Research: rooted R420 target forests save 4,784 XORs with exact class floor

[ofennell-quipex]

This finite bit-circuit construction with an exact scoped cost floor reduces PR543 R420's complete native word from 26,916 to 22,132 XOR additions, saving 4,784 with all preparation and restoration charged. No new κ is claimed.

Twelve rooted target trees reduce frame-zero D from 19,922 to 15,138 XORs. Helpers remain untouched during D; every arbitrary target input is restored apart from its required correction. All frame paths, 420 helpers, nine COPYs and rank mass 5,112 are unchanged. The reverse positive-prefix bound rises, so it is recomputed: the submitted circuit fits the inherited 104-bit allowance.

The rooted-forest recipe class is exactly a minimum-spanning-tree problem. A certificate checking all 7,260 cycle inequalities, corroborated by an independent Prim implementation, proves a 14,921-XOR unconstrained D floor. The best nominated tree at that floor needs 106 bits. Eight cuts yield the submitted 104-bit circuit; neither precision-constrained optimality nor failure of every minimum-cost tree is claimed.

Credit: #713 (Scott Woolsey Biggart) and #716 (Sagar Thakrar) supply target-pair factoring. Concurrent #742, prepared for @ellipsi2, supplies shared-donor stars. This extends to multiple tree levels, adds the exact full-forest floor and saves another 1,496 XORs beyond #742. No priority for pair factoring or shared donors is claimed. #694 and #733 achieve larger reductions in different helper-changing classes and are not superseded. No pending implementation is imported or composed.

python3 -I -B research/r420-target-forest/verify.py
python3 -I -B research/r420-target-forest/independent.py
# Optional unchanged C++17/Boost checks:
BOOST_INCLUDE=/path/containing/boost python3 -I -B research/r420-target-forest/verify.py --native

Expected primary result: PASS_R420_TARGET_FOREST D=19922->15138 ADDs=26916->22132 saved=4784 forest_floor=14921 payload_bits=104 columns=660 controls=8 no_new_kappa.

Validation passed locally on Linux/Python 3.14.7/GCC 16.2.1/Boost 1.92:

• All 672 E8 frames and 660 arbitrary-input columns; a separate set interpreter replays both directions of the official native streams.
• Unchanged C++ legality and 5/9/11-stage replay for both words on 900/1,380/1,620 columns; 96 omitted-bridge controls, three missing-restoration controls and the over-budget-tree rejection.
• Eight local controls; optimized/isolated Python; relocated/recompressed-source replay; source, certificate and forest corruption rejection; seed-5 bounded nomination reproduction; selected-record integrity and whitespace checks.

The original literal H2 and K extractor reject this syntax. The package proves and checks their sufficient semantic replacement, without patching predecessor checkers. Hosted-parent reconstruction, global invoices, semantic-interface adoption and changed-word outer assembly remain integration obligations. All inherited compiler/common-label/mutable-basis/hosting/precision/tape/analytic assumptions remain. Full repository regeneration, unchanged selected cold replay, Lean and remote CI were not run or claimed.

Base: 66849e1475c930fd7732b1a0067ffee7e1e95f60. PR543 input: 2065fe4f72a914d907f1046603f5a36f17018535, with matching original Git blobs. Only research/r420-target-forest/ is added. Proof, exact recipe/certificate, pins, validation and the completed local seed-5 claim are included. No pinned coordination issue existed; final overlap refresh covered 575 open PRs through #756.

Contributed by Oscar Fennell (@ofennell-quipex) with substantial OpenAI Codex assistance, under Apache-2.0. Full original and concurrent attribution is retained in the README. No independent human review or full multiplication theorem is claimed.

[Mister-Meeseeks]

100-PR batch review at 467b9da6fe130a12d0d6edda310cd6b0efd22a7e.

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

Reviewed scope: Target-forest compensation: 4,784 native XOR saving, complete map/prefix checks and MST dual; unconstrained cost optimum differs from feasible prefix-limited witness.

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.