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)

[DaysSky]

κ = 995656966441037/10¹⁸ ≈ 9.95656966441037e-4 (conditional), above 2⁻¹⁰ = 9.765625e-4. That is +2.94 % over #380 (9.67261003170394e-4) and +4.34 % over #375.

This PR runs #370's E8 bit word in the nine-stage bridged word B₄, with its idle climbs unsplit at halving degree two, and pairs it with #377's compressed nine-stage E8 complex component, vendored unchanged.

It adds one self-contained package, research/e8-nine-stage-kappa/, with its own workflow. No existing file changes except one .gitattributes line.

Change and mathematical scope

side	supplier	layout	coarse saving
bit	#370's bitE8-r865, unchanged	B₄: m = 72, W = 1,825, D = 8v − 9·72 = 312, unsplit (max child 50, 72² > 2·50²)	c = 249162322049641/(25·10¹⁶) ≈ 9.96649e-4
complex	#377's compressed nine-stage component, vendored with its verifier	m = 73, banks, halving degree 2	b = 999240641319551/10¹⁸

The bit side binds. All 47 constraints of outer.assembly hold, and the adjacent 10⁻¹⁸ grid point is rejected.

Why B₄ unsplit. #380 uses the seven-stage split bit word (9.68198e-4). The same word in B₄:

bit layout	c
B₃ split (#373)	9.6820e-4
B₃ unsplit	9.9123e-4
B₄ split	9.5702e-4
B₄ unsplit	9.9665e-4

Only the unsplit forms gain from the fourth pair.

Banks. 865 residuals of width 8 per stage tile width-72 banks with 9 per bank. Since 865 ≡ 1 (mod 9), this needs T = 9 literal replicas.

Halving degree two on the bit side. finite_k.py computes the least t with mᵗ > 2·(max child)ᵗ and charges the internal borrowed-row reserve t·(d + 1) = 10,370. At t = 1 it is the original bill. This is PR234's treatment ("The proper child ratio 5/6 requires halving degree four. The one-level weighted recurrence is retained; its histogram is not multiplied by four"). #375 and #377 use degree two on the complex side. The external row coefficient 20,161 is unchanged.

What is checked

bash research/e8-nine-stage-kappa/verify.sh <dir> runs the following.

Unchanged from #370:

• word regeneration, the independent all-column checker and its controls;
• legality (NCOPY = 9);
• E8-Gram nondegeneracy, with prime set {2}.

B₄, using #373's K-parameterized checkers:

• Formal columns (columns_k.cpp -DKPAIRS=4): all 1,825 physical columns over F₂, through 9 invocations, 18 twin bridges and the terminal exchange. All 18 omitted-bridge controls and κ = 8.75481145306751e-4: first E8 bit word (complex-bound; +9.29% over #352) #370's 6 record-level controls are rejected.
• PR234 geometry (geometry_k.py 4 … unsplit), on all 120 ports with W(E8) reflection words:
  • 9 blocks, routes involutive, 8 stage boundaries chained;
  • idle gap ranks equal to bridged.py's;
  • the 9-window entrance completion exact;
  • largest child 50, halving degree 2.
• Banks: an exact width-72 tiling with 9 replicas.
• Bit coarse saving and outer assembly: two engines with the 10⁻¹⁶ fallback, the adjacent grid point rejected, and 47 constraints with κ = 9.98243154231047e-4: nine-stage complex E8 (≈2^-9.9683; complex +14.036%, bit side pending) #377's b.
• Finite bill at m = 72: coefficient 2⁴⁸ < 2⁸⁰, q-power bound q⁵¹⁸⁵, halving degree 2.

Regressions. Each K-parameterized checker also runs at K = 2 and reproduces #370's receipt exactly: columns, tiling, κ fields and the finite bill byte for byte. #370's p10b regressions also run.

Complex. #377's own verify.py runs unchanged on its vendored, manifest-pinned package. It checks the Clifford frames, transitions and adapters, the data and cover phases, the global template and two moment engines. It gives b = 999240641319551/10¹⁸.

Provenance. Git blob ids are recorded for 162 files from #377 (d0a620f), 52 from #370 (b7fe9cd), 4 from #373 and 1 from #363.

Open obligations

• No Lean proof of B₄ on the bit side. It rests on the all-column F₂ check of the whole nine-stage word and on PR234's exact geometry on 9 blocks.
• Halving degree two on the bit side. The inherited bit bill states degree one. Here the internal row term is charged at degree two, per PR234's degree-four precedent.
• 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;
  • the bit interfaces.
• Everything κ = 9.98243154231047e-4: nine-stage complex E8 (≈2^-9.9683; complex +14.036%, bit side pending) #377 admits: the local E8 theorem, the Clifford/compiler, common-grid, row and analytic interfaces.
• The finite bill takes κ = 8.75481145306751e-4: first E8 bit word (complex-bound; +9.29% over #352) #370's formulas with B₄'s stage counts.
• This is a conditional finite construction, not a formal verification of the multiplication theorem.

Validation

python -m pip install -r research/e8-nine-stage-kappa/requirements.txt    # numpy==2.5.3 mpmath==1.3.0 sympy==1.14.0; g++ C++17 + Boost headers
bash research/e8-nine-stage-kappa/verify.sh /tmp/e8-nine-stage-kappa-verify

Local result (macOS, Python 3.12), selected lines:

   B_4 formal columns PASS: all 1825 physical columns over F2 (dirty helpers) through 9 invocations, 18 twin bridges, terminal exchange; 18 omitted-bridge controls rejected
PASS E8 geometry in B_4 (9 stages, m = 72): all 120 ports (W(E8) reflection words), routes involutive, 8 stage boundaries chained, idle gap ranks [14, 14, 7, 7, 28, 28, 7, 7, 42, 42, 7, 7, 50, 50, 36, 36, 22, 22, 8, 8] (= bridged.py), paid idle children [14, 14, 7, 7, 28, 28, 7, 7, 42, 42, 7, 7, 50, 50, 36, 36, 22, 22, 8, 8] (un
   unsplit: largest idle child 50 of m = 72, least halving degree t = 2 (72^2 > 2 * 50^2)
PASS exact width-72 tiling with 9 replicas: (((8, 8, 8, 8, 8, 8, 8, 8, 8), 865),)
   PR #377 complex coarse saving b = 999240641319551/1000000000000000000 (two moment engines, adjacent grid point rejected, halving degree two)
KAPPA = 995656966441037/1000000000000000000 = 9.956569664410e-04 | 47 strict constraints positive, adjacent grid point rejected
PASS finite bill (E8, K = 4, m = 72): coefficient 219390399358525 (48 bits) < 2^80, q-power bound q^5185, payload prefix 60 bits < 104, delta_tau 21235252823236, reference route {'swap': 7}; all 120 port routes exact elementary programs, at most 77 factors, denominators [1, 2, 4] (powers of two: units modulo every odd q)
   internal row reserve {'internal_coefficient': 10370, 'external_complex_coefficient': 20161, 'halving_degree': 2}
PASS conditional kappa = 995656966441037/1000000000000000000 = 0.000995656966441037 (bit-bound: E8 bit word in B_4 at halving degree two, c = 249162322049641/250000000000000000 = 0.000996649288198564 ; PR #377 complex b = 0.000999240641319551 ); above 2^-10 = 0.0009765625

These are finite certificate checks only; the obligations listed above remain open.

Credits and AI disclosure

• Dylan Crow (utcorvusvolat-dotcom), κ = 8.75481145306751e-4: first E8 bit word (complex-bound; +9.29% over #352) #370: the E8 bit word and the generalized bit-side checkers.
• Zhihao Chen (jacklightChen), κ = 9.98243154231047e-4: nine-stage complex E8 (≈2^-9.9683; complex +14.036%, bit side pending) #377: the compressed nine-stage complex component, vendored unchanged. κ = 9.98243154231047e-4: nine-stage complex E8 (≈2^-9.9683; complex +14.036%, bit side pending) #377 states it was prepared with OpenAI Codex assistance.
• rohanarun, Conditional κ = 9.67261003170394e-4: seven-stage E8 bit with nine-stage complex (+1.32149% over #378) #380: first paired κ = 9.98243154231047e-4: nine-stage complex E8 (≈2^-9.9683; complex +14.036%, bit side pending) #377 with an E8 bit word.
• Jacob Sussman: the E8 family and unit, and the bridged five-stage word.
• 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 (DaysSky): B_K, the K-parameterized bit checkers, and halving degree two.
• 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/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 NOTICE.md.

The nine-stage bit step, its halving-degree generalization and the package were prepared by DaysSky, assisted with Claude.

[Mister-Meeseeks]

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

Contribution: Nine-stage E8 bit/complex composition above 2^-10.

Review and limits: Dedicated full finite checks, guards and generalized invoice. Historical conditional milestone; no update to the current selected bound.

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: 00765b291d0f924f337298eb1e2930cb0878b95e. Publication: 4040a0f46d67b41c36a472a7cd4af791fd4c36cc.