All-length R420 quad4 cleanup amplification floor (seed 0)

[1337isnot1337]

CLAIM — bit supplier, OS-random assignment seed 0.

Base: dadfea6. Object: the fixed six-register positive integer transfer of R420 quad 4 from PR1058 (d148d20).

Question: is PR1266's attained reverse-positive infinity norm 9 optimal among all finite positive-ADD factorizations, including longer words? PR1266 (8edfb5d) proves only the shortest-word optimum and leaves 7/8 open. This is an all-length algebraic proof obligation, not more search for fewer gates.

Method/domain: combine the entrywise absolute-inverse bound with diagonal positivity and mod-2 inverse parity. Six ordered registers, only dst += src with distinct operands, no workspace, permutations, subtraction or alternate lift. Replay the norm-9 witness with two exact matrix engines and scope controls.

Expected artifact: a scoped proof and exact attaining certificate, or a concrete counterexample to the proposed floor. Expected finish 2026-10-13 22:05 UTC; claim expires 23:36 UTC. No new kappa or full native/hosted-supplier admission is claimed. Existing PR1058 and PR1266 credit is retained. Prepared for Cinna Davis (@1337isnot1337) with substantial OpenAI Codex assistance.

DONE — 2026-10-13 21:55 UTC. PR1296 proves the all-length minimum reverse-positive infinity norm is exactly 9 for the fixed target. The absolute-inverse floor of 7 is strengthened by a positive even cleanup diagonal at the zero inverse-diagonal coordinate. The existing 11-ADD witness from PR1266 attains the floor; no expanded gate search was required.

The registered Python 3.12.15 replay and local Python 3.14.7 checks pass, including two exact arithmetic engines, all 64 local Boolean inputs, 9,331 finite regression words, the actual 19-ADD predecessor word and eight mathematical/package tests. The arbitrary-length claim follows from the written proof, not those finite tests. Reproduce with python3.12 scripts/research_package.py --package research/r420-all-length-cleanup-floor.

Credit: Chris Barber (@cbarber1), PR1058, for the target and minimum-length construction; Cinna Davis (@1337isnot1337), PR1266, for the reused witness/envelope; eumemic, PR543, for the underlying R420 architecture, with retained lineage and AI disclosures. No native whole-word invoice, selected supplier adoption or new kappa is claimed. The exact next task is to evaluate the two norm-9 witnesses against the original word's nonuniform entering budgets under unchanged full native/exporter checks.