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)

[chafreaky]

κ = 9.98634656132003e-4

Change and mathematical scope

κ = 998634656132003/10¹⁸ ≈ 9.98634656132003 × 10⁻⁴. This is a conditional finite construction. It is complex-bound and sits +3.91502·10⁻⁷ (+0.0392 %) above our #382 (998243154231047/10¹⁸).

jacklightChen's #377 nine-stage complex component uses "the same pinned E8 local word". That is Sussman's gcert1-e8-r783, sha256 4b92f00f…. In our #378 we re-timed this unit by far piece relocation. The full rational operator on all 1,023 registers, the retained totals, h/v/R/cst/N, the ports, the start/final frames and the scatter are all unchanged. This PR substitutes #378's certificate for Sussman's inside #377's construction. The bit side is unchanged: it is our #382 word.

side	supplier	coarse saving
complex	#377's nine-stage component (m = 73, W = 14,727, deficit 2,496, max child 48, halving degree 2) with #378's unit	b = 999632926176283/10¹⁸ (was 999240641319551/10¹⁸, +3.92285·10⁻⁷)
bit	#382 (#370's E8 bit word + our retime + kernel, B₃ unsplit)	c = 251726760077483/(2.5·10¹⁷) ≈ 1.006907e-3

Why it transfers. #377's paid profile is the sum of three terms (from its complex_candidate.py/global_nine.py, read as data):

• 72 invocations × the certificate's child histogram, pooled over all blocks;
• 2vT idle advances at the ranks 16, 8, 32, 8, 48, 8, 48, 32, 16;
• 9R rank-1 paddings.

Only the pooled histogram depends on the gate schedule. It changes by {1:+2, 2:+2, 3:+6, 4:−21, 6:+10} per invocation, so the profile changes by 72 times that. Rank mass, W, the deficit and the max child 48 are unchanged. Our nine.py rebuilds #377's profile from Sussman's certificate, checks it equals GLOBAL-REVIEW.json for every rank, and re-certifies #377's b exactly. It then prices the swapped profile with both rational engines (10⁻¹⁶ fallback, next grid point excluded) and runs PR #315's outer.assembly (47 strict constraints, adjacent κ rejected).

Further moves. We re-ran our full move family with the χ ledger and price at #377's m = 73: far relocation, single/block/sub-block retiming and in-group pieces. From Sussman's original it converges to exactly #378's certificate, byte for byte. From #378 it finds nothing. The B₃ optimum carries over unchanged.

Inherited from #377 (its claims, not re-proved here):

• the m = 73 compressed window lift (beyond κ = 9.5424356568133e-4: seven-stage E8 suppliers with the complex idle climbs at halving degree two (+2.40 % over #373) #375's m ≤ 72 cover bound) and the arbitrary-bank dirty composition;
• the scalar chain, the idle advances and the paid padding;
• the exact Clifford data and cover phases (X_q, X_δ(g)) and the owner routes;
• the finite invoice;
• two-level halving (73² > 2·48²; row 15,445 < 20,161);
• everything κ = 9.98243154231047e-4: nine-stage complex E8 (≈2^-9.9683; complex +14.036%, bit side pending) #377 retains from κ = 8.01083628465007e-4: #346's w3 bit word with Sussman's Lean-checked E8 unit as the complex supplier (+0.96 % over #350) #352: the local E8 invocation theorem, the Clifford/compiler interfaces, tape primitives, precision/rows and the analytic interfaces.

None of #377's scripts were run. Only the receipt's profile, m, live and deficit are read.

Not re-run for the new unit (open obligations of this PR):

• κ = 9.98243154231047e-4: nine-stage complex E8 (≈2^-9.9683; complex +14.036%, bit side pending) #377/κ = 8.01083628465007e-4: #346's w3 bit word with Sussman's Lean-checked E8 unit as the complex supplier (+0.96 % over #350) #352's certificate-specific finite checks: emitter/splice, scalar words and lifetimes, the 946-frame/3,120-transition Clifford frame checks, and the invoice's local terms;
• these become necessary because the new unit has 775/1,226 gates (+0.4 %) and 2 new nested frames. Those frames fall under the general compiler interface, and the invoice exponents have ample slack, but neither is re-checked here;
• Conditional κ = 9.5464544449649·10⁻⁴: Sussman's E8 complex unit re-timed by far piece relocation, in #375's unsplit B₃ (+0.042 % over #375) #378's own condition: there is no Lean proof of the re-timed certificate. It is checked by Sussman's gx.check1/gxcore and by the full operator identity.

The bit side is #382's claim, including its bit-side halving-degree-two admission. Here c is only re-derived from the word.

Prerequisite: this branch is stacked on #382 (015f00b). The package itself is self-contained.

Credits:

• jacklightChen (Zhihao Chen), κ = 9.98243154231047e-4: nine-stage complex E8 (≈2^-9.9683; complex +14.036%, bit side pending) #377: the entire nine-stage complex construction used here.
• Jacob Sussman: the E8 unit, gx/gxcore and the stage geometry.
• aliromman (Reduce E8 recursive cost with four coupled frame changes #365): the first coupled re-timing of the unit, which Conditional κ = 9.5464544449649·10⁻⁴: Sussman's E8 complex unit re-timed by far piece relocation, in #375's unsplit B₃ (+0.042 % over #375) #378 generalises.
• DaysSky (Complex supplier b = 9.32783e-4: E8 in a seven-stage bridged word (+6.45 % over #352; no new κ, raises the complex cap) #363/κ = 9.3191395651209e-4: #370's E8 bit word and Sussman's E8 complex unit in the seven-stage bridged word (+6.45 % over #370) #373/κ = 9.5424356568133e-4: seven-stage E8 suppliers with the complex idle climbs at halving degree two (+2.40 % over #373) #375): B_K, the seven-stage bit construction and the halving-degree-two guard.
• utcorvusvolat-dotcom (κ = 8.75481145306751e-4: first E8 bit word (complex-bound; +9.29% over #352) #370): the E8 bit word.
• DreamingOfClouds (Conditional κ = 7.64230861320245e-4: the paired-cube bit word at cube size p = 10 (h = 20, m = 100), with a new centre-sharing complex supplier (+1.21% over #314) #315): the engines and the outer assembly.

Prepared by Chafik Boukhalfa (chafreaky) with substantial Anthropic Claude assistance. Apache-2.0.

Validation

• Commit fd81a16a1e4e82acfb4d45752cba17450644fde6 on research/e8-far-nine-complex (parent 015f00b = Conditional κ = 9.98243154231047·10⁻⁴: our retimed + kernel E8 bit word in B₃ (unsplit) with #377's nine-stage complex component (+0.26 % over #381) #382).
• bash research/e8-far-nine-complex/verify.sh --quick <dir> (~20 s) and the full mode (~40 s) both PASS locally, including from a fresh clone of the commit with a clean tree afterwards. The full mode adds the m = 73 regeneration. The checks cover:
  • the manifest;
  • git-blob pins for Sussman's certificate/gx/gxcore, Conditional κ = 9.5464544449649·10⁻⁴: Sussman's E8 complex unit re-timed by far piece relocation, in #375's unsplit B₃ (+0.042 % over #375) #378's certificate, κ = 9.98243154231047e-4: nine-stage complex E8 (≈2^-9.9683; complex +14.036%, bit side pending) #377's receipt and Conditional κ = 9.98243154231047·10⁻⁴: our retimed + kernel E8 bit word in B₃ (unsplit) with #377's nine-stage complex component (+0.26 % over #381) #382's word, plus the sha256 match against κ = 9.98243154231047e-4: nine-stage complex E8 (≈2^-9.9683; complex +14.036%, bit side pending) #377's pin;
  • gx.check1 (E0–E7, exact scalar), the gxcore mirror, the flipped-sign control and the source-span rule on both certificates;
  • the full operator identity;
  • κ = 9.98243154231047e-4: nine-stage complex E8 (≈2^-9.9683; complex +14.036%, bit side pending) #377's profile rebuilt field for field and its b reproduced exactly;
  • the new b, the bit c, and κ through Conditional κ = 7.64230861320245e-4: the paired-cube bit word at cube size p = 10 (h = 20, m = 100), with a new centre-sharing complex supplier (+1.21% over #314) #315's outer assembly. The regression gives Conditional κ = 9.98243154231047·10⁻⁴: our retimed + kernel E8 bit word in B₃ (unsplit) with #377's nine-stage complex component (+0.26 % over #381) #382's κ with κ = 9.98243154231047e-4: nine-stage complex E8 (≈2^-9.9683; complex +14.036%, bit side pending) #377's b.
• CI: .github/workflows/e8-far-nine-complex.yml runs quick and full on Python 3.12 and 3.14.

Lean (added)

The re-timed E8 certificate now has a kernel-checked theorem for the five-stage (B₂) unit, generated with Jacob Sussman's unchanged generators at 9c94857:

OAI.PowerSaving.WHT.wht_main_block_B2Ge8FRx : ∃ solve W, WHTProgram solve W ∧ WHTTimeBoundsAt (1 - 8766835/(10:ℝ)^10) W

The original unit's theorem is at 1 − 8762479/10¹⁰. lake build succeeds, Compare.lean reports RESULT: PASS, and the only axioms are propext, Classical.choice and Quot.sound. The evidence is in lean/LEAN-B2-E8FR.txt. The seven-stage and #377 nine-stage layouts priced in this PR still have no Lean proof.

Repository CI. This branch includes PR #318's unchanged three-file joint-dual source-pin repair (certificates/joint-dual-kappa.json, certificates/joint-dual-validation.json, research/joint-dual/SOURCE.json), as #366 and #368 do, so the repository-wide verify (joint) job passes. No mathematics is changed by it.

[Mister-Meeseeks]

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

Contribution: Retimed nine-stage complex supplier with kernel bit.

Review and limits: Both quick and regeneration modes are covered. Local Lean/checker evidence does not certify the global bank/layout machinery.

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: 3fa794621ee566edcad56f86d2e4d0ab53da490e. Publication: 4040a0f46d67b41c36a472a7cd4af791fd4c36cc.