Scoped negative result: R420 phase-B frames are moment-optimal at fixed gate order

[yuj1r0]

Change and mathematical scope

Scoped negative result; no new κ. The selected PR543 R420 bit word already minimizes its paid recursive moment among all simultaneous phase-B frame reassignments that preserve its 837 scalar gates, operand lists and order, with D/A/scatter/C and boundary frames fixed. The bound admits arbitrary rational subspaces, including frames outside the existing catalogue.

Forward spans and backward exact subspace intersections give necessary gate-dimension intervals. Dropping shared-gate geometry yields independent register rank paths. The baseline majorizes every one of the 918 relaxed partitions across 660 registers, so concavity proves the bound for every exponent 0 < p < 1. The attained variable moment is 720 + 451*2^p + 120*5^p; adding the fixed exterior and COPYs recovers the published local profile and rank mass 5112.

This adds a written proof, small exact certificate, pinned source identities, two separate standard-library implementations and 12 controls. It excludes reordering, factoring, pruning, circuit resynthesis, changed boundary/COPY interfaces, other words, and changed hosting. It is weak optimality of this paid moment, not uniqueness or a bound on integer multiplication generally. No selected-frontier or upstream file changes.

The distinction from PR589 is simultaneous all-frame optimality for the specific R420 B schedule; PR589 already searches both A and B and disclaims optimality. PR612's R420 scalar/B-XOR formalization addresses a different obligation. Live overlap metadata was refreshed through #660; no pinned coordination issue existed, so the seed-8 bit-supplier claim and its result are local in CLAIM.json.

Validation

Exact base: 66849e1475c930fd7732b1a0067ffee7e1e95f60 (also latest main at final check). PR543 dependency: 2065fe4f72a914d907f1046603f5a36f17018535; both word and baseline-checker Git blobs match that PR. SOURCES.json carries SHA-256 pins and PR589 context.

python3 -B research/r420-phase-b-frame-bound/verify.py

Expected final line: PASS_FIXED_ORDER_B_FRAME_BOUND gates=837 registers=660 partitions=918 gaps=0.

Passed locally on macOS / Python 3.12.8:

• Pinned PR543 exact labels, all 672 frames, all 660 symbolic dirty-input columns and H1–H5.
• Complete per-register equality between rank-state DP and independent ordered-composition/event-cut enumeration; exact certificate and all 12 controls.
• A legal rank-3 promotion outside the original catalogue is detected as suboptimal at source 13 and target 128. Omitting its mandatory rank-2 read exposes the false unsplit rank-7 saving.
• Corrupt certificates, an illegal low frame, changed cleanup, and an incorrect input pin are rejected; 1,000 small interval domains agree with brute force.
• Isolated independent auditor, optimized-Python run, selected-record binding check, and staged whitespace check pass.

VALIDATION.json records the local scope. No remote CI result is claimed. The optional dedicated workflow was removed because the available GitHub token lacks workflow scope; the full local reproduction command is included. The unchanged full frontier assembly and repository-wide suite were not rerun for this standalone addition. Finite execution is not presented as a Lean proof or discharge of inherited compiler/physical/hosting/precision/tape/analytic obligations.

Credit is retained for eumemic (PR543/493), Dylan Crow (PR370), DreamingOfClouds (PR510/589), Dugongue (PR541/548), Rohan Arun (PR542), and the cumulative selected-construction authors. Prepared for yuj1r0 with substantial OpenAI Codex assistance, including separate Codex workers for computational cross-checking and review; Apache-2.0.

[Mister-Meeseeks]

Thank you @yuj1r0. This contribution is now retained on main in the scoped 100-PR review and contributor credits.

Contribution: Simultaneous phase-B frame optimality for the R420 word.

Review and limits: Allows arbitrary rational subspaces but freezes scalar gates, order, boundary/COPY interfaces and all other phases. The majorization bound does not forbid circuit resynthesis or changed hosting.

The complete original source, notices and relevant execution evidence are retained through the pinned source archive. The review distinguishes fresh local runs, source-bound historical Linux CI and prior exact-head review; it does not claim 100 fresh full mathematical audits. Closing this historical PR as reconciled. This preserves parallel and cumulative credit without replacing the selected frontier or promoting priced research targets to completed constructions.

Reviewed head: 9ca7aa3ab0a3260a2f36ac86b9911a31fd88bd89. Publication: 64039f1bd3b6a523b96083fd858ad040a75e3030.