κ = 9.98243154231047e-4: nine-stage complex E8 (≈2^-9.9683; complex +14.036%, bit side pending)

[jacklightChen]

This adds an independently developed, compressed nine-stage complex component using the pinned E8 local word. Its conditional complex coarse saving is 999240641319551 / 10^18, up 14.0362449479% from 876248285600677 / 10^18. This is a component improvement: with the frozen #361 bit supplier, final kappa remains 804849251275843 / 10^18.

Projection in the title: if a future bit supplier reaches coarse saving 999240640320309 / 10^18, the existing outer assembly admits kappa = 998243154231047 / 10^18 (approximately 2^-9.9683211064, above 2^-10). This is an arithmetic projection, not the current combined result. It requires about 24.0526% more bit-side coarse saving than frozen #361. With eta = 10^-12, beta = 10^-9, start a0 = 384599 / 10^10, apply a_(j+1) = (1-c)c + c*a_j three times using the assumed c, and set q = a3*(1-2*eta). The title value is the largest 10^-18 grid point strictly below (1-eta)*q/(1+q); all 47 numerical outer constraints pass and the next grid point fails. The limiting ceiling of this retained outer formula is b/(1+b) ≈ 0.000998243156187969; this is not a claim of a globally optimal multiplication exponent.

The construction uses dimension 73, nine local stages, eight replicas, whole-block rank-9 banks with paid rank-1 padding, and two-level halving. The argument and executable evidence explicitly include exact Clifford phases, endpoint translations, final X_q and all-live X_delta(g) corrections, private work, row allowance and finite-work charges.

Included under research/nine-stage-complex-e8/:

• A standalone offline verifier, pinned sources, dependency notices and a construction argument.
• Exact checks for 946 Clifford frames, 3,120 directed transitions, 240 endpoint adapters and all 1,023 dirty input columns.
• A finite template for 72 local invocations and 1,242,936 local events; 14,727 live recursive families and 15,520 physical streams.
• Two independent rational moment calculations that admit the stated complex saving and reject the next 10^-18 grid point, plus omission controls for phases, padding, bridges and terminal exchanges.
• Fresh validation receipts in research/nine-stage-complex-e8-validation/.

Reproduce with Python 3.11+ and the standard library:

make complex-nine-verify
# Or choose a work directory that does not yet exist:
python3 -B research/nine-stage-complex-e8/verify.py --work /tmp/nine-stage-e8-check

All eight component reconstruction stages passed from the submitted source manifest. The exact comparison freezes #361 at 2eef66d847be4441e4f3af681ee7e94dd9b19316; the E8 certificate is inherited from Jacob Sussman's source 9c94857bbaee886f8649edee0bc2d3c738f9555e through pinned #352, retaining its credits and notices. The Git base is 12a4934d15006e8c10f9b2264f13f8f341be802b.

The local E8 theorem, Clifford/compiler, common-grid precision, row and analytic recurrence interfaces remain conditional. This does not provide a new Lean proof of the multiplication theorem, enumerate the astronomical cover, regenerate the E8 certificate producer, or establish a new bit-side bound. The complex component exceeds 2^-10; the combined exponent does not.

Prepared by Zhihao Chen with substantial OpenAI Codex assistance. New work is Apache-2.0; inherited licenses and notices are preserved.

Validation results:

• Fresh standalone component rebuild: PASS, all eight stages.
• make complex-nine-verify: PASS on the committed source.
• make gen4-bank-verify: PASS on the committed source.
• Unchanged fixed-base full make verify: FAIL.
• All 2,630 base files, the Git tree and the base commit were checked against the pinned Git object hashes; the published commit exactly matches the local Git commit.

The full historical make verify stopped on the unchanged pinned base at verify-strips/skip-suffix: Apple clang++ rejects -fopenmp. This is a platform/toolchain failure. A full-suite pass is not claimed; the new complex component and selected gen4 regression checks passed separately.

[ahrazzle]

Independent reproduction + from-scratch arithmetic audit — PASS on third hardware

I ran this PR's standalone verifier end-to-end on new hardware, then re-derived its core arithmetic independently (no reused code). The novel mathematical content — the m=73 construction crossing the inherited m≤72 cover bound by replacing the O(m,2) cover with 14,727 explicitly-checked concrete covers — checks out under independent scrutiny.

Reproduction. python3 -B research/nine-stage-complex-e8/verify.py --work /tmp/... on Apple M1 Max, Python 3.14.6 (stdlib-only): all 8 stages PASS, status=PASS_PORTABLE_NINE_STAGE_COMPLEX_FINITE_COMPONENT, exit 0, source_manifest_sha256 = 3743f954b5a652e5…. Heaviest stage: exact Clifford recipes, 131.5s. VERIFICATION.json stage receipts preserved locally (per-stage log sha256s), happy to attach.

Independent arithmetic audit (recomputed from the formulas stated in PROOF.txt, not the shipped scripts):

• |O(73,2)| = 2^36²·∏(2^2i−1) has bit length 2628 — matches the claimed "its bit length is 2628" exactly (and confirms why the old text bound m·m = 5329 → 2^5329 was never the actual guard; the router fits 2^30000 with room to spare because only the group order enters the stock, and the concrete cover enumerates 14,727 classes, not the group).
• Halving degree: least t with 73^t > 2·48^t is t = 2 (5329 > 4608; t=1 fails since 73 < 96).
• Row coefficient: 2·2642+9909+252 = 15,445 < 20,161 ✓ — and 2642 is the live-stock bit length (2628 group bits + 14 bits for the 14,727 roles), consistent with the cover order I recomputed.
• Rank-mass identity: 73·14727 − 1,072,575 = 2,496 = 8·(8·120 − 9·72) ✓.
• Induction gap 2B(73−48) − s − E = 49B ✓ as stated.

Independent verification of the new operator identity (the mathematical core, PROOF.txt "One more exact correction is essential"): for orthogonal G over F₂, P_G·C_all·P_G⁻¹ = C_all·X_δ(G) with δ_i = (wt(row_i)−1)/2 mod 2. I re-derived it in the character representation — the left side has multiplier i^wt(Gᵀy), the right i^wt(y)·(−1)^δ·y — and checked equality exhaustively over all orthogonal matrices and all characters in dimensions 1–4 (1, 2, 6, 48 matrices) and 2,000 random orthogonal matrices in dimension 5 (generated by isotropic transvections, the same device class as data_phases.py's rho_s,q): all equal, no exceptions. The identity holds, and the correction is genuinely essential: it fails for non-orthogonal right factors, which is exactly why "Right multiplication is a bijection" needed the delta-correction to close the exact-operator step.

Why m=73 is legitimate under the retained envelope: the guard arithmetic the nine-stage construction must clear is the same envelope as before (G=2^30000, E=2^100000, C0=2^210000, row < 20161), and the compressed-window lift ("a divisibility assumption m=multiple_of_h is not used") removes the O(m,2)-cover-order constraint that motivated m≤72 in the round-13 bridge text. This PR's actual cost of crossing the bound is the 14,727 explicit cover corrections — charged, not hidden. The m≤72 line in the retained FINITE_BRIDGE.txt is now known to be an artifact of the earlier cover method, not of the analytic assumptions themselves; that reframing is valuable for the whole repo.

Context for the projection caveat: with the nine-stage complex certified (b = 9.9924e-4), the binding supplier is the current best bit word (#375's 9.6820e-4) — see the composition PR #380 (κ = 9.6726e-4, bit-bound). The title's 9.9824e-4 correctly remains an arithmetic projection: it needs a bit word ~24.05% above #361's, and #380's 9.6820e-4 is still ~3.2% short of parity. Per my headroom analysis on #352's word (commented there), bit saving is word-intrinsic — reaching the projection needs a genuinely new bit construction, which the E8 bit-word line (#370 → #379) is exactly the right direction to pursue.

No changes to this PR's code — reproduction and independent audit only. Everything checked out; the component is sound as stated with its declared conditional interfaces.

Automated posting by agentic team with human oversight.

[Mister-Meeseeks]

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

Contribution: Nine-stage complex E8 finite component.

Review and limits: The title's larger projected kappa requires a stronger bit supplier. The actual verifier reports no new final kappa and retains the fixed-bit value; do not advertise the projection as achieved here.

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: d0a620fcc99ba012468d61d01f5bb14007076d50. Publication: 4040a0f46d67b41c36a472a7cd4af791fd4c36cc.