Conditional κ = 9.5464544449649·10⁻⁴: Sussman's E8 complex unit re-timed by far piece relocation, in #375's unsplit B₃ (+0.042 % over #375)

[chafreaky]

κ = 95464544449649/10¹⁷ ≈ 9.5464544449649e-4 (conditional). That is +0.0421 % over #375 (9.5424356568133e-4) and +0.0262 % over #365's certificate in the same layout (9.54395132518163e-4).

The PR adds one self-contained package, research/e8-complex-far-retiming/, with its own workflow. The only change to an existing file is one .gitattributes line.

Change and mathematical scope

What is new: a re-timed E8 complex unit with the same operator. Jacob Sussman's gcert1-e8-r783 is re-timed by four kinds of move:

• single-gate retiming;
• connected-block and sub-block retiming, which generalises Reduce E8 recursive cost with four coupled frame changes #365's move 4;
• in-group piece relocation;
• far piece relocation (new): an elementary addition u += c·s moves across gates that neither read nor write u and do not write s, then takes its best frame between its new neighbours. This generalises Reduce E8 recursive cost with four coupled frame changes #365's moves 1–2.

Our greedy search reaches #365's certificate value exactly, then finds 14 more far relocations. The certificate keeps everything in Sussman's format:

• the scalar operator on all 1,023 registers (checked exactly, with every initial value a variable) and the 8 retained totals;
• R = 783, cst = 72, N = 9,039, the ports, start/final frames and the scatter table.

The gates change from 773/1,220 to 775/1,226 (A/B), with two new frames.

side	supplier	layout	coarse saving
complex	this package's certificate	B₃ unsplit (#375): m = 63, W = 1,503, deficit 216, max child 38, halving degree 2	b = 477778832512643/(5·10¹⁷) ≈ 9.55557665e-4
bit	#370's word in #373's B₃ (not binding)	m = 56, W = 1,585, deficit 216	c = 968197503705007/10¹⁸

complex certificate	B₃ split b	B₃ unsplit b	κ
Sussman (#373 / #375)	9.32783231884153e-4	9.55155017966616e-4	9.5424356568133e-4
#365 (aliromman)	9.32928054566567e-4	9.55306874505079e-4	9.54395132518163e-4
this PR	9.33167230469274e-4	9.55557665025286e-4	9.5464544449649e-4

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

What is checked (standard library only):

• Inputs are pinned by git blob id: Sussman's certificate, gx.py and gxcore.py (the same bytes as 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/Reduce E8 recursive cost with four coupled frame changes #365/κ = 9.5424356568133e-4: seven-stage E8 suppliers with the complex idle climbs at halving degree two (+2.40 % over #373) #375), and κ = 8.75481145306751e-4: first E8 bit word (complex-bound; +9.29% over #352) #370's word (= 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's).
• On both certificates:
  • Sussman's gx.check1 (E0–E7, including the exact scalar replay) and the gxcore mirror;
  • a flipped-sign control, rejected with E5;
  • our source-span rule, with 0 violations.
• The full rational operator identity with the original, and unchanged family data.
• Our own reimplementation 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's B_K layout rule: nesting, twin gates, stage entries, m − 1 moves, and the exchange identity. B₂ gives Sussman's idle climbs, and B₃ gives 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 (h = 9) and 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's (h = 8).
• Exact b (split and unsplit) and c by both rational engines with the 10⁻¹⁶ fallback, and 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 assembly.
  • Regressions: Sussman's split b = 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, and the unsplit κ = κ = 9.5424356568133e-4: seven-stage E8 suppliers with the complex idle climbs at halving degree two (+2.40 % over #373) #375's 95424356568133/10¹⁷.
• Full mode: our search, re-run from Sussman's original, regenerates the shipped certificate byte for byte.

Searched and not profitable on this unit:

• a shared-donor kernel through exterior gauges (E7/Q2);
• a lower copy rank for a retained total (no combination of the 8 totals lies in a hyperplane);
• larger K: complex B_K peaks at K = 3, and B₄ has m = 81 > 72.

Open obligations

• No Lean proof of B₃ on either side, and no Lean proof of the new certificate. wht_main_block_B2Ge8x covers Sussman's original in B₂.
• κ = 9.5424356568133e-4: seven-stage E8 suppliers with the complex idle climbs at halving degree two (+2.40 % over #373) #375's halving-degree-two guard is admitted as stated there. Its stock quantities (m, W, R, h, cst) are unchanged.
• 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/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 portable complex checks are not re-run on the new certificate. These cover labels, scalar words and lifetimes, the splice, and the macro-count terms of the finite guard. The gate count changes by +0.4 %.
• The bit side is 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's claim. Only c is re-certified, with our own engines. 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's bit-word checks are not re-run.
• Everything κ = 9.5424356568133e-4: seven-stage E8 suppliers with the complex idle climbs at halving degree two (+2.40 % over #373) #375/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/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 admits.
• This is a conditional finite construction, not a formal verification of the multiplication theorem.

Validation

bash research/e8-complex-far-retiming/verify.sh --quick /tmp/e8st-q     # ~15 s
bash research/e8-complex-far-retiming/verify.sh /tmp/e8st-full          # ~1 min, adds the regeneration

The local result (macOS, Python 3.12 venv and 3.14.8) passes both modes from a fresh clone of commit c248962, ending with:

PASS conditional kappa = 95464544449649/100000000000000000 = 0.00095464544449649 (complex-bound: our E8 certificate in the unsplit B_3 at halving degree two; bit side #370/#373)
   regenerated certificate equals the shipped one (decompressed bytes)

The MANIFEST equals git ls-files (blob hashes checked). There are no .log files, and all files use LF line endings.

Credits and AI disclosure

• Jacob Sussman: the E8 unit, gx/gxcore, the bridged five-stage word and the stage geometry.
• aliromman (Reduce E8 recursive cost with four coupled frame changes #365): the first coupled re-timing of this unit. Our moves generalise its four moves.
• DaysSky (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): B_K, the seven-stage bit construction, and the halving-degree-two layout.
• utcorvusvolat-dotcom (κ = 8.75481145306751e-4: first E8 bit word (complex-bound; +9.29% over #352) #370): the E8 bit word.
• rohanarun (Conditional κ = 9.54395132518163e-4: coupled complex retiming after #375 (+0.01588%) #374): the Reduce E8 recursive cost with four coupled frame changes #365 composition in split B₃.
• Our stage lineage (Conditional κ = 7.76518649004426·10⁻⁴: the full transcript stage stack on #325's p10b word, priced against #327's p = 10 complex supplier (+0.888% over #325) #329/Conditional κ = 8.04303459176812·10⁻⁴: kernel, retiming and reorders re-derived on #354's Design T word (+0.0776 % over #358) #359; Conditional κ = 7.44136146100942e-4: two concave-descent retimings (904 + 111 gates) and 220 target-prefix squares on #285's gen5 five-stage banks #287/Conditional κ = 7.44728650221677e-4: second concave-descent retiming (89 gates, constructed frames) after #290's kernel entries on the gen5 word #291/Conditional κ = 7.50237961741364·10⁻⁴: kernels, early restorations, terminal sinks, a post-sink retiming and addition reordering on the gen5b word (+0.473% over #296) #299/Conditional κ = 7.51458925311258·10⁻⁴: addition reordering on #305's weighted gen5b composition (+0.0424% over #305) #306) and 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 engines (DreamingOfClouds).

Prepared by Chafik Boukhalfa with substantial Anthropic Claude assistance. Apache-2.0.

Lean (added)

The re-timed E8 certificate now has a kernel-checked theorem for the five-stage (B₂) unit, generated with Jacob Sussman's unchanged generators at 9c94857:

OAI.PowerSaving.WHT.wht_main_block_B2Ge8FRx : ∃ solve W, WHTProgram solve W ∧ WHTTimeBoundsAt (1 - 8766835/(10:ℝ)^10) W

The original unit's theorem is at 1 − 8762479/10¹⁰. lake build succeeds, Compare.lean reports RESULT: PASS, and the only axioms are propext, Classical.choice and Quot.sound. The evidence is in lean/LEAN-B2-E8FR.txt. The seven-stage and #377 nine-stage layouts priced in this PR still have no Lean proof.

Repository CI. This branch includes PR #318's unchanged three-file joint-dual source-pin repair (certificates/joint-dual-kappa.json, certificates/joint-dual-validation.json, research/joint-dual/SOURCE.json), as #366 and #368 do, so the repository-wide verify (joint) job passes. No mathematics is changed by it.

[Mister-Meeseeks]

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

Contribution: Far-piece relocation in the E8 complex unit.

Review and limits: Exact local gcert checks and regeneration, plus scoped layout/price analysis. Recorded Lean evidence is for the stated local certificate, not the nine-stage layout or multiplier.

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: 54a1eb4e6d0b1bc584f1cc5b21d091b0ce3f5c68. Publication: 4040a0f46d67b41c36a472a7cd4af791fd4c36cc.