Conditional κ = 7.11032238200698e-4: concave-descent frame retiming of 25 gates on #259's multi-cut condensation

[rohanarun]

Conditional κ = 355516119100349/(5·10^17) = 7.11032238200698e-4, +2.85 × 10⁻⁹ over #259 (7.11029386054035e-4) and above #261's fixed-prime refinement of #259 (7.110293899874e-4). Stacked on #259's unmerged main7 branch at a53ec0c.

Full replay: PASS. verify.py --output /tmp/multicut-descent-replay on the committed package regenerated the #249 baseline from source, ran the transform, the new stage and all seven native checkers, and matched every pinned receipt: PASS: kappa = 0.000711032238200698, certificate status PASS_ALL_SEVEN_NATIVE_CHECKERS_AND_PINNED_RECEIPTS, new_record_sha256 = 80af525cfdc547090015dc9ab1eaf14fdffe5cacb5b9d0efcb05a4f908e2cc6d (macOS/clang + libc++; the stable_sort fix makes the transcript identical under libstdc++).

Change

One additional stage, code/descent_retiming.py, runs between #259's cohort-transform and its seven native checkers. It reassigns the frames of twenty-five internal forward_gate ADDs (frozen in inputs/descent-selection.json with explicit rational bases, bound by ADD ordinal, scalar content and old-basis equality) found by local descent on the concave paid-rank objective Σ r·ln(120/r). The local paid histogram changes by {2: −2, 4: +24, 5: −48, 6: +25, 11: −1, 13: +2, 17: −1}: one local recursive call disappears at unchanged rank mass 432,833. Normalized retained profile: stock 173,435, 3,956,800 calls (was 3,956,840), rank mass 20,777,000, deficit unchanged. Banks, charts, entrances, prefix kernels, the scalar word and every endpoint are unchanged. The twenty-five registers are disjoint from all 518 pivots and their donors, so the moves are the same ones as in #260 (on #251) and #258 (on #254).

Why these 25 and no more: for a register window (prev, cur, next) the objective is concave in dim(cur), so each gate is optimal at the minimal frame (join of the operands' previous frames and their integer source spans) or the maximal one (meet of the next frames); on this word 99.7 % of gates have both coincide with the current frame, the fully eager greatest-fixpoint assignment is worse, and run-level block moves are blocked by partner chains or spans.

Verification

• descent_retiming.py rebuilds every MOVE from the retimed gate needs and checks: byte-identical scalar/COPY projection, nested chains, unchanged initial/final frames and COPY lifetimes, nondegenerate endpoint bases, the exact integer source span of every non-target operand inside its new frame (recomputed from the actual transcript), the expected histogram delta and unchanged rank mass. It writes lead/DESCENT-RETIMING.json (pinned in expected/) and folds its delta into the replay receipt, so the native price checker still reconstructs and regresses the κ = 7.10392186446856e-4: transport dirty compensations to 21 entrance frames #249 baseline exactly.
• All seven native checkers then run on the retimed transcript unchanged: independent chronological legality (PASS pairs 518 ADD 760393 mass 432833 COPY24), 911,680 prefix kernel equalities, 518 charts and 3,317,400 bank assignments, 23,627 five-stage columns, exact rational price (baseline 88799023305857/125000000000000000 reproduced; candidate 355516119100349/500000000000000000; 47 strict constraints; adjacent grid point rejected), finite invoice (stock 867,175, selector bill 626,937,381,600). Receipts re-pinned in expected/.
• cohort-transform.cpp: sort → stable_sort for the cut gauges. std::sort is unstable, so Multi-cut kernel condensation: 518 entrances, kappa = 7.11029386054035e-4 #259's pinned transcript hash is libstdc++'s tie order and does not reproduce under libc++ (the replay receipt does). The cohort transcript is now pinned as cohort_record_sha256 and the retimed one as new_record_sha256 in RESULT.json; verify.py checks both.

python3 -m pip install -r research/multicut-kernel-condensation/requirements.txt
python3 research/multicut-kernel-condensation/verify.py --output /tmp/multicut-descent-replay

Python 3.11+, SymPy 1.14.0, C++17 GNU-compatible compiler and Boost headers; new output directory outside the package; no network.

Scope and credits

Exactly #259's hypotheses and conditional interfaces; no unconditional multiplication bound, benchmark or Lean certificate is claimed. #259's own caveat applies: the public eight-stage Python command is not replayed on this modified word; the native checkers are.

Dugongue (#259, #254): the multi-cut kernel condensation, native checkers and packaging; eumemic (#249, #244, #210) and the retained PR234/197/200/193/207/211/216/230/237 lineage. The concave-descent retiming stage, the determinism fix and this package's rebinding were prepared by Rohan Arun with Anthropic Claude assistance.

🤖 Generated with Claude Code

[Mister-Meeseeks]

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

Contribution: 25 concave-descent retimings on the multi-cut construction.

Review and limits: Fresh full native replay passed all independent checkers and assembly. Stable equal-rank ordering is explicit. Same 25-gate mechanism as PR258/260, not additive independent savings.

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: 87129308f762f86a9d8e9a60f5d6c8ef23ce1f6e. Publication: 50d768818f579d4e165b4c257c6ee25bca300802.