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)

[chafreaky]

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)

One self-contained package, research/multicut-kernel-composition/ (renamed from the first version of this PR), with its own workflow. No existing file is changed.

κ = 142805349285993/(2·10¹⁷) = 7.14026746429965·10⁻⁴ on the pinned #249 word: +0.4216% over #259, +0.0510% over #271's 7.13662156·10⁻⁴ (99 sandwiches on #272) and +0.0472% over #275's 7.13690108·10⁻⁴, the best rungs on this word before it. Before the sandwich stage, with #259's seven native checkers alone: κ = 7.12005174354669·10⁻⁴ (+5.6·10⁻⁸ over #273). #276's new bit word (gen4, 7.2287·10⁻⁴) stands above every rung on #249's word, this one included; the levers composed here will have to be re-derived on that word.

Result

stage (each on the previous word)	families / entrance rank	normalized stock W	κ
#259	518 / 1,344	173,435	7.11029386054035·10⁻⁴
kernel families: early-cut twins, headroom, 54 shared-donor families	890 / 1,719	173,310	7.11462218·10⁻⁴
+ #263's 25 descent gates + #270's 29 plateau blocks	890 / 1,719	173,310	7.11482139228173·10⁻⁴
+ #273's 209 target-prefix squares (native stage) — seven native checkers' word	890 / 1,719	173,310	7.12005174354669·10⁻⁴
+ 126 cleanup sandwiches (#271), after swapping the pivot role in 83 pair families	890 / 1,719	172,470	7.14026746429965·10⁻⁴

Four levers by five authors on one transcript. Three rounds of kernel families at #259's earliest cut (record 676,559, the last initial compensation read): 333 twin pairs disjoint from #259's helpers (319 on lines eᵢ − eⱼ, admitted by #259's generic chart audit and excluded by #254's norm == 4 rule), 32 more from a census of every response class at twelve cuts (nothing at any later cut, no net-positive XOR triple), and 54 shared-donor families — a new pivot whose twin partner already serves another family as donor, lifted once on a nested entrance chain (the idea of #270's 24 reuse pairs and #267's chronological donor sharing; #259's transform admits it as is, since it only requires pivots to be unique and never donors), with 26 plain pairs dissolved where a shared-donor family is strictly better. Then #263's 25 descent gates and #270's 29 plateau blocks (57 ADDs), both selections rebound to this transcript by #270's scalar-core alignment (every ADD/COPY/ERASE outside the kernel categories identical). Then eumemic's target-prefix compression as rohanarun's native stage (#273, verbatim; its 209 squares rebound by post-cut scalar alignment, recomputing every F₂ dependency from the actual word: +5.2·10⁻⁷), and evmckinney9's cleanup sandwiches (#271): on this word 83 two-member kernel families have the sandwich shape on the member that was not the pivot, so the pivot role is moved there (cost-neutral for the kernel transform: identical histogram, rank, reads and native receipts), which raises the admissible sandwiches from 45 to 128; 126 are retired (the bank re-tiling needs a multiple of three), +2.0·10⁻⁶ — the largest single lever here, and the reason #271's 99 on #272's word was the previous best. Every family, retiming, square and sandwich is admitted by the package's exact pricer on the whole word; the deficit 35,200 per replica is invariant, so the gain is in the log-weighted rank mass (and, for the sandwiches, the stock).

Argument and checks (PROOF.md)

#259's mechanism untouched: the pivot starts at the common entrance (its initial reads deleted), the donor moves there and absorbs it, the inverse is paid at the full frame; the omitted response cancels against the induced one because the responses coincide. Checked on the native word (sha eaff3c98…, behind sha gates cohort 4458dce8… → descent a6d793d7… → plateau a1cf58b3…): all-column F₂ replay forward and inverse (890 pivots, 60,520 reads removed, 2,286 gates, raw mass 434,177 → 432,458; omitted-Q / omitted-Q⁻¹ controls fail nontrivially), the descent, plateau and target-prefix stages with their audits (operand source spans, added frames with nondegenerate Grams, both all-column replays of the compression with the omitted-setup control), chronological legality (1,903 helpers, 0 owner/donor intersections, 24 COPY/ERASE windows), prefix kernels (4 cuts, 1,566,400 target-kernel equalities), bank review (919 exact charts ≤ 262 factors, 3,317,400 assignments), the 23,627 five-stage F₂ columns, the exact price (47/47 constraints, adjacent grid rejected, pinned baseline regression exact) and the finite invoice. The sandwich word (sha 9556bfe3…) is certified as #271 certifies it, because #259's native bank review and price driver assume every helper ends at the full frame: native legality with the stage's final frames, native prefix and five-stage columns on the new word, exact SymPy endpoint charts (126 rank-3 partial swaps) with the bank re-tiling, the unchanged price engine through #271's histogram wrapper (which first reproduces the native price bit for bit), and the native finite invoice; RESULT.json and the certificate carry both κ values.

Verify

sudo apt-get install -y g++ libboost-dev
python -m pip install -r research/multicut-kernel-composition/requirements.txt    # sympy 1.14.0
python3 -I research/multicut-kernel-composition/verify.py --output /tmp/mkc-replay   # about 4 minutes; fresh directory outside the package

#259's verifier with the stages inserted (02b descent, 02c–02e plateau, 02f target prefix, the seven native checkers, then 09–17 the sandwich stage and its checks), each gated by the sha256 of the transcript it consumes; prints PASS: kappa = 0.000714026746429965 (seven native checkers alone, before the sandwich stage: 0.000712005174354669). Passes locally (macOS/clang) and on Linux/g++ with identical transcript hashes (native word eaff3c98…, sandwich word 9556bfe3…); the workflow runs it on Ubuntu with Python 3.11 and 3.13.

Scope and limits

• Same conditional status as κ = 7.10392186446856e-4: transport dirty compensations to 21 entrance frames #249/Response-kernel entrance gauges: 132 helper-pair rewrites, κ = 7.1046520063283e-4 #254/Multi-cut kernel condensation: 518 entrances, kappa = 7.11029386054035e-4 #259 (finite construction under the retained public all-size hypotheses; no Lean certificate). κ is bit-bound; the complex supplier of this lineage is at 7.5474·10⁻⁴ five-stage (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, Lean-checked).
• Not composed: Frame-filtered collective kernels + target-prefix sharing: kappa = 0.000712074434532274423885827 #272's frame-filtered collective kernels (925 families), Finite assembly theorem and source-bound PR799 application #275's zero-frame target restores, the fixed-prime refinement of Fixed-prime refinement of PR259 (kappa = 0.000711029389987426878345881) #261/Conditional κ = 7.11032242134072e-4: fixed prime, eight finite levels and η = 10⁻²⁴ on #263's descent-retimed multicut word #265 (+4·10⁻¹²), the two sandwiches beyond 126 (mod-3 rule). Conditional κ = 7.22869827347495e-4: a new paired-cube bit word (arc-aware local circuit, centre-sharing pair module) in the five-stage completed banks #276's gen4 word is a different bit word; none of these levers has been re-derived on it yet.
• Diff against Multi-cut kernel condensation: 518 entrances, kappa = 7.11029386054035e-4 #259's package (discovery/README.md): the frozen selection, the retiming stages and their rebound selections (verbatim from Conditional κ = 7.11032238200698e-4: concave-descent frame retiming of 25 gates on #259's multi-cut condensation #263 and Conditional κ = 0.000711457853376510: 887 entrances and connected-block retiming #270), Conditional κ = 7.11948953756315e-4: target-prefix compression (209 square groups) composed with the multi-cut kernel condensation #273's target-prefix stage (verbatim) with its rebound selection, Cancel cleanup sandwiches of 99 helpers on #272: conditional κ = 7.1366215638961e-4 #271's sandwich mechanism ported as three files, verify.py with the stages and sha gates, regenerated receipts, and two platform fixes (std::stable_sort on the gauge order — the tie order of std::sort differs between libstdc++ and libc++ and changes the transcript's sha256 — and the compiler-dependent memo-cache counter containment_pairs_checked dropped from the compared receipt fields). Multi-cut kernel condensation: 518 entrances, kappa = 7.11029386054035e-4 #259's own README/PROOF are kept as README-PR259.md, PROOF-PR259.md.

Credits

Prepared by Chafik Boukhalfa with substantial Anthropic Claude assistance. Apache-2.0; lineage in NOTICE.md and vendor/UPSTREAM-NOTICE.md. The mechanism, chart audit, checkers and pricer are Dugongue's (#254, #259; donor sharing #267); the descent retiming stage and its 25 gates, and the native target-prefix stage with its 209 squares, are rohanarun's (#263, #273); the target-prefix compression is eumemic's idea (#268); the plateau retiming stage, the 24 reuse pairs, the two e = 2 planes and the selection rebinding are utcorvusvolat-dotcom's (#270); the cleanup-sandwich rewrite, its endpoint charts, re-tiling and price wrapper are evmckinney9's (#271); the word is eumemic's #249 (on #244/#251, #210's helper, #168's modules) in the five-stage layout of hcg890 (#234) and rohanarun (#237/#209). This package contributes the censuses, the 387 families, the pivot-role swap that exposes the sandwiches, the composition of the four levers in one verifier, the receipts and the discovery scripts.

[Mister-Meeseeks]

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

Contribution: Multi-cut kernel families, target-prefix groups and cleanup sandwiches.

Review and limits: Source-bound Linux full replay passed all 17 stages including cleanup-sandwich legality, prefixes, columns, endpoint banks and pricing. Local run exhausted disk during that stage; the failed local run is not represented as a pass. Independent mechanism gains are not added.

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: 8cff894317648ae8d4f170b3d33456a76a7822c7. Publication: 50d768818f579d4e165b4c257c6ee25bca300802.