Conditional κ = 7.11426001276118e-4: 365 net-positive supplementary equal-response pivot families on #259's multi-cut condensation

[rohanarun]

Conditional κ = 355713000638059/(5·10^17) = 7.11426001276118e-4, +1.25 × 10⁻⁶ (+0.176 %) over #259 (7.11029386054035e-4), and above #266's 7.11421337e-4 (same lever; see below). Superseded by #273, which adds target-prefix compression on top of this word. Stacked on #263's branch (PR259's package + the descent retiming) at 8712930.

Full replay: PASS (final package, commit d4ca9aa): PASS: kappa = 0.000711426001276118, PASS_ALL_SEVEN_NATIVE_CHECKERS_AND_PINNED_RECEIPTS. An earlier revision of this PR admitted every intersecting pair (416 pairs, 7.11419e-4); pairs are now kept only when net-positive under g(r) = r·ln(120/r) with the pivot chosen to minimize the total, which is the same frontier #266 reports (365 families).

What is new

416 supplementary pivot families, found where #259's search stopped. #259's mechanism (a pivot helper starts at a nondegenerate entrance frame E, its initial compensation reads are omitted, donors are sheared by Q = I + Σ c_j e_ij e_pᵀ at the cut and un-sheared at the full frame) needs an F₂ dependency among untouched helpers' dirty responses and a common nondegenerate subframe of their first required frames. Measuring at #259's first cut (record 676,559): the 13,093 untouched zero-start helpers that #259 leaves unused have response columns of F₂ rank only 1,760, so dependencies are abundant (12,500+); geometry is the bottleneck. Among the 618 leftover exact equal-response pairs, 365 disjoint net-positive pairs have first required frames whose exact intersection is nondegenerate with an admissible prime witness (362 lines, 3 planes), adding 366 entrance-rank units (363 pairs kept by the transform's mod-3 normalization): raw mass 434,177 → 432,467, 59,184 initial reads removed in total, 881 cohorts over 1,920 helpers.

They are supplied through cohort-transform's existing extra-candidate-file interface (inputs/candidates-supplementary.json.gz); the C++ is unchanged apart from the stable_sort determinism fix from #263. The transform validates each family exactly as it validates #259's own (untouched before its cut, prefix reads only zero-frame dirty reads into targets, exact first-frame containment of E, independent entrance basis, nilpotent basis support, 40-replica integrality) and every native checker re-examines the combined word.

The 25-gate concave-descent retiming, re-searched on the 934-family transcript (code/descent_retiming.py, inputs/descent-selection.json): one local recursive call per helper invocation disappears at unchanged rank mass; the stage checks nested chains, fixed endpoints, COPY lifetimes, nondegenerate bases and exact integer operand source spans, and the native checkers run after it.

Normalized retained profile: stock 173,313, 3,971,320 calls, rank mass 20,762,360, deficit 35,200.

Reproduce

python3 -m pip install -r research/multicut-kernel-condensation/requirements.txt
python3 research/multicut-kernel-condensation/verify.py --output /tmp/multicut-934-replay

Python 3.11+, SymPy 1.14.0, C++17 GNU-compatible compiler and Boost headers; new output directory outside the package; no network. cohort_record_sha256 (after the transform) and new_record_sha256 (after the retiming) are pinned in RESULT.json.

Scope and credits

Exactly #259's hypotheses and conditional interfaces; #259's caveat applies (the public eight-stage Python command is not replayed on this word; the native checkers are). No unconditional multiplication bound, benchmark or Lean certificate is claimed. There is more here: the equal-response pairs are the simplest dependencies; three-member dependencies and additional cuts are being searched.

Dugongue (#259, #254): the multi-cut kernel condensation, native checkers and packaging, including the supplementary-candidate interface; eumemic (#249, #244, #210) and the retained lineage. The supplementary family search, the retiming stage 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: Supplementary equal-response pivot families.

Review and limits: Fresh full native replay passed, including the actual-cut compatibility, paid frames, prefix and column checks. Supplementary pair gains are assessed on one final word.

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: d4ca9aacd885375e6b6ce555bfb2d228ca71e1c7. Publication: 50d768818f579d4e165b4c257c6ee25bca300802.