Research: signed CK target forests save 66,520 primitives with an exact class floor (no new kappa)

[dd-hemant]

Finite circuit construction / research note with an exact scoped cost floor; no new κ.

The #626 CK checkpoint prepares 8,704 frame-zero compensation additions independently. A restored signed forest of its 112 target rows reduces that panel to 3,342 additions, including 106 preparation and 106 restoration gates. Under the unchanged binary-Horner lowering, each complete local orientation saves 66,520 real scalar primitives:

CK local program	Before	After
Forward real primitives	224,938	158,418
Backward real primitives	224,986	158,466
Scalar macros, either orientation	21,936	16,574

All helper inputs are restored. Frames, ordered wrappers and paid recursive children are unchanged. The full positive prefix envelopes are 40/65 bits, within the inherited 192-bit local allowance. These are circuit counts, not runtime measurements or a new exponent.

The signed-unit forest class reduces exactly to a 113-vertex minimum spanning tree. Its attained compensation floor is 16,364 complex primitives, with every coefficient preparation, source access, basis operation and restoration charged. All 12,544 signed/root edge cycle inequalities pass; an independently implemented Prim solver with different coefficient arithmetic agrees. The floor excludes general resynthesis, non-unit parent coefficients, helper transformations and cross-macro coefficient sharing.

Only research/complex-ck-target-forest/ is added: proof, executable construction, exact recipe/certificate, separate integer checker, source pins, local claim, validation and attribution.

Reproduce with Python 3.10+ and the standard library:

python3 -I -B research/complex-ck-target-forest/verify.py

Expected final line:

PASS_CK_SIGNED_TARGET_FOREST compensation=8704->3342 saved=66520/66520 forest_floor=16364 columns=716 controls=11+25 kappa=unchanged

The unchanged native emitter and scalar/frame/wrapper auditor check both complete programs on all 716 arbitrary input columns, with all 25 inherited adverse cases rejected. Eleven new controls include omitted basis preparation: that mutant passes helper-only replay but fails arbitrary target-input replay. All 206 distinct coefficient lowerings are checked with arbitrary private work input. A separate integer engine checks 366 active columns in each orientation and verifies the remaining 350 are untouched.

Local Linux validation passes on CPython 3.10.12 and 3.12.11, including optimization, minimal relocation, paths with spaces, alternate gzip packaging, output preservation and four source/recipe/certificate corruption checks. The original base's selected-record check and staged whitespace checks pass. Full repository/cold regeneration, global admission, Clifford wrapper regeneration, Lean and hosted CI were not run.

Original base: 66849e1475c930fd7732b1a0067ffee7e1e95f60. OS assignment seed 39, complex-supplier track. No pinned sprint coordination issue existed, so the claim is local.

During verification, upstream advanced to 7ee02b7c96d85f22b38d28bd7ed49bb8c1aeb261 and selected #741/#589/#555 at κ = 0.001661202643272239. This remains a #626 checkpoint contribution, with no claimed transport to #741. All seven runtime input pins are unchanged on refreshed main; its preserved #626 record matches the initial selected record exactly. The patch applies cleanly to refreshed main, and a minimal checkout using its actual inputs and new selected pointer passes the complete verifier.

Credit #749 (Sricharan Vagvala) for rational JP8 target pairs and #759 (Oscar Fennell) for binary target forests and the superroot reduction. This adds signed rational coefficient-dependent forest prices, CK coverage, an attained exact class floor and full native replay. Concurrent #807 (David Awad) changes CK helper pairs; #769 changes CK dyadic runs; #793/#811 change JP8 coefficient routes. No pending code is imported and savings are not assumed additive. The #622/#626 lineage, icekylinx, DreamingOfClouds, Jeremy Kothe, Chafik Boukhalfa, eumemic and retained predecessors are credited in the package.

Prepared for @dd-hemant with substantial OpenAI Codex assistance, Apache-2.0. Both new implementations were written by one AI agent; no independent human review is claimed. A typed production rewrite, refreshed bindings and global owner/invoice/assembly remain open, together with inherited all-size/compiler/realization/hosting/precision/tape/analytic assumptions. Finite replay is not a proof of the complete multiplication theorem.