κ = 9.5424356568133e-4: seven-stage E8 suppliers with the complex idle climbs at halving degree two (+2.40 % over #373)

[DaysSky]

κ = 95424356568133/10¹⁷ ≈ 9.5424356568133e-4 (conditional). That is +2.40 % over #373 (9.3191395651209e-4), which this PR extends, and +9.00 % over #370.

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

Change and mathematical scope

Everything is #373's: #370's E8 bit word and Sussman's E8 complex unit, both unchanged, both in the seven-stage bridged word B₃. There is one change, on the complex side.

What changes. #363 split the complex supplier's two longest idle climbs below m/2 (38 → 31 + 7, 32 → 31 + 1). It did this only to meet the half-shrink guard (2 · max child < m) of the inherited complex finite bridge. Here they are left whole. The largest complex child becomes 38 of m = 63, so one level no longer halves the size but two levels do: 63² = 3969 > 2 · 38² = 2888, i.e. halving degree two.

side	layout	max child / m	coarse saving
bit (#370 word, #373 checks)	B₃, split	27 / 56, degree 1	c = 968197503705007/10¹⁸
complex (Sussman's E8)	B₃, unsplit	38 / 63, degree 2	b = 119394377245827/(125·10¹⁵) ≈ 9.55155e-4 (+2.40 % over the split 9.32783e-4)

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.

Why halving degree two is admissible. The inherited bridge (FINITE_BRIDGE.txt §4) uses half-shrink in exactly one place: "The complex child maximum below m/2 gives halving degree 1, while log2 W < 5208 < 10000", inside the external row coefficient A = 10000 + 9909 + 252 = 20161.

With halving degree t, the recursion depth is t·⌈log₂ n⌉ and the row term is t·log₂ W. The generalized guard does three things:

• it computes the least t;
• it charges t times the actual physical stock logarithm: 2·1903 + 9909 + 252 = 13,967 < 20,161, so A and the outer assembly are unchanged;
• it keeps m ≤ 72 (the cover bound) and the completed-child induction gap (49·B > 0).

At t = 1 it is the original guard.

The repository already uses this generalization:

• Paired cubes and shared completed cores for conditional kappa = 4.609169e-4 #144's integration review (research/pr-review-20261009/check_paired_cube_arithmetic.py): complex child 60 at m = 72, halving degree 4, row coefficient 4*2571+9909+252;
• the copied-centers assembly (copied-centers-assembly.tex, research/copied-fixed-reversed/balanced_assembly.py): degrees 9 and 20, reserve D_j(e) = t_j⌈log₂ e⌉;
• PR234 on the bit side: "The proper child ratio 5/6 requires halving degree four. The one-level weighted recurrence is retained; its histogram is not multiplied by four."

The moment equation is the one-level histogram either way.

Why not further. Unsplit complex B₄ prices at 9.5673e-4 but has m = 81 > 72. Unsplit B₅ needs degree 3 and prices lower. The bit side stays split because it does not bind; unsplit at degree 2 it would give 9.9123e-4.

What is checked

Every step of #373's verify.sh runs:

• κ = 8.75481145306751e-4: first E8 bit word (complex-bound; +9.29% over #352) #370's checks;
• the B₃ formal columns over all 1,585 columns, the PR234 geometry on 7 blocks, width-56 banks with T = 7, and the finite bill at m = 56;
• each with its K = 2 regression against κ = 8.75481145306751e-4: first E8 bit word (complex-bound; +9.29% over #352) #370.

On the complex side certify_b3.py runs the guard three times:

• B₂, against 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 (degree 1);
• split B₃, against 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 pins (degree 1, b₃ = 932783231884153/10¹⁸);
• unsplit B₃, against the new pins-e8-b3h.json: degree 2, rows 13,967 < 20,161, induction gap 49·B, m ≤ 72.

The circuit binding is unchanged: Sussman's Lean data for wht_main_block_B2Ge8x regenerate byte for byte, and his replay accepts at 8762479.

The κ step reproduces #373's κ = 93191395651209/10¹⁷ with the split b₃, then gives 95424356568133/10¹⁷ with the degree-2 b₃.

Provenance: 142 files are identical to #370 (b7fe9cd), 4 to #363 (9df55da) and 7 to #373 (3ff87b5), recorded by git blob ids. The edited files are portable_complex.py (halving degree), certify_b3.py, verify.sh, provenance.py, README and NOTICE. The new file is pins-e8-b3h.json.

Open obligations

• Halving degree two on the complex side. The inherited bridge text states degree 1. Here its row term is charged at degree 2, as in the repository's degree-4 and degree-9/20 precedents above. The precision/recovery interface is inherited at that depth.
• Everything 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 lists:
  • no Lean proof of B₃ on either side;
  • κ = 8.75481145306751e-4: first E8 bit word (complex-bound; +9.29% over #352) #370's admissions (9 retained totals at h = 8; PR234's banks for a non-paired-cube family; the inherited bit interfaces);
  • the complex-side interfaces of 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;
  • the finite bill's seven-stage counts.
• This is a conditional finite construction, not a formal verification of the multiplication theorem.

Validation

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

Local result (macOS, Python 3.12 and 3.14), selected 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
complex checks, halving degree two: B_3 unsplit PASS_FRESH_COMPLEX_LABEL_SCALAR_SPLICE_PRECISION | m 63, max child 38 (63^2 > 2 * 38^2), halving degree 2, rows 13967 < 20161, induction gap 49 B
PASS halving degree two: B_3 b = 119394377245827/125000000000000000 = 9.551550179666e-04 (+2.398 % over the split B_3, +9.005 % over B_2)
   regression: with the split complex b_3 the seven-stage bit word gives PR #373 kappa 93191395651209/100000000000000000
KAPPA = 95424356568133/100000000000000000 = 9.542435656813e-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 = 95424356568133/100000000000000000 = 0.00095424356568133 (seven-stage bit word, complex-bound by the seven-stage E8 complex unit at halving degree two; 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 and the generalized bit-side checkers.
• Jacob Sussman: the E8 family and the E8 complex 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 (DaysSky): B_K, the seven-stage complex unit and the seven-stage bit construction.
• The halving-degree generalization follows Paired cubes and shared completed cores for conditional kappa = 4.609169e-4 #144's integration review, the copied-centers assembly and PR234 (hcg890 / Henry Grant).
• The 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 lineage: see NOTICE.md and NOTICE-PR352.md.

The halving-degree-two complex step 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 E8 with halving degree two.

Review and limits: A changed precision/recurrence bill with explicit regression and guard checks, not merely parameter substitution in an unchanged theorem.

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: 79874623cf86d92325c26ccafc31413d12e9d248. Publication: 4040a0f46d67b41c36a472a7cd4af791fd4c36cc.