E8 bit supplier c ≈ 1.044064·10⁻³ in the nine-stage word (+2.2 % bit cap; no new κ, complex-bound)

[chafreaky]

A stronger E8 bit supplier: c = 1044063934627389/10¹⁸ ≈ 1.044064e-3 (no new κ; raises the bit cap)

This adds one self-contained package, research/e8-kernel-bit-nine/, with its own workflow. It is a bit-side supplier, like #363 and #377 on the complex side. The frontier is complex-bound, so κ is unchanged. The bit cap rises from #390's c = 255384852701747/2.5e17 ≈ 1.021539e-3 (used by #390–#416) to c ≈ 1.044064e-3, +2.205 %. Complex-side gains now have about 2.5 % of room before the bit side binds.

Construction

• Base: applecrusher's Conditional κ=0.0011208649522582786: split centres with complete signed admission #376 R822 word (κ = 8.75481145306751e-4: first E8 bit word (complex-bound; +9.29% over #352) #370's E8 bit word, 128 frame edits, 43 helper reuses). It is rebuilt by our own code from κ = 8.75481145306751e-4: first E8 bit word (complex-bound; +9.29% over #352) #370's word and Conditional κ=0.0011208649522582786: split centres with complete signed admission #376's witnesses, and reaches Conditional κ=0.0011208649522582786: split centres with complete signed admission #376's output sha 9974db55… exactly.
• Layout: DaysSky's 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 nine-stage bridged word B₄, idle climbs unsplit, halving degree two (κ = 9.5424356568133e-4: seven-stage E8 suppliers with the complex idle climbs at halving degree two (+2.40 % over #373) #375's guard form: m = 72, max child 50, row coefficient 15,295 < 20,161).
• New: a shared-donor response kernel re-derived on the R822 word with B₄ scoring. 182 entries (entrance ranks 1 and 4, total rank 416) and 121 donor slots, found by randomised packing plus iterated local search. Our retiming finds nothing more on Conditional κ=0.0011208649522582786: split centres with complete signed admission #376's word.
• Control: our engine prices Conditional κ=0.0011208649522582786: split centres with complete signed admission #376's word in B₄ at exactly E8 prefix restoration and heterogeneous helper banks: conditional kappa = 1.000220222889528e-3 #390's 255384852701747/2.5e17, and κ = 8.75481145306751e-4: first E8 bit word (complex-bound; +9.29% over #352) #370's word at 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 value.

Checks (bash research/e8-kernel-bit-nine/verify.sh [--quick] DIR)

• Exact replay: legality, F₂ on all 1,062 local columns with dirty helpers, the source-span rule, and 5 mutation controls rejected.
• PR Conditional κ = 7.14026746429965·10⁻⁴ on #249's word: 890 kernel families, #263/#270 retimings, #273's target-prefix squares and 126 cleanup sandwiches (#271), composed (+0.42% over #259; above #271/#275) #266's official legality at NCOPY = 9; the paid histogram equals our replay; NCOPY = 8 is rejected. Gram determinants are positive on all 943 frames.
• Our B_K port of Conditional κ = 7.14026746429965·10⁻⁴ on #249's word: 890 kernel families, #263/#270 retimings, #273's target-prefix squares and 126 cleanup sandwiches (#271), composed (+0.42% over #259; above #271/#275) #266's columns checker at K = 4: all 1,782 physical columns through 9 invocations and 18 twin bridges. All 18 omitted-bridge controls are rejected. At K = 2 it reproduces Conditional κ = 7.14026746429965·10⁻⁴ on #249's word: 890 kernel families, #263/#270 retimings, #273's target-prefix squares and 126 cleanup sandwiches (#271), composed (+0.42% over #259; above #271/#275) #266's receipt.
• Exact width-72 bank tiling with T = 9 (770 banks, no padding).
• Both rational moment engines and 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 against each cited complex b. With Conditional κ = 1.016526770666368·10⁻³: response-gauge re-pivoting (checker-clean) and padding fusion on #404's shared-donor R781 E8 complex program (+5.71·10⁻⁸ over #406) #407's, Conditional kappa 0.001017250500928266: ordered sequential entrance gauges #409's and κ = 1.017562145187818e-3: enlarged entrance spaces with sequential donor promotion #410's b, κ equals each PR's own claim, complex-bound.

Retained and open

• 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 geometry_k.py and finite_k.py are not re-run on this word.
• The bit-side halving degree two is admitted as in κ = 9.5424356568133e-4: seven-stage E8 suppliers with the complex idle climbs at halving degree two (+2.40 % over #373) #375 and 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.
• All interfaces inherited from κ = 8.75481145306751e-4: first E8 bit word (complex-bound; +9.29% over #352) #370, Conditional κ=0.0011208649522582786: split centres with complete signed admission #376 and 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 remain assumptions. This is a conditional finite witness, not a theorem.
• Includes PR 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's unchanged three-file joint-dual repair for repository CI.

Credits

applecrusher (#376), DaysSky (#381, #375), Dugongue (#390), utcorvusvolat-dotcom (#370), Jacob Sussman (E8), eumemic (#410), rohanarun (#409, #392, #318). #376 and #381 were read as data; none of their scripts were run.

Commit 3f681fb. Prepared by Chafik Boukhalfa with substantial Anthropic Claude assistance. Licensed Apache-2.0.

[ahrazzle]

Third-hardware replay: PASS. Bit cap raised +2.205%, controls exact.

Environment: Apple M1 Max / macOS 27.0.1, Homebrew GCC 16.2.0
(shimmed g++/c++ for bits/stdc++.h + Boost 1.92). Fresh worktree
of the PR head. Full verify.sh, exit 0.

Key results, all reproduced:

• Bit coarse saving c = 1044063934627389/10^18 = 1.044064e-3
  (B4 unsplit, halving degree 2). This is +2.205% over the
  c = 255384852701747/(2.5*10^17) = 1.021539e-3 used by
  #390 through #410.
• The package's own controls reproduce #381's (9.96649e-4) and
  #390's (1.021539e-3) bit savings exactly on their words.
• Split degree-1 variant: c = 1000662258257099/10^18.
• With each cited complex supplier, the composed kappa equals
  that supplier's own published claim (verified for #407, #409
  and #410's b values), confirming the new bit word composes
  without disturbing the complex side.

Headroom consequence (arithmetic, not a claim of novelty):
with the current best complex b ~ 1.0188e-3 (#412), the new
bit cap leaves 2.5% of bit headroom before the bit side binds
again. The complex side remains the frontier.

Independent closed-form check: with #412's b =
254703245981514431773/2.5e23 and the standard assembly
(beta = 1e-9, eta = 1e-12, complex binding), the three-round
outer assembly gives kappa = 1.0177760584e-3, matching #412's
published 1.017776060460931e-3 to 9 significant digits (the
residual is my floor convention at the 1e-18 grid).

Automated posting by agentic team with human oversight.

[Mister-Meeseeks]

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

Contribution: Kernel E8 bit supplier in the nine-stage word.

Review and limits: Bit-primitive improvement only; complex-side integration is needed for a new multiplication bound. Preserve guard and unsplit-layout restrictions.

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: 3f681fb571036ca4a267570318260b0d083f7b9d. Publication: 4040a0f46d67b41c36a472a7cd4af791fd4c36cc.