Finite construction: selected X4 target forests save 47,690 primitives with an exact class floor (no new kappa)

[RossRichardson]

Finite complex-supplier construction with an exact scoped cost floor; no new kappa.

The selected PR799 X4 compensation panel uses 6,320 separately prepared additions. A restored signed target forest reduces it to 2,324 additions, including all 106 preparation and 106 restoration gates. Under the unchanged binary-Horner lowering:

Complete scalar projection	Original real primitives	Candidate	Saved
Forward	193,162	145,472	47,690
Inverse transpose	193,210	145,520	47,690

This is 13,890 additional primitives saved per orientation beyond #901's reported disjoint-pair result on the same panel. The complete candidate is repriced; predecessor savings are not added. All source access, coefficient work, boundary operations and restoration are charged. Paid children and non-scalar events compare unchanged.

The 113-vertex minimum spanning tree attains 10,883 compensation primitives per real component (21,766 real primitives). The complete 12,544-edge census, independently implemented Prim solver and every cycle inequality certify the optimum only within the signed-unit single-parent forest class with independently lowered coefficients. Shared work, other lowerings, helper changes and boundary folding are outside that floor.

Reproduce offline, standard-library Python 3.9+:

python3 -I -O -B research/complex-x4-target-forest/verify.py

Expected: PASS_X4_TARGET_FOREST compensation=6320->2324 saved_real=47690/47690 forest_floor=10883 columns=737 orientations=2 no_new_kappa.

Validation includes full literal scalar-primitive replay on 737 arbitrary input columns in both orientations, independent compensation operators on 328 active columns, eight microcircuits, five primary and seven independent controls, nine package tests, relocated source loading/census, and the selected-record check plus its five tests. Isolated optimized CPython 3.14.6 and 3.9.6 reproduce the exact receipt. The missing-preparation mutant passes a helper-only replay but fails arbitrary-target checking. Forward positive-prefix budget stays 55 bits; reverse rises from 72 to 74 bits, within the inherited 192-bit allowance. Exact fractional depths remain 3/19.

Exact base and refreshed main: 91a2094f2e022930069e8cfdc810a3ea1b81cd09. OS assignment seed 34, complex-supplier track. Inputs are pinned accepted replay members, bound to PR799's canonical and emitted-program hashes. Only research/complex-x4-target-forest/ is added; the selected certificate and upstream/ are unchanged. No pinned sprint issue existed. The final overlap inventory contained 721 open PRs through #931; #901 and #816 retained their inspected heads.

Credit @dd-hemant #816 (133735a2cc2319c12ba5d786214b2217e083e7da) for this signed rational forest/MST mechanism on historical CK, Oscar Fennell #759 for binary forests/superroot reduction, Sricharan Vagvala #749 for rational pairs, and ToikY #901 (dbe8dc01f8306bd7dab8d861a953b23e5cda6a70) for selected-X4 disjoint pairs. This transports the existing forest mechanism to selected X4 and checks its additional saving and family floor; it does not claim invention of that mechanism. No pending implementation is imported. #909's nonunit disjoint-pair exclusion leaves overlapping bases outside scope; #917 concerns historical CK/helper bases, and #883/#916 concern preceding X4 preconditions. Credit Jeremy Kothe #799, Chafik Boukhalfa #741, icekylinx and the retained supplier lineage.

Scalar interpreters do not execute physical Clifford/frame/phase events. Typed production admission, refreshed bindings, global owner/hosting/invoice/assembly and inherited all-size/compiler, realization, precision/tape and analytic obligations remain. Full repository/cold supplier suites, hosted CI and Lean were not run. The selected conditional kappa remains 0.001661634653669567; no full multiplication theorem or runtime speedup is claimed.

Prepared by Ross Richardson with substantial OpenAI Codex assistance, including separate AI implementations and mathematical review. Apache-2.0; no independent human review claimed. Proof, exact recipe, source pins, receipt, local claim and validation/handoff are included.