Complex supplier b = 9.32783e-4: E8 in a seven-stage bridged word (+6.45 % over #352; no new κ, raises the complex cap)

[DaysSky]

Complex supplier: E8 in a seven-stage bridged word, b = 932783231884153/10¹⁸ ≈ 9.32783e-4. That is +6.45 % over the five-stage E8 of #352 (8.76248e-4). No new κ: the bit side binds at about 8.05e-4 (#361/#362). This PR raises the complex cap, so bit-side improvements count toward κ up to about 9.33e-4 instead of 8.76e-4.

This PR adds one self-contained package, research/e8-seven-stage/, with its own workflow. No existing file changes except one .gitattributes line.

Change and mathematical scope

The layout. Sussman's bridged word B₂ (Work/Bridge/Net.lean at 9c94857) gives every class two bank pairs. Pair 1 runs forward, backward, forward. Pair 2 runs backward, forward and borrows its first forward invocation from pair 1, through three layers of free ±1 additions between twin roles at equal frames. The result is two exchanges in five invocations, on H ⊕ 4·H.

B_K repeats the borrowing along a chain: pair p borrows from pair p − 1 the same way. That gives K pairs in 2K + 1 invocations on H ⊕ 2K·H, with m = (2K+1)h, W = 2Kv + R and D = 2Kv − (2K+1)·cst.

For E8 (v = 120, R = 783, cst = 72):

	stages	m	W	D	certified b
B₂ (#352)	5	45	1,263	120	876248285600677/10¹⁸
B₃	7	63	1,503	216	932783231884153/10¹⁸
B₄	9	81	1,743	312	9.19445e-4 (lower)

The helper circuit is E8, unchanged. Only the word around it changes.

What is checked:

• Layout. All frames are coordinate sets in the one orthonormal basis per class, as in Sussman's general stage geometry (Work/BridgeGeom/Stage.lean, proved for any number of blocks). bridged.py verifies, for K = 2 and K = 3:

  • every climb is nested;
  • every twin gate is at equal frames;
  • every invocation is entered at its stage frame;
  • the final frames are correct;
  • every data role makes m − 1 moves;
  • the exchange identity X_i = −y_i, Y_i = x_i holds (sympy).

  For K = 2 it reproduces Sussman's B₂ idle climbs exactly. For B₃ the two largest idle climbs (38 and 32) are split below m/2 = 31.5, so the finite guard's half-shrink condition holds (2 · 31 < 63).

• Complex checks. 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 portable checks take the layout as one parameter K.

  • At K = 2, every 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 pin holds, and the full receipt equals 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 except for the timing field.
  • At K = 3 they cover gx.check1 with the exact scalar identity, gxcore, labels, both scalar words and lifetimes, the seven-window splice, and the finite precision guard at m = 63. The guard gives rows 12,064 < 20,161, induction gap 63·B, local bill 2⁴² < 2⁴⁸, global groups 2¹⁹³⁶ < 2⁶⁰⁵⁰, router 2⁵⁷⁶⁵ < 2³⁰⁰⁰⁰, and m ≤ 72.

• The circuit is Sussman's E8. His generators regenerate the Lean data of wht_main_block_B2Ge8x byte for byte, and his independent replay accepts the certificate at 8762479.

• Exact b. It is computed from the guard's own histogram with 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 rational moment engines and the 10⁻¹⁶ fallback. The adjacent 10⁻¹⁸ point is rejected, and B₂ 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 b exactly.

Open obligations.

• No Lean proof of B₃. Sussman's bridge2_certificate covers B₂. A Bridge3 would need the same algebra (checked here symbolically) and the group-theoretic wiring on Orth(H ⊕ 6·H) proved in Lean. Here the layout rests on the finite coordinate-frame check and the exchange identity at the abstraction of Work/Bridge/Net.lean's header.
• Inherited from 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: the complex symbolic correctness, the precision/recovery and row interfaces (including the additive row constants 9909 + 252), and the analytic transfer.
• This is a conditional finite construction, not a formal verification of the multiplication theorem.

Validation

python -m pip install -r research/e8-seven-stage/requirements.txt    # sympy==1.14.0
bash research/e8-seven-stage/verify.sh /tmp/e8-seven-stage-verify    # ~10 s

Local result (macOS, Python 3.12.13 and 3.14.8, sympy 1.14.0), identical on both:

== 0. package manifest and PR #352 provenance
   all files match MANIFEST.sha256
   93 files byte-identical to PR #352 at 04b4c3c (git blob ids); 8 edited
== 1-4. layout B_2 / B_3, complex checks (B_2 regression, B_3), circuit binding, exact b
layout: B_2 idle climbs {1: [20], 2: [16, 8, 4]} (Sussman: 2h+2 | 2h-2, h-1, 4); B_3 idle climbs {1: [38], 2: [16, 8, 22], 3: [32, 8, 6]}; 7 invocations, 12 twin gates; nested climbs, equal-frame gates, m-1 moves per data role, exchange identity: PASS
complex checks: B_2 regression PASS_FRESH_COMPLEX_LABEL_SCALAR_SPLICE_PRECISION (#352 pins); B_3 PASS_FRESH_COMPLEX_LABEL_SCALAR_SPLICE_PRECISION | m 63, live 1503, deficit 216, max child 31, rows 12064 < 20161, induction gap 63 B
circuit: the Lean data of wht_main_block_B2Ge8x regenerate byte for byte; stand-alone replay accepts at 8762479
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 %)
PASS E8 in the seven-stage bridged word: complex coarse saving b = 932783231884153/1000000000000000000 = 0.000932783231884153

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

Credits and AI disclosure

• Jacob Sussman: the bridged five-stage word B₂ and the general stage geometry that B₃ extends; the E8 unit; gx, gxcore, the Lean-data generators and the replay (wht-power-saving-lean, Apache-2.0).
• Complex checks: 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, on LJH-217's κ = 7.92580093369946e-4: w3 cube layer on #325's p10b word with a stronger p = 10 complex supplier #346 port of 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 portable checks (The source-assisted complex word of #233 as one explicit numbered program (gcert/1): reference-checked, 7.547361·10⁻⁴ in the Lean-checked five-stage layout (no new κ) #256/A stronger complex supplier: PR200's reuse pairing on the source-assisted v4 word, complex saving 7.0991·10⁻⁴ (+1.28% over #194; no new κ, bit binds) #233/Compose the v4 source-assisted complex word with the PR200 bit supplier: conditional κ = 6.768823e-4 #202/A stronger bit supplier: fixed-coordinate and face-diagonal bit local circuit on PR168 v4 (effective bit saving 6.7734e-4) #200/Source-assisted v4 complex word with PR189's bit supplier: conditional κ = 6.674324e-4 #194/Source-assisted common-frame compression for conditional kappa = 6.566436e-4 #184 lineage).
• E8's devices from this repository: jamesyc Birth-read slot recycling: conditional kappa 1.1038332094346e-4 #124 and eumemic Late compensation and joint frame refinement: conditional κ = 4.24830828577e-4 #143; Chafik Boukhalfa A stronger bit supplier: fixed-coordinate and face-diagonal bit local circuit on PR168 v4 (effective bit saving 6.7734e-4) #200/A stronger complex supplier: PR200's reuse pairing on the source-assisted v4 word, complex saving 7.0991·10⁻⁴ (+1.28% over #194; no new κ, bit binds) #233; icekylinx Source-assisted common-frame compression for conditional kappa = 6.566436e-4 #184, carried in ikeboy's Source-assisted complex word on PR168 v4 modules: conditional κ = 6.626307e-4 #191/Pair donors to avoid parity erasure on the source-assisted v4 word: conditional κ = 6.647872e-4 #193.

See NOTICE.md.

The seven-stage generalisation, 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: Seven-stage bridged E8 complex supplier.

Review and limits: A stronger complex component, not a new multiplication bound by itself. Local unit, global layout and all-size compiler assumptions remain separate.

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: 9df55dad6ddd05c3458793dfd82d1c2fe0ddbbb4. Publication: 4040a0f46d67b41c36a472a7cd4af791fd4c36cc.