Conditional κ = 8.03137165163490·10⁻⁴: w3 Design T + twins on #329's transcript-staged p10b word with Sussman's E8 complex unit (+0.256 % over #352)

[chafreaky]

Change and mathematical scope

Conditional κ = 80313716516349/10¹⁷ ≈ 8.03137165163490e-4. That is +0.2563 % over #352 (8.01083628465007e-4) and +0.196 % over #353 (8.01598358635541e-4). The bit side binds.

side	supplier	coarse saving
bit	#325's p10b word → our #329 transcript stages (descent, target, restore, sink) → #310's w3 Design T and twin condensation (h = 20 as in #346), reimplemented	c = 160756542666561/(2·10¹⁷) ≈ 8.03782713332805e-4
complex	Jacob Sussman's E8 unit gcert1-e8-r783 (wht-power-saving-lean 9c94857), as in #352	b = 876248285600677/10¹⁸

• Reimplementation. Design T, the completion reads and twin condensation are our own code. On 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's unstaged parity word they rebuild κ = 7.92580093369946e-4: w3 cube layer on #325's p10b word with a stronger p = 10 complex supplier #346's published final word byte for byte (records c6a9311a…); verify.sh runs this as a control.
• Composition. The stages and Design T overlap.
  • 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's descent retimes the four 011 X-mix pairs per cube. Design T routes those pairs through M, so that gain is given back.
  • 1,190 of 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's 1,339 shared-donor kernel families use cube-layer helpers that Design T rewrites or frees, so the kernel is left out.
  • Target squares (Y registers only), early restorations (240) and terminal sinks (10, R → 8,090) compose additively: Δκ over 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 is +0.090 %, +0.086 % and +0.080 %.
• Retained hypotheses. Unchanged from κ = 7.92580093369946e-4: w3 cube layer on #325's p10b word with a stronger p = 10 complex supplier #346/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 all-size compiler, common charts, restored rows, selectors, routing, prime supply, precision/recovery, complex symbolic correctness and analytic transfer. Conditional κ = 0.000752893499605334868002344: source-bound w3 word and complete finite admission (+0.1855% over #309) #310's native downstream chain is not ported; only bank feasibility is shown. This is a conditional finite witness, not a theorem.

Validation

bash research/w3-design-t-staged-p10b/verify.sh /tmp/out takes about 3 min (--quick skips the #346 control). It checks:

• the manifest;
• the staged and parity input hashes;
• the rebuild of the final word, which must equal the shipped word;
• our exact legality and full F₂ replay;
• 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's official cohort legality and five-stage columns checkers, ported to h = 20 (11,930 columns, six omitted-bridge controls);
• exact G = I − J/9 nondegeneracy of all 1,000 new frames;
• an exact width-100 tiling with 60 replicas;
• Sussman's gx.check1 + gxcore on E8, with a flipped sign rejected, and b by both engines with the 10⁻¹⁶ fallback;
• the bit c by both engines 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 outer assembly (47 strict constraints, adjacent κ rejected).

Commit 3c59b8c on upstream main 12a4934.

Credits and AI disclosure

• LJH-217 (κ = 7.92580093369946e-4: w3 cube layer on #325's p10b word with a stronger p = 10 complex supplier #346): h = 20 port of Design T/twin, on 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.
• utcorvusvolat-dotcom (Conditional κ = 0.000752893499605334868002344: source-bound w3 word and complete finite admission (+0.1855% over #309) #310): w3 Design T and twin condensation.
• DaysSky (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): E8 as the complex supplier.
• Jacob Sussman: the E8 unit, gx and gxcore.
• DreamingOfClouds (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/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): bit word and five-stage pipeline.
• 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's official checkers.
• Our 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 stage stack (κ = 7.11191385398487e-4: compose kernel entrances with 217 target-prefix groups #268/Conditional κ = 0.000724189082265949: gen4 retiming, 440 early restorations and 60 dirty entrances #280/Conditional κ = 7.26111344123732762670617e-4: gen4 response kernels and endpoint-aware composition of #279/#280 #283/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 = 7.45513573133379·10⁻⁴: early restorations and terminal sinks on the gen5 kernel word, with #233's complex word as supplier (+0.107% over #290) #295 lineage).

#310 and #346 were read as data and not run. Prepared by Chafik Boukhalfa with substantial Anthropic Claude assistance. Licensed Apache-2.0.

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: Design-T plus twins on staged p10b with E8.

Review and limits: Literal stage and scalar checks plus complete paid-profile pricing; native bank/projector/prime/invoice extensions identified by the source remain conditional obligations.

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: 2e3f3cf37d2d53d902cf78200c5d59668280cf5b. Publication: 4040a0f46d67b41c36a472a7cd4af791fd4c36cc.