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)

[DaysSky]

κ = 93191395651209/10¹⁷ ≈ 9.3191395651209e-4 (conditional). This is +6.45 % over #370 (8.75481145306751e-4) and +3.17 % over #370's bit word in five stages with #363's seven-stage complex unit (9.03256e-4, which #370 reports but does not claim).

It uses #370's E8 bit word and Sussman's E8 complex unit, both unchanged, and runs both in the seven-stage bridged word B₃ of #363. #370 priced its bit word in B₃ (c ≈ 9.682e-4) but noted that "a seven-stage bit construction does not exist yet". This PR builds that construction and checks it.

This PR adds one self-contained package, research/e8-bit-seven-stage/, 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 (h = 8, v = 120, R = 865, 9 centres)	B₃: m = 56, W = 1,585, D = 6v − 7·72 = 216	c = 968197503705007/10¹⁸
complex	Sussman's gcert1-e8-r783 (#363)	B₃: m = 63, W = 1,503, D = 216	b = 932783231884153/10¹⁸

The bit side is 3.8 % above the complex side, so the complex side binds. All 47 constraints of outer.assembly hold, the complex leaf cap itself is rejected, and the adjacent 10⁻¹⁸ grid point is rejected.

The bridged word B_K (#363's bridged.py). Pair p borrows its first forward invocation from pair p − 1. That exchanges K bank pairs with 2K + 1 invocations on H ⊕ 2K·H. One invocation of either unit is unchanged; only the word around it changes.

On the bit side, B₃ means:

• Idle climbs of [34] | [14, 7, 20] | [28, 7, 6]. The two climbs above m/2 are split at explicit intermediate frames (34 → 27 + 7, 28 → 27 + 1), so 2 · 27 < 56.
• Banks. All 865 helper residuals have width 8, and width-56 banks hold 7 of them. Since 7 ∤ 865, there are T = 7 literal replicas, giving 865 exact banks. The moment arithmetic is invariant under this normalization.

What is checked. Every step of #370's verify.sh runs, and so does the B₃ version of each stage-dependent step. Each new checker is a K-parameterized copy of a #370 checker. At K = 2 it reproduces that checker's receipt, and verify.sh compares them.

• Formal columns (columns_k.cpp -DKPAIRS=3). The bank schedule, twin bridges and terminal exchange are generated from 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's rule. The check covers all 1,585 physical columns over F₂, dirty helpers included, through 7 invocations, 12 twin bridges and the terminal exchange. All 12 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. At K = 2 the receipt equals κ = 8.75481145306751e-4: first E8 bit word (complex-bound; +9.29% over #352) #370's columns_h, and on the PR Conditional κ = 7.696840393833340e-4: a stronger p = 10 bit word — re-annealed pair module, p = 10 local design, balanced cube orientation, tiered reuse matching (+0.71% over #315's word) #325 p10b word it equals PR κ = 7.92580093369946e-4: w3 cube layer on #325's p10b word with a stronger p = 10 complex supplier #346's unmodified checker.

• PR234 geometry (geometry_k.py 3), on all 120 ports with W(E8) reflection words:

  • 7 blocks, routes involutive, 6 stage boundaries chained;
  • every idle climb a commuting nested pair whose gap rank equals bridged.py's;
  • the splits at dyadic intermediate frames, built from sums of orthogonal E8 root projectors, so the prime set stays {2};
  • the 7-window entrance completion exact.

  At K = 2 it gives κ = 8.75481145306751e-4: first E8 bit word (complex-bound; +9.29% over #352) #370's ranks (14, 14, 7, 7, 18, 18, 4, 4).

• Banks (tile_k.py): an exact width-56 tiling. At K = 2 it is byte-identical to tile_g.py.

• Bit coarse saving (certify_bit_k.py): histogram 7 × invocation + 240 × B₃ idle blocks, 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 two engines with the 10⁻¹⁶ fallback, and the outer assembly. At K = 2 its fields equal κ = 8.75481145306751e-4: first E8 bit word (complex-bound; +9.29% over #352) #370's κ receipt.

• Finite bill at m = 56 (finite_k.py). κ = 8.75481145306751e-4: first E8 bit word (complex-bound; +9.29% over #352) #370's formulas with the stage counts of B₃:

  • 7 invocations' additions plus 12v bridge additions;
  • route families 2·7·(2v + R) + 6v;
  • 7·NCOPY·T copy/erase episodes;
  • the selector bill 2·7·T·(…).

  The coefficient is 2⁴⁶ < 2⁸⁰, with q-power bound q³¹³⁷. At K = 2 the bill is byte-identical to κ = 8.75481145306751e-4: first E8 bit word (complex-bound; +9.29% over #352) #370's.

• Unchanged from κ = 8.75481145306751e-4: first E8 bit word (complex-bound; +9.29% over #352) #370: word regeneration, the independent all-column checker and its controls, legality (NCOPY = 9), and E8-Gram nondegeneracy with prime witnesses.

• Complex side (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's certify_b3.py): the B₂ regression reproduces 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 pins. At B₃ it runs the portable checks at m = 63, regenerates Sussman's Lean data byte for byte, runs his replay, and computes exact b₃.

Open obligations

• No Lean proof of B₃ on either side. Sussman's bridge2_certificate covers B₂. On the bit side B₃ rests on the all-column F₂ check of the whole seven-stage word and PR234's exact geometry on 7 blocks. On the complex side it rests on 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's checks.
• 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 bank construction for a non-paired-cube family, and the inherited bit-supplier interfaces.
• Everything 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 = 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 admit on the complex side: symbolic correctness, the precision/recovery and row interfaces, and the analytic transfer.
• The finite bill takes κ = 8.75481145306751e-4: first E8 bit word (complex-bound; +9.29% over #352) #370's formulas with seven-stage counts. Its proof text is the source527/p10 lineage's and is not re-derived here.
• This is a conditional finite construction, not a formal verification of the multiplication theorem.

Validation

python -m pip install -r research/e8-bit-seven-stage/requirements.txt    # numpy==2.5.3 mpmath==1.3.0 sympy==1.14.0; g++ C++17 + Boost headers
bash research/e8-bit-seven-stage/verify.sh /tmp/e8-bit-seven-stage-verify  # about 4 min

Local result (macOS, Python 3.12 and 3.14), last lines:

   B_3 formal columns PASS: all 1585 physical columns over F2 (dirty helpers) through 7 invocations, 12 twin bridges, terminal exchange; 12 omitted-bridge controls rejected
PASS E8 geometry in B_3 (7 stages, m = 56): all 120 ports (W(E8) reflection words), routes involutive, 6 stage boundaries chained, idle gap ranks [14, 14, 7, 7, 28, 28, 7, 7, 34, 34, 20, 20, 6, 6] (= bridged.py), paid idle children [14, 14, 7, 7, 27, 1, 27, 1, 7, 7, 27, 7, 27, 7, 20, 20, 6, 6] (explicit dyadic intermediate frames, 2 x 27 < 56); entrance census {0: 865 helpers}: 7-window completion exact, duplicate window rejected
PASS exact width-56 tiling with 7 replicas: (((8, 8, 8, 8, 8, 8, 8), 865),)
PASS coarse savings (two engines, 10^-16 fallback, adjacent grid point excluded): B_2 b = 876248285600677/1000000000000000000 = 8.762482856007e-04 (= #352); B_3 b = 932783231884153/1000000000000000000 = 9.327832318842e-04 (+6.452 %)
KAPPA = 93191395651209/100000000000000000 = 9.319139565121e-04 | 47 strict constraints positive, adjacent grid point rejected
PASS finite bill (E8, K = 3, m = 56): coefficient 57685376070645 (46 bits) < 2^80, q-power bound q^3137, payload prefix 60 bits < 104, delta_tau 37945757723910, 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)
PASS conditional kappa = 93191395651209/100000000000000000 = 0.00093191395651209 (seven-stage bit word, complex-bound by the seven-stage E8 complex unit; bit coarse 968197503705007/1000000000000000000 = 0.000968197503705007 )

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, its producer and independent checker, and the generalized bit-side checkers this package extends.
• Jacob Sussman: the E8 family and the E8 complex unit; the bridged five-stage word, which B₃ generalizes.
• 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 (DaysSky): B_K and the seven-stage complex unit.
• 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 and NOTICE-PR352.md.

The seven-stage bit construction, its checks 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: E8 bit with seven-stage complex composition.

Review and limits: Full shipped verification includes the changed finite bill and p10 regression. Local supplier proofs and global analytic assumptions are not conflated.

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: 3ff87b5dc0b22a57958547ee347d5da25b7f9401. Publication: 4040a0f46d67b41c36a472a7cd4af791fd4c36cc.