Finite bit construction: R439 frame 267 reaches a 12-ADD positive-transfer minimum (no new kappa)

[RossRichardson]

Finite bit-supplier construction and restricted-class minimum; no new kappa. PR589's six-register phase-A frame-267 circuit uses 21 ADDs. This complete-frame replacement uses 12, and its required literal reverse cleanup gives 24,305 → 24,287 ADDs in the local word.

The full positive integer transfer is identical. The reverse positive matrix is entrywise no larger; forward bound stays 49,347 and reverse falls 1,113,070 → 1,107,434. All 661 frame paths, 439 dirty helpers, nine COPY/ERASE pairs, original H1–H5, paid profile and rank mass 5,129 are preserved. Payload remains 93 bits below the 104-bit cap. All remaining source reads, moves and restoration operations are explicitly priced; no helper or recursive child is added.

The exact minimum is 12 for this fixed positive integer transfer on six ordered wires, allowing repeated reads and writes. One implementation exhausts the nonnegative predecessor DAG (416,508 states / 2,366,680 edges). A separately written flat-matrix breadth-first checker fully expands depths 0–11 (2,271,945 transitions) and first reaches identity at depth 12. This is not unrestricted XOR optimality or a formal proof of the multiplication theorem.

Reproduce with standard-library Python 3.11+:

python3 -I -B research/r439-frame267-seed01/verify.py

Expected: PASS_R439_FRAME267 gates=21->12 native_adds=24305->24287 columns=661 rank_mass=5129 controls=8.

Completed verification includes exact integer matrices, all 64 local inputs, all 661 symbolic dirty-input columns, unchanged geometry/interface checks, a separate record interpreter, eight mathematical controls, and seven package tests including optimized minimal offline relocation and corruption rejection. Fresh unchanged C++17/Boost legality and 5/9/11-stage checks pass on both words: 883 / 1,327 / 1,549 formal columns, all 96 bridge-omission controls rejected. Full proof-relevant native receipts are retained. make selected-record-check, exact source-object checks and whitespace checks pass.

Repository-wide coverage is incomplete: one isolated make verify invocation reached its predeclared 240-second budget in inherited endpoint-gauge-producer (h=40). No full-suite pass is claimed. No selected complex cold replay, full changed hosted-parent invoice, outer assembly or hosted CI pass is claimed. Integration still requires supplier binding and inherited all-size/compiler, common-label/mutable-source, whole-call hosting, precision/tape and analytic obligations.

Overlap was refreshed through #924 across 714 open PRs immediately before publication. Gareth Davidson's #904 already supplies two small-window substitutions in this frame, reaching 19 ADDs. This complete six-wire witness reaches 12, adding 14 further complete ADD savings on this frame and a restricted-class minimum. The comparison is pinned and replayed; savings are not combined across packages. mtpsl #691 is credited for the positive-transfer/reverse-domination approach and adapted verifier; Hao Yan #823 treats a different historical R480 matrix. #728/#792 concern separate mini-nodes. The native wrapper adapts Ali Romman's #688 work.

Exact base: 91a2094f2e022930069e8cfdc810a3ea1b81cd09. Selected PR799 conditional κ remains 0.001661634653669567 with PR589 supplier head a5afc8c6169ea61ba7c51dfd3cade02cc7b13b46. All three execution inputs match the selected source inventory; 20 source/context pins match base Git objects. Only research/r439-frame267-seed01/ is added. OS assignment seed 01, bit track; no pinned coordination issue existed, so the claim is local. No merge or selected-frontier change is requested.

Credit eumemic (PR543/555), DreamingOfClouds (PR545/589/510), Dylan Crow, Zhihao Chen, DaysSky and the retained predecessor lineage. Prepared by Ross Richardson (@RossRichardson) with substantial OpenAI Codex assistance, one AI agent; no independent human review claimed. Apache-2.0.

Proof and complete cost · Reproduction and scope · Executed validation.

Published head: eb85d40a1ba80c2f776481fbeed1bae8cf04cfde. The mathematical contribution is at 8541db4f7aff70fe8ab4ecd5a12277b696a532de; the final commit only records this PR link and result in the local sprint claim. Remote head, OPEN state and the exact 18-file scope were verified. All five hosted workflows at the published head report action_required, so there is no hosted CI pass. Research-artifact workflow.