 10^18. This is approximately 24.0286% above the frozen #361 combined value. The complex supplier now binds; this is no longer the bit-side-pending projection in #377. No claim of priority is made.

Construction

• Reproduce κ = 8.75481145306751e-4: first E8 bit word (complex-bound; +9.29% over #352) #370 at b7fe9cdf2da009c947ff6b334c562bc190b30373, including its original word, regeneration, independent F₂ checker, official record/formal-column checks, exact geometry, negative controls and regressions. Its original five-stage bit saving alone is insufficient to cross 2^-10 with κ = 9.98243154231047e-4: nine-stage complex E8 (≈2^-9.9683; complex +14.036%, bit side pending) #377.
• Lift that local word into nine windows in Q^8 ⊕ (Q^7)^8, so m = 64. Use four data pairs, eight replicas and 7,785 whole-block helper banks, with 15,465 live families and nine reserved private copy slots. Each bank contains eight rank-8 residuals and needs no padding child.
• Defer endpoint completion until the final stage. Exact port isometries align complete frames at every bridge. Reflected calls use the actual reversed record stream and a freshly derived rank-8 COPY; they do not invert erasure.
• Retain κ = 9.98243154231047e-4: nine-stage complex E8 (≈2^-9.9683; complex +14.036%, bit side pending) #377's exact complex Clifford/Z₄ checks, terminal X_q and every-live-family X_delta(g) signed phase corrections. Binary matrix equality alone is not used to identify complex operators.

Quantity	Exact certified value
New bit coarse saving	1057833103534317 / 10^18
Complex coarse saving	999240641319551 / 10^18
Selected capped bit saving	999240640320309 / 10^18
Final conditional κ after three bootstrap steps	998243154231047 / 10^18

The bit maximum child is 42/64, so the proof uses and charges two-level halving: 2*42² < 64². The actual nine-stage payload prefix needs 117 bits; it receives a 192-bit allowance rather than reusing the old five-stage 104-bit guard. The two-level finite coefficient has 269 bits and is charged 512 bits. Its row coefficient is 18,355, below the inherited 20,161 allowance. See research/e8-bit-complex-nine/PROOF.txt for the cover-owner, arbitrary-dirty-bank, precision and finite-cost arguments.

Validation

Run make e8-bit-complex-nine-verify, or choose a fresh nonexistent output directory:

python3 -B research/e8-bit-complex-nine/verify.py --work /tmp/e8-bit-complex-nine-fresh

Requirements: Python 3.11+, NumPy, SymPy, mpmath, a C++17 compiler and Boost headers. The standard-header shim permits Clang without changing the pinned upstream sources. The complete source and dependency manifests are checked before rebuilding.

Fresh validation receipts are included in research/e8-bit-complex-nine-validation/. They cover the complete #370 and #377 reconstruction followed by the new exact geometry, source-bound emitted program, separate template audit and rational assembly:

• all 120 port routes and 944 exact local frame projectors;
• all 3,289 nested and reflected frame pairs;
• 8,745 independent scalar columns, 62,280 bank assignments and 72 local invocations;
• 1,355,328 emitted local events and the literal child profile;
• 18 omitted-shear controls and eight structural template mutations, plus inherited negative controls;
• two rational moment engines with the 10^-16 fallback, all 47 positive strict outer constraints, and rejection of the adjacent bit-saving and final-κ grid points.

The earlier make complex-nine-verify and unchanged make gen4-bank-verify passed. The full historical make verify stopped on the unchanged pinned base at verify-strips/skip-suffix: Apple clang++ rejects -fopenmp. A full historical-suite pass is not claimed.

Scope and attribution

This remains a conditional finite construction, not a new Lean proof of the complete multiplication theorem. The all-size weighted compiler, prime/tape/precision, common-grid/restored-row and analytic-transfer interfaces remain explicit inherited obligations. #370's admission of nine copied totals at h = 8 is retained. Neither the whole astronomical orthogonal cover nor an independent human review is claimed by these finite checks.

The local E8 bit word is Dylan Crow's contribution, with Anthropic Claude assistance, from #370. Jacob Sussman's E8 unit and all upstream licenses and notices are preserved. The compressed bit composition, independent audit and combined package were prepared by Zhihao Chen with substantial OpenAI Codex assistance under Apache-2.0. #377's existing complex commit is included in this branch so the combined PR can be reproduced independently of whether either dependency has merged.

[Mister-Meeseeks]

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

Contribution: Compressed nine-stage windows and whole-block banking.

Review and limits: A separate finite native composition with explicit restoration and guard obligations; its updated wrapper is not automatically covered by a parent's replay.

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: dd11cf8069554f0333afde398f12a56c3a7c29dd. Publication: 4040a0f46d67b41c36a472a7cd4af791fd4c36cc.