κ = 8.75481145306751e-4: first E8 bit word (complex-bound; +9.29% over #352)

[utcorvusvolat-dotcom]

E8 bit word: κ = 8.75481145306751e-4 (complex-bound)

Claim. We claim a conditional κ = 875481145306751/10¹⁸ ≈ 8.75481145306751·10⁻⁴. That is +9.29 % over #352 and
+8.54 % over the best p = 10 bit package.

What is new. This is the first bit supplier that is not a paired-cube word. Its labels are the 120 root lines of E8
in Q⁸, with the E8 form, which is positive definite, so every frame is nondegenerate:

• h = 8, m = 40, v = 120, R = 865;
• 9 retained totals at the full frame, so copy loss 72 and deficit 120;
• the junk of each target is the E7 in its cap;
• a side network built on D4 blocks and π-coset pairings, after a local search.

Bit coarse saving. c = 9.04072820242522·10⁻⁴. That is +12.0 % over the previous bit root and 3.2 % above
Sussman's E8 complex unit (b = 8.76248285600677·10⁻⁴), which therefore binds. With #363's seven-stage b₃ the same bit
word would bind at κ = 9.0326e-4; this package reports that number but does not claim it.

How it is checked (bash verify.sh DIR, about 2.5 min):

1. The word is regenerated from its producer parameters.
2. An independent checker verifies:
   • exact rational frame legality;
   • the exact F₂ identity on all 1105 columns, with dirty helpers;
   • the Gram determinants and the price.
     It rejects 6 word-level mutants.
3. The 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 official legality and five-stage formal-columns checkers run with h = 8 and NCOPY = 9. 6 record-level
   mutants are rejected.
4. Exact nondegeneracy under the E8 Gram and PR 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 prime-witness rule. The only prime is 2.
5. PR234's five-stage route and idle geometry is checked on all 120 ports, with W(E8) reflection words as port
   isometries. The idle ranks are (14, 7, 18, 4).
6. Banks: 173 exact width-40 banks.
7. PR 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 complex checks, unchanged.
8. PR 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 moment engines and the outer assembly with all 47 constraints. The adjacent grid point is rejected.
9. The finite bill at m = 40: coefficient 41 bits < 2⁸⁰.

Regressions. Every bit-side generalization has a parameter whose default is the original. On the PR #325 p10b word,
and on agent G's p = 10 loans receipts, each one reproduces the original checker's output:

• legality: byte-identical receipt;
• five-stage columns: identical receipt;
• prime witnesses: identical counts;
• banks: byte-identical output;
• certify_bit: identical fields;
• finite bill: the same coefficient, gaps, chain and route hash;
• geometry: the same routes.

Admitted explicitly. 9 retained totals at h = 8: a layout pin of the paired-cube words. It is charged as NCOPY = 9
copies, temporaries and episodes. All inherited bit-supplier interfaces, and #352's complex obligations, are retained.
PR234's bank construction for a non-paired-cube family is checked on the exact matrices; it is not re-proved.

Author: Dylan Crow, with Anthropic Claude assistance. Credits:

• Jacob Sussman: the E8 family and unit;
• DaysSky: 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, and 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 for the secondary step;
• κ = 7.92580093369946e-4: w3 cube layer on #325's p10b word with a stronger p = 10 complex supplier #346/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/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/source527/PR234 lineage: see README.md and NOTICE.md.

[Mister-Meeseeks]

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

Contribution: First E8 bit word.

Review and limits: The bit-side mechanism is distinct from the E8 complex unit. Retain the complete shipped word/guard/price checks and the source's common-grid, lowering and all-size assumptions.

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: b7fe9cdf2da009c947ff6b334c562bc190b30373. Publication: 4040a0f46d67b41c36a472a7cd4af791fd4c36cc.