Conditional κ = 8.04303459176812·10⁻⁴: kernel, retiming and reorders re-derived on #354's Design T word (+0.0776 % over #358)

[chafreaky]

Change and mathematical scope

Conditional κ = 201075864794203/(2.5·10¹⁷) ≈ 8.04303459176812e-4.

• +1.16629·10⁻⁶ (+0.1452 %) over 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 (8.03137165163490e-4).
• +0.0776 % over κ = 8.0367971186626189473809e-4: reordered stages with 471 twins and signed18 #358 (8.03679711866262e-4).
• +0.0807 % 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 builds on our #354, the commit 3c59b8c it ships, and updates the same package (research/w3-design-t-staged-p10b/).

side	supplier	coarse saving
bit	#354's word (#325 p10b → our #329 descent/target/restore/sink → reimplemented #310/#346 Design T + twins) → shared-donor kernel, retiming and two reorder rounds, re-derived on the Design T word	c = 804950884359437/10¹⁸ ≈ 8.04950884359437e-4
complex	Jacob Sussman's E8 unit gcert1-e8-r783 (wht-power-saving-lean 9c94857), as in #352/#354	b = 876248285600677/10¹⁸

• Why it was missing. In 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 the kernel, post-sink retiming and reorder of 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 stack were dropped, because Design T rewrites or frees the cube-layer helpers that 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 kernel families used.

• What is new. Here those stages are re-derived after Design T, on the Design T word itself (code/stages.py, code/stagelib.py, frozen selections in stages/):

  • Shared-donor response kernel (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 mechanism; 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 lineage).
    • On the Design T word all completion reads come first.
    • The census reads each σ = 0 helper's F₂ response from them (5,930 helpers).
    • The 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/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 closure over line and intersection pools packs 323 entries: total entrance rank 435, 553 donors, φ −386.03.
    • Each pivot starts at its entrance and loses its completion reads. Each donor pays d += p at the entrance after the completion block and d −= p at FULL at the end.
  • Retiming (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 descent2; 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): 47 gates, φ −49.68.
  • Reorder (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 reorder; 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): 130 moves (φ −296.89), then 2 (φ −5.55). A third round and a retiming re-search on the final word both find nothing.

• Gain by stage.

  stage	κ
  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	8.03137165163490e-4
  kernel	8.03745281493820e-4
  retiming	8.03823969373136e-4
  reorder	8.04294652399288e-4
  reorder2	8.04303459176812e-4

• 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: 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 8 min; --quick skips only the #346 rebuild control. Both exit 0 locally (Python 3.14, numpy 2.5, Apple clang, Boost 1.86). The script checks:

• the manifest;
• the staged and parity input hashes;
• the Design T rebuild, which must equal 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's final word;
• the four frozen stages, each:
  • bound to its input records hash;
  • followed by a full replay: exact nesting, ADD frames, COPY window, final frames, exact frame validity and nondegeneracy, formal F₂ of every column, and the source-span rule;
  • required to give its frozen output hash.
• the final word, which must equal the shipped word (4512d646…);
• 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 1,064 new frames;
• an exact width-100 tiling with 60 replicas (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 bank template, with its economy fallback);
• 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).

== 2b. transcript stages on the Design T word: shared-donor kernel, retiming, reorder, reorder2 (frozen selections)
kernel: 323 entries, total entrance rank 435, 553 donors, 2740 completion reads removed, 1099 setups + 1099 restores
retime: 47 gates moved to new frames
reorder: 130 ADDs moved
reorder: 2 ADDs moved
   rebuilt final word equals the shipped word
PASS pairs 0 ADD 357650 mass 168685 COPY24
   official five-stage formal columns PASS: 11930 columns, prefix payload 93 bits
new/changed frames 1064 degenerate 0
complex b = 876248285600677/1000000000000000000 0.000876248285600677
bit coarse c = 804950884359437/1000000000000000000 0.000804950884359437 | binding bit
PASS conditional kappa = 201075864794203/250000000000000000 = 0.000804303459176812

Commit 2cdb716 on top of #354's 3c59b8c.

Credits and AI disclosure

• 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: the Design T reimplementation on the staged word.
• 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.
  • Descent 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.
  • Also eumemic's 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 and Rohan Arun's descent and Conditional κ = 7.49982567267028e-4: 1,734 kernel entries (restart-packed), second descent, 440 restorations and 8 sinks on #296's gen5b word #300 restart loop.
• 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.

eumemic's #330–#347 stages are not used here. #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: Kernel, retiming and reorders after Design-T.

Review and limits: Finite exact transcript checks and pricing, with source-stated downstream native obligations retained. Bounded search exhaustion is not a general barrier.

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: e428efc9bea88c06ab3df89cb3b48eca13757813. Publication: 4040a0f46d67b41c36a472a7cd4af791fd4c36cc.