Conditional κ = 1.128983115440183e-3: an E8 bit word with 740 helpers under #433's complex supplier

[eumemic]

#433 (icekylinx) is bit-bound: its complex supplier has b = 1183986549206329/10¹⁸, and its R758 bit word gives c ≈ 1.122123e-3. This PR replaces the bit word with an E8 bit word with R = 740 helpers, in #386's compressed nine-stage layout. Pricing uses #433's own bit engine, finite invoice and six-level tail.

κ = 1128983115440183/10¹⁸ ≈ 1.128983115440183e-3, bit-bound. That is +0.7243 % over #433 (560432476129139/(5·10¹⁷)). It is also above #435 (1.128832830332521e-3, #417's kernel on #433's R758 word) and #434, which were posted while this was being prepared. The kernel of #417/#435 and this word's reuse are different mechanisms; stacking them is not attempted here.

layout	bit c of this word	best published in that layout	gain
compressed nine-stage (#386; used by #425, #431, #433)	282564789736676592463/(2.5·10²³) ≈ 1.130259e-3	#433 (R758)	+0.725 %
B₄ unsplit (#381)	1064525457834209/10¹⁸ ≈ 1.064525e-3	#417	+1.960 %

One self-contained package, research/e8-bit-elimination-reuse/, with its own workflow, plus one .gitattributes line. No existing file changes.

Mechanism

1. κ = 8.75481145306751e-4: first E8 bit word (complex-bound; +9.29% over #352) #370's producer with new parameters (67 of 520 entries changed) gives R = 863.
2. Elimination reuse of dead helper slots gives R 863 → 740.
   • A dead helper still holds a leftover x-content, an F₂ sum of source lines.
   • Before a later helper is born at frame F, take an F₂ relation among the leftovers of dead slots inside F. One gate at the join of their frames empties one slot, and the new helper takes it over.
   • The 9 retained totals also take dead slots; their leftover lines are read at the full frame.
   • 114 slots are reused by helpers and 9 by totals.
3. Single-gate frame retiming, Complete far-retimed E8 replay and coupled bit frames; conditional kappa 0.000998634658090835 #389's idea: 37 gate moves.

Overlap.

• Conditional kappa 0.001018766833114736: checker-clean chronological gauges and R767 reuse #426 (ai-nanachi) published the same slot-reuse mechanism first: 55 "clean resets" on Conditional κ=0.0011208649522582786: split centres with complete signed admission #376's R822 word, giving R767. We developed this pass independently. On a different base word it finds 114 reuses plus the 9 totals.
• Split centres and source-basis reuse for conditional κ = 0.001120864952258278 #433's nine-sum source-basis change frees no register here, because the totals already sit on dead slots.

Checks

bash research/e8-bit-elimination-reuse/verify.sh [--quick] DIR: about 40 s quick, 67 s full.

• Provenance. Manifest; 52 vendored files pinned by git blob id. Every cited complex b and κ must equal that PR's own receipt.
• Rebuild. The word is rebuilt from pinned inputs and must equal the shipped word. κ = 8.75481145306751e-4: first E8 bit word (complex-bound; +9.29% over #352) #370's producer regenerates κ = 8.75481145306751e-4: first E8 bit word (complex-bound; +9.29% over #352) #370's word as a control.
• κ = 8.75481145306751e-4: first E8 bit word (complex-bound; +9.29% over #352) #370's independent checker: exact F₂ on all 980 columns with dirty helpers, and Gram determinants > 0 on every frame (only prime 2). 6 word-level and 6 reuse-specific mutants are rejected.
• Official checkers. 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 legality at NCOPY = 9; NCOPY = 8 is rejected. Five-stage and B₄ formal columns, with 18 omitted-bridge controls rejected.
• Compressed nine-stage E8 windows and whole-block banking for a conditional κ = 9.98243154231047e-4 > 2^-10 bound #386 layout. Geometry, source-bound template, audit and price.
• Split centres and source-basis reuse for conditional κ = 0.001120864952258278 #433's engine, invoice and tail. These reproduce Split centres and source-basis reuse for conditional κ = 0.001120864952258278 #433's own c and κ exactly, and Compressed nine-stage E8 windows and whole-block banking for a conditional κ = 9.98243154231047e-4 > 2^-10 bound #386's, κ = 1.048005508028206e-3: mutable-basis Hamming packets (≈2^-9.89814; conditional; complex +4.97%) #425's and Candidate conditional kappa 0.001056975315151441: Hamming complex and retimed bit composition #431's κ. The tail uses 47 strict inequalities; the next grid point, zero η, zero β and unit β are rejected.
• B₄. 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, banks (T = 9), finite bill and price.
• Local build. The C++ checkers were built against an overflow-checked 128-bit stand-in for boost::multiprecision::cpp_int, because the preparing machine has no Boost. In CI, the package workflow built them with libboost-dev and passed on Python 3.12 and 3.14.
• Repository CI. This branch includes rohanarun's 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 three-file joint-dual pin repair (byte-identical to a45747d), as E8 bit supplier c ≈ 1.044064·10⁻³ in the nine-stage word (+2.2 % bit cap; no new κ, complex-bound) #417 and Conditional κ = 0.001133928043872860: R740 shared-donor kernel with explicit residual bank routes #439 do, so the repository joint check passes.

Retained assumptions

• Everything κ = 8.75481145306751e-4: first E8 bit word (complex-bound; +9.29% over #352) #370 admits: 9 retained totals at h = 8, PR234's banks for a non-paired-cube family, and the all-size bit compiler, prime, tape, precision and analytic interfaces.
• Compressed nine-stage E8 windows and whole-block banking for a conditional κ = 9.98243154231047e-4 > 2^-10 bound #386's compressed-layout arguments, including two-level halving (max child 42/64).
• Split centres and source-basis reuse for conditional κ = 0.001120864952258278 #433's complex supplier is cited, not rebuilt. Its proof rests on κ = 1.048005508028206e-3: mutable-basis Hamming packets (≈2^-9.89814; conditional; complex +4.97%) #425's written scalar-interface extension.
• This is a conditional finite construction, not a Lean proof.

Credits

utcorvusvolat-dotcom / Dylan Crow (#370: format, producer, checkers); applecrusher (#376); ai-nanachi (#389 retiming; #426 clean-reset reuse, published first); icekylinx (#433 complex supplier, bit engine, invoice and tail); DaysSky (#381, #375, #363); Zhihao Chen / jacklightChen (#386, #425); Chafik Boukhalfa / chafreaky (#417); Rohan Arun (#392, #431); Jacob Sussman (E8, bridged word); the PR #266 and PR #315 (DreamingOfClouds) lineage. Vendored files are byte-identical and listed in NOTICE.md and SOURCE-pr*.json.

Prepared by eumemic with substantial Anthropic Claude assistance. Apache-2.0.

🤖 Generated with Claude Code

[Mister-Meeseeks]

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

Contribution: Dead-helper elimination reuse yielding the R740 bit word.

Review and limits: Preserve independent parallel credit for ai-nanachi's earlier reuse and this different-base construction. Full dirty-column, frame and compressed-layout obligations remain.

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: 8f6b8297b08ad863940d5168a2db5402fad3690f. Publication: 64039f1bd3b6a523b96083fd858ad040a75e3030.