Research: R420 quad 4 reaches an 11-ADD positive-transfer minimum (no new kappa)

[cbarber1]

This finite construction replaces phase-A quad 4 in the historical PR543 R420 word with an 11-ADD circuit instead of 19. Rebuilding literal reverse cleanup reduces the complete native word 26,916 → 26,900 ADDs. No new kappa. Only research/r420-quad4-optimal/ is added.

Eleven is the exact minimum for this fixed positive integer transfer on the same six ordered registers, using only r[d] += r[s], d != s. The complete inverse-subtraction DAG has 165,828 states and 917,020 edges; a separately written packed-row BFS independently first reaches identity at depth 11. This is a scoped circuit minimum, not unrestricted XOR optimality or a proof of the multiplication theorem.

The forward integer map is identical and the reversed positive map is entrywise no larger. All 660 symbolic dirty-input columns, H1–H5, ordered register frame paths, 420 helpers, nine COPY/ERASE pairs and paid rank mass 5,112 are preserved. The forward positive prefix stays 144,651; reverse falls 2,270,064 → 2,261,508. Payload remains 100 bits.

Reproduce with Python 3.9+ and the standard library:

python3 -B research/r420-quad4-optimal/verify.py
python3 -I -B research/r420-quad4-optimal/independent.py

The package includes the literal witness, written completeness argument, exact receipts, source pins, independent record interpretation and ten rejection controls. Local verification passed on Python 3.9.6 and 3.14.7, including optimized and minimal relocated runs. The primary and independent checks were rerun before submission. Full repository verification, C++ multistage replay, hosted invoice/outer assembly, remote CI and Lean are not claimed. Inherited compiler, physical hosting, precision/tape, exact recovery and analytic obligations remain.

Overlap: all 1,051 open/closed PRs through #1052 were refreshed. PR925 (Ross Richardson), PR1002 (@woklerx) and PR1027 (Hanson Sun / @bbMisk) already use the same DAG/BFS method: no technique novelty is claimed. The exact targets in #691/#925/#986/#1002/#1027 are different under all 720 register relabelings; matrices and hashes are recorded in OVERLAP.json. #789 concerns phase-B block 7; #904/#936/#952 optimize the later R439 word. Their savings are not combined here. This retains its historical R420 scope.

Experimental base: 66849e1475c930fd7732b1a0067ffee7e1e95f60; PR543 source: 2065fe4f72a914d907f1046603f5a36f17018535. All seven runtime/provenance pins remain unchanged at upstream 2921874911956718fcaaaf9396fe8267b3cfa9f0. OS assignment seed 4, bit track. No pinned coordination issue existed; the completed claim is local. Selected sources and upstream/ are unchanged.

Prepared for Chris Barber (@cbarber1) with substantial OpenAI Codex assistance. Credit eumemic, Dylan Crow, DreamingOfClouds, Zhihao Chen, DaysSky, chafreaky, mtpsl and the complete predecessor lineage retained in NOTICE.md. Related exact-search and circuit contributions are explicitly credited there. Apache-2.0; no independent human review is claimed. No automatic merge requested.

[Mister-Meeseeks]

100-PR batch review at d148d200f2d64c90504fa3a9608e455aaa4c0c21.

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

Reviewed scope: Eleven-gate R420 quad witness and exact monotone-integer minimum on the same six registers; no unrestricted F2 minimum.

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.