Conditional κ = 8.04849251275843·10⁻⁴: post-Design-T kernel, retiming and reorders on #357's reordered base word with all 471 twins (+0.0679 % over #359)

[chafreaky]

Change and mathematical scope

Conditional κ = 804849251275843/10¹⁸ ≈ 8.04849251275843e-4.

• +5.458·10⁻⁷ (+0.0679 %) over our Conditional κ = 8.04303459176812·10⁻⁴: kernel, retiming and reorders re-derived on #354's Design T word (+0.0776 % over #358) #359 (8.04303459176812e-4).
• +1.170·10⁻⁶ (+0.1455 %) over κ = 8.0367971186626189473809e-4: reordered stages with 471 twins and signed18 #358 (8.03679711866262e-4).
• +1.195·10⁻⁶ over κ = 8.03654539036015e-4: #354's staged Design T + twins on a reordered-design p10b base word (+0.064%) #357 (8.03654539036015e-4).

The bit side binds.

This stacks two pieces:

• our Conditional κ = 8.04303459176812·10⁻⁴: kernel, retiming and reorders re-derived on #354's Design T word (+0.0776 % over #358) #359 post-Design-T stages: shared-donor kernel, retiming, reorder;
• rohanarun's reordered-design PLAIN12 base word (κ = 8.01598358635541e-4: #352 with Design T + twins on a reordered-design p10b base word (+0.064%) #353/κ = 8.03654539036015e-4: #354's staged Design T + twins on a reordered-design p10b base word (+0.064%) #357).

It updates the same package (research/w3-design-t-staged-p10b/) on top of #359's commit 2cdb716.

side	supplier	coarse saving
bit	#357's staged word (reordered PLAIN12 base + our #329 descent/target/restore/sink, vendored) → our Design T + completion reads + all 471 twins → kernel (334 entries, rank 439), retiming (93 gates), reorder (130 + 2), each re-derived on this word	c = 12585899309117/(1.5625·10¹⁶) ≈ 8.05497555783488e-4
complex	Jacob Sussman's E8 unit gcert1-e8-r783 (wht-power-saving-lean 9c94857), as in #352/#354/#359	b = 876248285600677/10¹⁸

• Base. data/staged and data/parity are κ = 8.03654539036015e-4: #354's staged Design T + twins on a reordered-design p10b base word (+0.064%) #357's snapshots, byte-identical, with κ = 8.03654539036015e-4: #354's staged Design T + twins on a reordered-design p10b base word (+0.064%) #357's reordered local design kept as provenance. 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 own parity word stays as data/parity325 for the κ = 7.92580093369946e-4: w3 cube layer on #325's p10b word with a stronger p = 10 complex supplier #346 byte-for-byte control.
• Twins. All 471 eligible twins are kept, as in κ = 8.0367971186626189473809e-4: reordered stages with 471 twins and signed18 #358. κ = 8.03654539036015e-4: #354's staged Design T + twins on a reordered-design p10b base word (+0.064%) #357 dropped one to keep the residual sum ≡ 0 (mod 5); here the kernel is trimmed to total rank ≡ 4 (mod 5) instead.
• Stages. These are Conditional κ = 8.04303459176812·10⁻⁴: kernel, retiming and reorders re-derived on #354's Design T word (+0.0776 % over #358) #359's mechanisms with fresh selections:
  • kernel: φ −402.05 (≈ +6.37e-7 κ);
  • retiming: φ −48.79;
  • reorder: φ −296.89 and −5.55.
• Not used.
  • κ = 8.0367971186626189473809e-4: reordered stages with 471 twins and signed18 #358's eighteen signed pairs and its T = 300 / eight-level arithmetic. They are orthogonal and could be stacked later.
  • eumemic's κ = 7.7652200910342944936985e-4: cross-entrance retiming and operational PR327 supplier #330–κ = 7.78097715735654505738899e-4: coherent component frame retimings #347 stages.
• 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/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) #354/Conditional κ = 8.04303459176812·10⁻⁴: kernel, retiming and reorders re-derived on #354's Design T word (+0.0776 % over #358) #359: 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.

word	κ
#357	8.03654539036015e-4
+ kernel	8.04291732418766e-4
+ retiming	8.04369110315724e-4
+ reorder	8.04840432528145e-4
+ reorder2	8.04849251275843e-4

Validation

bash research/w3-design-t-staged-p10b/verify.sh /tmp/out takes about 8 min; --quick skips only the #346 control. Both exit 0 locally. The script checks:

• the manifest and all input hashes;
• the Design T rebuild (471 twins), which must equal data/designt;
• the four frozen stages, each bound to its input hash, fully re-checked (legality, exact frames and nondegeneracy, F₂ of every column, source spans) and required to give its frozen output hash;
• the final word, which must equal the shipped one;
• κ = 7.92580093369946e-4: w3 cube layer on #325's p10b word with a stronger p = 10 complex supplier #346's word rebuilt from 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 parity word (c6a9311a…, full run only);
• 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 legality and five-stage columns checkers (11,930 columns);
• nondegeneracy of all 1,085 new frames;
• an exact width-100 tiling with 60 replicas;
• E8 checked by gx.check1 + gxcore and priced by both engines;
• 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).

PASS pairs 0 ADD 357390 mass 168690 COPY24
   official five-stage formal columns PASS: 11930 columns, prefix payload 93 bits
new/changed frames 1085 degenerate 0
bit coarse c = 12585899309117/15625000000000000 0.000805497555783488 | binding bit
PASS conditional kappa = 804849251275843/1000000000000000000 = 0.000804849251275843

Commit 2eef66d.

Credits and AI disclosure

• rohanarun (κ = 8.01598358635541e-4: #352 with Design T + twins on a reordered-design p10b base word (+0.064%) #353/κ = 8.03654539036015e-4: #354's staged Design T + twins on a reordered-design p10b base word (+0.064%) #357): the reordered-design PLAIN12 base word and its staged snapshot (vendored).
• eumemic (κ = 8.0367971186626189473809e-4: reordered stages with 471 twins and signed18 #358): keeping all 471 twins; also the κ = 7.11191385398487e-4: compose kernel entrances with 217 target-prefix groups #268/Conditional κ = 7.11948953756315e-4: target-prefix compression (209 square groups) composed with the multi-cut kernel condensation #273 stage lineage.
• Our 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) #354/Conditional κ = 8.04303459176812·10⁻⁴: kernel, retiming and reorders re-derived on #354's Design T word (+0.0776 % over #358) #359: the Design T reimplementation and the post-Design-T kernel/retiming/reorder stages.
• 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: the stage stack.
  • Kernel lineage: Frame-filtered collective kernels + target-prefix sharing: kappa = 0.000712074434532274423885827 #272/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/κ = 7.68257328708259e-4: shared-donor kernels after frozen PR317 #319/Conditional κ = 7.69553898621543·10⁻⁴: the full transcript stage stack on #315's p = 10 word, with a shared-donor closure kernel (+0.697% over #315) #320.
  • Retiming and reorder lineage: 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.
• LJH-217 (κ = 7.92580093369946e-4: w3 cube layer on #325's p10b word with a stronger p = 10 complex supplier #346) and utcorvusvolat-dotcom (Conditional κ = 0.000752893499605334868002344: source-bound w3 word and complete finite admission (+0.1855% over #309) #310): 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 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.

#357, #358, #310 and #346 were read as data; none of their scripts were 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: Post-Design-T stages on the reordered base.

Review and limits: Exact stage composition and controls; native downstream qualifications from this proof interface remain open rather than being erased by a PASS marker.

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: d7b59136fa7dc6e2abfcd9fe41c6be6e886382f7. Publication: 4040a0f46d67b41c36a472a7cd4af791fd4c36cc.