Research: certify R420 forest cuts under the 104-bit cap

[anthonylei]

This finite bit-circuit construction with exact scoped certificates and a generic Lean proof saves 4,897 binary XORs in PR543's R420 word: 26,916 → 22,019 overall, with compensation D 19,922 → 15,025. Every preparation and restoration operation is charged. The unchanged native 104-bit prefix cap passes. No new κ is claimed.

Concurrent #759 by Oscar Fennell is credited. It appeared during final validation with the same forest class and 14,921-XOR unconstrained floor. This independently corroborates that floor and adds an exact coefficient-constrained cut optimization, 113 fewer XORs than its submitted 15,138-XOR D circuit, independent verification and a generic Lean proof. No priority for forest factoring or the floor is claimed. The package uses the distinct path research/r420-forest-cut-budget/ to avoid colliding with #759.

The contribution has three linked parts:

• A minimum spanning tree and all 7,140 non-tree cycle inequalities certify the 14,921 unconstrained optimum for the explicitly defined single-parent forest recipe. The particular seed-6 optimum needs 106 payload bits; all three unchanged native column checkers reject it. It is an explicit failing calibration.
• An exact dynamic program checks all 2^116 cut subsets of the fixed oriented tree using 724 ancestor states. Cutting at targets 5, 34, 39, 57, 78 attains the exact minimum D cost 15,025 in this class. The independently reimplemented recurrence agrees. Its complete positive-prefix maxima are F=144651, B=9800790, with 64 F^3 B^2 = 18606576056688220135806852422400 < 2^104.
• Lean 4.21.0 proves generic XOR-circuit linearity/reversal, children-before-parent forest execution, and compensation for arbitrary helpers and arbitrary initial targets. Seven axiom reports contain only propext and Quot.sound; no sorry or extra axioms. Lean does not bind the concrete word or prove MST/DP, physical, or asymptotic claims.

Optimality is deliberately narrow. A positive scope-control test changes only parent[32] from 37 to 34 and obtains 15,023 D XORs with the same exact scalar prefix bound and valid 660-column/frame replay. It is outside the frozen cut family and correctly fails its cut certificate. That separate word has not received native or hosted replay. No best-known circuit or global constrained-optimality claim is made.

python3 -B research/r420-forest-cut-budget/reproduce.py
# Optional scoped formal and unchanged native checks:
python3 -B research/r420-forest-cut-budget/reproduce.py --lean /path/to/lean-4.21.0/bin/lean
python3 -B research/r420-forest-cut-budget/native.py --out /absolute/fresh/output --boost /directory/containing/boost

Expected primary result: PASS_R420_TARGET_FOREST D=19922->15025 ADDs=26916->22019 saved=4897 columns=660 payload_bits=104 cut_states=724; uncut_optimum=14921 rejected_at_106_bits.

Validation passed locally on macOS arm64 and on Ubuntu 24.04 / Python 3.13 at f0935b4ab2fd1e051d7508b56b44f143e23211e8: dedicated Linux CI run. The run retains its exact head, file inventory, environment, logs and native receipts.

Coverage:

• Source-pinned canonical reconstruction; all 660 arbitrary-input columns, exact geometry and unchanged MOVE/COPY/ERASE paths; rank mass 5,112, 420 helpers, nine copies.
• 37 adversarial/positive tests, including missing preparation/restoration, wrong coefficients, underpriced/nonminimal certificates, numeric-type source-binding mutations, recompressed source, paid neutral operations, and the out-of-class improvement.
• Independent literal unsigned-ADD enumeration of 620 small ancestor states / 7,904 cut assignments, with corrupt-certificate controls; source-pin compatibility controls; credited semantic-D chronology checks on all 660 columns.
• Fresh unchanged C++17/Boost checks at 5/9/11 stages: 900/1,380/1,620 formal columns, all 6/18/24 bridge controls on both original and selected words. A frame-legal missing restoration fails all three endpoint checks. The uncut calibration fails the unchanged payload check.
• Lean compilation and axiom audit; isolated minimal relocated replay; original selected-record integrity; Python compilation and documentation/whitespace checks.

The original helper-only H2 syntax and literal K extractor reject target mixing. The semantic replacement proves and checks the same complete D boundary map; no predecessor checker is patched. The conditional chronological substitution bridge from woklerx #1194 and Arham Tahir #1311 is now checked for this exact word: the original-D splice is literally the original checked word, and the actual D supplies identical values and frames to its continuation. Full hosted-parent/whole-owner reconstruction, global finite invoices, maintained supplier adoption of semantic D and changed-word outer assembly remain open. The inherited uniform/compiler/common-label/mutable-basis/hosting/precision/tape/analytic assumptions remain. The dedicated Linux workflow runs reconstruction, all controls, Lean and both native modes; its checked revision and receipts are retained. No repository-wide suite or full multiplication theorem pass is claimed.

Base: 66849e1475c930fd7732b1a0067ffee7e1e95f60; PR543 source: 2065fe4f72a914d907f1046603f5a36f17018535. Assignment seed 6, drawn once from OS randomness. No pinned coordination issue existed; the completed claim is local. The PR adds this research package and its dedicated CI workflow; it does not alter upstream/ or select a new frontier.

Canonical main at 0fdd7d82de03e43c9e5a1000dfbcd380d1b687d2 is merged without conflicts (1,817 upstream commits since the scientific base). The live selected result is now #1094 using PR589/555 R439; this certificate intentionally retains its historical R420 baseline. The old selected-record hash is checked against the byte-identical #626 checkpoint, with no expected hash changes or live-record fallback. All original data, source hashes, claim, validation and Lean files are unchanged. REVIEW.md records the priority-revision-4 guidance, live overlap refresh, credited upstream comparison and added coverage.

Credits include eumemic's R420 construction and its cumulative community lineage; #713/#716 target pairs; #742 shared donors; #759 forests; #686/#694 helper bases and #733's stronger two-sided composition. The latter is not superseded or combined here. Tree-based parity synthesis by Vandaele, Martiel and Goubault de Brugière (arXiv:2104.00934 §3.1) is acknowledged explicitly; no literature-wide priority claim.

Prepared by Anthony (@anthonylei) with substantial OpenAI Codex multi-agent assistance, under Apache-2.0. This is not independent human peer review.

[Mister-Meeseeks]

100-PR batch review at 50f8c366f27a9083e0655593a1be3d6810d1f053.

This remains open after scoped triage. Next review step: Constrained forest construction plus Lean proof needs both finite and formal coverage.

This is a specific continuation point, not a rejection or a request to redo already recorded checks. Completed local evidence is retained separately from outstanding whole-head obligations.

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