Scoped negative result: selected CK B frames are optimal at fixed boundaries

[thakrarsagar]

The selected CK complex supplier already minimizes its paid B-phase recursive moment among all simultaneous frame reassignments that preserve the scalar circuit, gate order and exact end-of-B frames. This is a scoped negative result / research note, with no new κ.

The new package exhausts all intermediate F₂ subspaces, including frames outside the retained catalogue. Of 1,930 B gates, 474 have frame freedom: 418 two-frame domains and 56 five-frame domains. Exact variable elimination retains all incomparable profiles and reduces the problem to 119 paid histograms. An exact four-rank derivative argument shows that the existing histogram beats all 118 distinct alternatives for every exponent 0 < p < 1. The fixed A, center, tail and final costs are recounted separately and recover the complete selected local profile.

Only research/complex-ck-b-frame-seed20/ is added. It includes the proof, complete finite-domain checker, exact certificate, source pins, focused independent checks, validation record and local sprint claim.

Reproduce

python3 -B research/complex-ck-b-frame-seed20/check.py

Python 3.11+, standard library, offline. Expected output includes:

PASS_CK_FIXED_END_B_FRAME_BOUND gates=1930 flexible=474 profiles=119 exponent=0<p<1

Passed on macOS arm64 with Python 3.11.17 and 3.14.8, including a clean archive containing only the package and eight pinned inputs. Direct enumeration matches 100 actual components across 1,544 assignments and a coupled synthetic case across 125 assignments. Negative controls detect an inferior split-jump baseline and equal-rank frames that fail containment. A second rational method independently confirms all 118 derivative inequalities. Optimized Python is rejected explicitly. The selected-record source check and staged whitespace checks pass.

Scope and provenance

• Exact base: 66849e1475c930fd7732b1a0067ffee7e1e95f60. Reviewed Conditional κ = 0.00163542231500688: K-triangle skeleton atom (492 helpers per call) on the PR617 cold-ended witness #622 source: f69477f4f6d3c7eb1f740b166881bffed4ffa9f6; selected Conditional κ = 0.001635527380375392: coordinated owner repair on PR623 #626 source: 660de893d27bb7abd1d5b01b1640207528a5c851. The canonical payload is bound to the retained Conditional κ = 0.001635527380375392: coordinated owner repair on PR623 #626 SPEC.
• OS assignment seed 20, complex supplier, selecting CK. No pinned coordination issue existed. The overlap refresh covered 513 open PRs through Research audit: signed block extraction and coefficient-alias controls #693, followed immediately by review of Research: save 6,934 R420 XORs with a restored helper basis #694–Research construction: direct-source E8 with 120 fewer helpers #697. Research: lower selected complex scalar primitives by 22–25% (no new kappa) #696 lowers scalar primitives without changing frames and Research construction: direct-source E8 with 120 fewer helpers #697 changes source access; neither overlaps this fixed-circuit class. The local claim now records completion and links this PR.
• Related Scoped negative result: R420 phase-B frames are moment-optimal at fixed gate order #661 establishes a fixed-order bit frame barrier. This complex case requires coupled subspace elimination and a further derivative certificate; its per-register relaxation alone has gaps. No Scoped negative result: R420 phase-B frames are moment-optimal at fixed gate order #661 code or result is imported. Research: the complex delivery layer is rigid — target hosting and raw-face sharing closed exactly #651, Research: eight signed-root K triangles with 480 helpers (no record claim) #646, Consolidate six circuit PRs: 31.72% fewer bit ADDs and 22–25% fewer complex primitives #681/Gauge44 factory and whole-family assignment for conditional κ = 0.001649483702558979 #645, and concurrent Scoped negative result: partial K/M face bypass requires four carriers #683/Scoped negative result: certify one K-triangle relocation neighborhood #687 investigate different delivery, factory or reorder mechanisms.
• Credit: Jeremy Kothe's K-triangle construction, Chafik Boukhalfa's selected owner package, DreamingOfClouds' cold-helper/native-verifier work, icekylinx's factory/direct-delivery lineage, Dominik Scholz's prefix work, and the recurrence/frame contributors retained in the maintainer review. @yuj1r0's related Scoped negative result: R420 phase-B frames are moment-optimal at fixed gate order #661 is credited. Prepared for @thakrarsagar with substantial OpenAI Codex assistance. Apache-2.0.

The result does not cover changed scalar circuits, gate reorderings, relaxed B boundaries, JP8, or changed ownership. It makes no uniqueness or global supplier optimality claim. The inherited all-size/compiler/common-label/mutable-basis/hosting/precision/tape/analytic assumptions remain. This finite certificate is not a Lean proof or a full multiplication-theorem audit. The full repository suite and selected cold/global replay were not rerun because no existing supplier, generator, verifier or certificate changes. upstream/ and the selected frontier are untouched.

[Mister-Meeseeks]

100-PR batch review at ab78f797cffcbeddbb17a7eb069741085ffe3351.

Merged to main, preserving the original contributor commits and package attribution.

Reviewed scope: Fixed-end CK B-frame optimum: complete finite variable elimination and exact four-rank derivative comparison; only the pinned circuit and endpoint constraints.

The pinned scoped command passed; the batch ledger contains its command, input hashes, environment and log. This incorporates the research/tooling contribution and does not change the selected κ or discharge the inherited multiplication-theorem assumptions.

Batch review, attribution and evidence. Thank you for the contribution.