Conditional κ = 1.000385913854264·10⁻³: bank padding fused into three new free sink helpers on #390's E8 complex program (+0.0166 % over #390)

[chafreaky]

e8-clean-padding: bank padding fused into clean-sink helper climbs on #390's E8 complex program; conditional κ = 1.000385913854264e-3

κ = 125048239231783/1.25·10¹⁷ ≈ 1.000385913854264 × 10⁻³, which is +1.65691·10⁻⁷ (+0.0166 %) over #390 (1.000220222889528e-3). It is complex-bound:

• b = 40055507598519/4·10¹⁶ ≈ 1.001387690e-3, up from E8 prefix restoration and heterogeneous helper banks: conditional kappa = 1.000220222889528e-3 #390's 1.001221667e-3.
• The bit side is our Conditional κ = 9.98243154231047·10⁻⁴: our retimed + kernel E8 bit word in B₃ (unsplit) with #377's nine-stage complex component (+0.26 % over #381) #382 word (c ≈ 1.006907e-3 > b). E8 prefix restoration and heterogeneous helper banks: conditional kappa = 1.000220222889528e-3 #390's larger c gives the same κ.

What is new

#377's banks pay a separate rank-1 C child on coordinate 72 to complete each width-73 bank; #390 keeps 3,123 per stage.

We fuse that padding into the final climb of a helper in the same bank: (M₁·C_[d]·M₂) ⊗ C₇₂ = (M₁⊗I)(C_[d]⊗C₇₂)(M₂⊗I), so rank d + rank 1 becomes one rank d+1 child. This is legal only if nothing touches the helper afterwards.

• 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 cleanup bracket rules it out for every helper. In 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) #378's word all 775 lifting helpers receive x → helper gates, and those can only be undone at FULL, after the final climb.
• E8 prefix restoration and heterogeneous helper banks: conditional kappa = 1.000220222889528e-3 #390's dirty-preconditioned bracket permits it for zero-row sinks. That bracket is target precancellation y −= B·A⁻¹·h, helper prewhitening h ← A⁻¹h, and FULL repair of N_r·x only. A helper with N_r = 0 is never touched after its final climb.

We search #390's word for free sinks: an exact combination of helpers already at the helper's exact frame, applied after its last influence on y. The search finds exactly three: 301, 324 and 387, each with final lift rank 2. For 301 the gates are h276 += h1021/2; h301 −= h276/2; h276 −= h1021/2 (frame 731), with analogous gates for the other two (frames 744 and 759). They need no climbs and leave the paid local histogram unchanged.

Each invocation then has three rank-2 + rank-1 pairs become rank-3. The 96 fused occurrences per stage sit in 96 distinct ordinary banks of #390's own assignment (routes fix coordinate 72). Profile delta: {1: −864, 2: −864, 3: +864}. Live (58,899), deficit (9,984), max child (48) and halving degree (2) are unchanged.

The 16 additional candidates that need paid donor advances (the same count #390 reports) all lower b, so none is used.

Checks (bash research/e8-clean-padding/verify.sh /fresh/dir; Python 3.11+, stdlib only)

• Pins: git-blob pins on E8 prefix restoration and heterogeneous helper banks: conditional kappa = 1.000220222889528e-3 #390's prefix18, sink-audit and prewhite, 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) #378's word, Sussman's gx, and Conditional κ = 9.98243154231047·10⁻⁴: our retimed + kernel E8 bit word in B₃ (unsplit) with #377's nine-stage complex component (+0.26 % over #381) #382's word.
• Words: Sussman's gx.check1 (E0–E7, exact scalar) and the gxcore mirror on E8 prefix restoration and heterogeneous helper banks: conditional kappa = 1.000220222889528e-3 #390's prefix18 word, on E8 prefix restoration and heterogeneous helper banks: conditional kappa = 1.000220222889528e-3 #390's main word (+355/403) and on ours. The x/y operator rows and retained totals of ours equal 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) #378's, and the five sink rows are exactly zero.
• Regeneration: the free-sink search over all 773 lifting helpers regenerates exactly our gates.
• Bracket: our reimplementation of E8 prefix restoration and heterogeneous helper banks: conditional kappa = 1.000220222889528e-3 #390's bracket reproduces E8 prefix restoration and heterogeneous helper banks: conditional kappa = 1.000220222889528e-3 #390's prewhite.json gate for gate.
• Local program: the fused program, in both orientations, is replayed on all 1,023 live columns. The replay checks private work erased, common exact labels, and no event on a fused sink after its padded climb. Five mutation controls are rejected.
• Price: E8 prefix restoration and heterogeneous helper banks: conditional kappa = 1.000220222889528e-3 #390's b and κ are reproduced from its own emitted program. Ours is priced with both 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 engines (adjacent grid point excluded) and 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 outer assembly (47 strict constraints).

Conditions

• Re-opened, with the replacement argument in PROOF.md:
  • 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 emitter/validator chronology (E8 prefix restoration and heterogeneous helper banks: conditional kappa = 1.000220222889528e-3 #390 had already replaced it);
  • κ = 9.98243154231047e-4: nine-stage complex E8 (≈2^-9.9683; complex +14.036%, bit side pending) #377's padding as a separate end-of-schedule cover sweep, for 96 banks per stage. This uses the retained arbitrary-dirt invocation property, and the compiler admitting a C child spanning local coordinates plus window coordinate 72.
• Not re-run:
  • E8 prefix restoration and heterogeneous helper banks: conditional kappa = 1.000220222889528e-3 #390's rank-8 endpoint adapters, Clifford edge audits and finite invoice. We add 9 scalar gates per invocation, the bracket sizes change from 1,550/7,800 to 1,556/7,776, and no new local edge is used;
  • all of κ = 9.98243154231047e-4: nine-stage complex E8 (≈2^-9.9683; complex +14.036%, bit side pending) #377/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 retained interfaces;
  • Conditional κ = 9.98243154231047·10⁻⁴: our retimed + kernel E8 bit word in B₃ (unsplit) with #377's nine-stage complex component (+0.26 % over #381) #382's bit admissions.

This is a conditional finite construction, not a formal verification.

Credits

• Dugongue (E8 prefix restoration and heterogeneous helper banks: conditional kappa = 1.000220222889528e-3 #390):
  • the prefix rewrites;
  • the dirty-preconditioned bracket, which is what makes cleanup-free sinks possible;
  • sinks 355/403;
  • the heterogeneous banks.
• jacklightChen (κ = 9.98243154231047e-4: nine-stage complex E8 (≈2^-9.9683; complex +14.036%, bit side pending) #377): the nine-stage component, banks and padding.
• 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: the local invocation, compiler and emitter.
• Jacob Sussman: the E8 unit and gx/gxcore.
• 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) and utcorvusvolat-dotcom (κ = 8.75481145306751e-4: first E8 bit word (complex-bound; +9.29% over #352) #370): the bit lineage.
• 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: the engines and outer assembly.
• Ours: 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) #378, Conditional κ = 9.98243154231047·10⁻⁴: our retimed + kernel E8 bit word in B₃ (unsplit) with #377's nine-stage complex component (+0.26 % over #381) #382, and here the fusion, the free-sink search and the checks.

Prepared by Chafik Boukhalfa (chafreaky) with substantial Anthropic Claude assistance.

Commit be2a06b; it builds on our #384 (8148cd0) and #382 branches.

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: Fusing bank padding into clean sink helpers.

Review and limits: Checker-clean finite local construction and scoped bank pricing. Owner placement, fused-climb lowering and other source-listed native obligations are not established by local gcert success.

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: baead918497269357c7da7eefc78eb56a9e4d9aa. Publication: 4040a0f46d67b41c36a472a7cd4af791fd4c36cc.