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)

[chafreaky]

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)

One self-contained package, research/five-stage-gen5b-kernels-restorations-sinks/, with its own workflow. No existing file is changed.

κ = 187559490435341/(25·10¹⁶) = 7.50237961741364·10⁻⁴: +0.473% over #296's gen5b word (7.46709720962478·10⁻⁴), +0.633% over #295, +0.822% over #287. (Earlier versions of this PR: 7.49899215319498·10⁻⁴ without the last two stages, 7.49920356821633·10⁻⁴ with the post-sink retiming.) Bit coarse 7.50462·10⁻⁴, complex coarse 7.54736·10⁻⁴ (#233's word); the bit side binds with a 0.57% margin — with #193's complex supplier (7.47455·10⁻⁴) this word would now be complex-bound.

What it is

#296's package (the gen5b bit word: #285's generator and pair module with an exactly optimal alternative local design and a debt-10 all-but-one module; #287's descent and target squares re-derived; #233's complex supplier) with the three transcript stages of #290 and #295, each re-screened on gen5b and priced on the whole word (the φ-ledger model reproduces #296's κ exactly and predicted each row below exactly):

stage (census on gen5b)	mechanism	φ	κ	gain
#296 base	—	—	7.46709720962478·10⁻⁴	—
1,720 collective kernel entries — 920 twin pairs (904 on a line, 16 on rank 3) and 800 families (375 quadruples, 425 triples); entrance rank 2,192	#254/#259/#272 (Dugongue), stage of #282/#290	−3,799.1	7.49121118627047·10⁻⁴	+0.323%
440 early helper restorations (cut at record 757,526)	#280 (utcorvusvolat-dotcom), endpoint banking #283	−886.6	7.49686102247225·10⁻⁴	+0.075%
7 terminal sinks (R 15,434 → 15,427)	#283 (Dugongue)	−333.7	7.49899215319498·10⁻⁴	+0.028%
second concave-descent retiming after the sinks — 113 gates (31 operand-chain frames, 34 constructed join/meet frames, 36 connected pair blocks), 77 new bases	#287/#291 (rohanarun), #286 (GamingPuzzled), #289 (gupt1156), #270 (utcorvusvolat-dotcom)	−33.3	7.49920356821633·10⁻⁴	+0.003%
addition reordering (two rounds, 237 moves): a single addition moved past its sibling incidences into the frame of a neighbouring incidence — 213 root deliveries y += h from a rank-21 frame into y's next rank-22 frame, 22 forward gates from a rank-18 back to an earlier rank-16 frame, 2 more in round 2; exact over Z because no crossed gate reads its destination or writes its source	found while testing Khattar–Gidney's conditionally clean ancillae (arXiv:2407.17966), whose literal class is empty here	−496.6	7.50237961741364·10⁻⁴	+0.042%

gen5b is much richer in kernels than gen5: 952 twin pairs share a cut and a nondegenerate common line (292 on gen5); helper responses have F₂ rank 1,760 and nullity 11,440 over 13,200 plain helpers; with a common cut and entrance there are 952 pairs, 2,104 triples and 3,807 quadruples.

Checks

python3 -B verify.py runs fifteen fresh stages — virtual, raw, bit (descent + targets), kernel, restore, sink, descent2, reorder, reorder2, scalar, primes, banks, complex, math, finite — with fifteen omitted-stage negative controls. The reorder stage re-derives every frozen move on the actual word (record content, the anchor frame's exact basis, the crossed-interval rule, disjoint operand sets), runs the F₂ replay of all columns forward and inverse, an omission control (932 wrong rows), an independent integer replay mod 2⁶¹ − 1 with random dirty inputs, the MOVE rebuild with both reflected ledgers, and #287/#291's exact source-span rule on every gate of its output. The post-sink retiming (descent2_transform.py, #287's transform with final frames read from the actual post-sink state so restored helpers keep their early endpoints) keeps every check of #287's descent (byte-identical scalar/COPY projection, nested chains with every MOVE rebuilt, fixed endpoints, COPY lifetimes, exact integer operand spans, nondegenerate frames, both reflected ledgers) and adds a forward and inverse F₂ replay; the deficit stays 4,400. Kernel: 38,036 compensation reads removed, 5,790 setup/restore gates, all columns forward and inverse, omission controls 3,190 / 1,438 wrong rows, prefix relation checked for every entry. Restore: 1,767,480 crossed-gate commutation checks, omission control 440 wrong rows, completion P_σ + I − P_E. Sinks: omission controls 28 / 21 / 21 wrong rows, each removed helper frees 24 rank, deficit re-checked at R = 15,427. Banks: literal stock 1,236,750 → 1,229,750, 807,350 banks, 910 exact charts, 3,954 entrances; the 1,574 residual-23 pivots are tiled as (r⁴, 24, 4²⁴⁻ʳ) — width 120 and full — because #283's (r⁴, 4³⁰⁻ʳ) rule runs out of rank-4 blocks on this word. Every changed literal is a recomputed pin; nothing is dropped. The two intermediate packages (kernel only; kernel + restore) also pass verify.py end to end.

Verify

python -m pip install -r research/five-stage-gen5b-kernels-restorations-sinks/requirements.txt    # sympy 1.14.0
python3 -B research/five-stage-gen5b-kernels-restorations-sinks/verify.py --output /tmp/gen5b-krs-verification   # about 5 minutes; not under -I

The workflow runs it and the bit-word regeneration on Ubuntu.

Scope and limits

• Conditional finite construction under the retained public all-size hypotheses (no Lean certificate for the bit side; the complex word is Lean-checked via The source-assisted complex word of #233 as one explicit numbered program (gcert/1): reference-checked, 7.547361·10⁻⁴ in the Lean-checked five-stage layout (no new κ) #256). Conditional κ = 7.42785663120535e-4: gen5 — compile-cost-annealed pair and all-but-one modules on #276's gen4 bit word, five-stage completed banks #285, 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 and Conditional κ = 7.46709720962478·10⁻⁴: gen5b — exactly optimal local design and a debt-10 all-but-one module on #285's word, #287's stages re-derived, #233's complex supplier (+0.348% over #287) #296 are open; inherited stages are reused unchanged.
• Not composed: Conditional κ = 0.000724189082265949: gen4 retiming, 440 early restorations and 60 dirty entrances #280's transported dirty entrances (almost no candidates once the target squares are in place), restorations whose join is the full frame (34), quintuple families (never net-positive), and the retiming rungs stacked on Conditional κ = 7.44714499357628·10⁻⁴: 450 collective response-kernel entries on #285's gen5 word, after #287's descent and target squares (+0.080% over #287) #290 (Conditional κ = 7.44728650221677e-4: second concave-descent retiming (89 gates, constructed frames) after #290's kernel entries on the gen5 word #291–Fixed-prime eight-level gen5 descent: conditional κ = 744728654228590183370616/10^27 #293), which were bound to the gen5 transcript.
• KERNEL-PROOF.md and LEVERS-PROOF.md describe the mechanisms with their gen5 counts (Conditional κ = 7.44714499357628·10⁻⁴: 450 collective response-kernel entries on #285's gen5 word, after #287's descent and target squares (+0.080% over #287) #290/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); the README carries this word's counts.

Credits

Prepared by Chafik Boukhalfa with substantial Anthropic Claude assistance. Apache-2.0; lineage in NOTICE.md. The response-kernel mechanism, collective kernels, sinks and endpoint banking are Dugongue's (#254/#259/#272/#283); early restorations are utcorvusvolat-dotcom's (#280); the gen4/gen5 generator, pair module and compile-cost method are DreamingOfClouds' (#276/#285); the descent and target-square stages are rohanarun's (#287; target-prefix mechanism eumemic's #268); the five-stage package is eumemic's source527 lineage on hcg890's #234 layout and Jacob Sussman's construction. This package contributes the gen5b word (#296), the censuses and selections on it, the generalised kernel stage (#282/#290), the ports of the restoration and sink stages to this pipeline (#295), the (r⁴, 24, 4²⁴⁻ʳ) tiling and the complex supplier (#233/#256).

[Mister-Meeseeks]

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

Contribution: Gen5b kernels, restorations, sinks, post-sink descent and reordering.

Review and limits: Source-bound full Linux verifier at the updated head and producer regeneration. Retain the newer two reorder passes, not only the earlier headline in PROOF.md.

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: 11b69525c7b2bb116cf29b17f976781003ef33b4. Publication: 50d768818f579d4e165b4c257c6ee25bca300802.