[ToikY]

For six formal query targets over any characteristic-two field, prove total paid rank eight is minimal with at most four square-zero calls of rank at most two, arbitrary constant free bases and any finite initialized-clean helpers returned zero. The endpoint is the complete clean-column contract W(x,0)=(Qx,0); helper input columns remain unspecified.

The new implication is the fixed-subspace extension of Nick Petalas #6800's credited projection/rank-factorization argument: if M=sum N_i fixes a d-dimensional space and N_i²=0, then S=sum rank(N_i) ≥ d+max rank(N_i). Actual I+QN calls normalize through I+QN=(I+N)(I+(1+Q)N). No projection assumption is imposed on M. Existing #6808 supplies attainment and helper minimality; neither is replayed or recounted. The four-call cap is essential: #6800/#6138 already give six targets with seven rank-one workspace-free calls, total seven.

Validation: first new diagnostic passed in 0.678s /21.672MiB. Exhaustive bounded rank-factor checks cover 23,998 ordered two-role square-zero sums over F2/F3/F5; three rank-two block fixtures; 54 new tiny F2 normalization words/270 instruction steps; 81 padded rank profiles; four mathematical controls, one legal-zero-padding counting convention and three parser controls. Final normal and optimized complete packets agree (26,723 bytes, SHA256 e913efb71bbeda6694cce9395b295967d2e13b66a750a91db6251a993d177e87). Four inert exact predecessor prose files are locally authenticated. No predecessor engine or eight-role attaining word runs. Registered replay and committed source-history/mode preflight pass. Read-only Git-free empty-PATH export replays match the whole packet in both modes. Ten isolated source/typed-False-to-0 corruptions reject for their intended reasons under normal/O. Original hosted run42325853361 passed on attempt1: 711 total helper tests including24skips, plus two focused scientific commands on CPython3.12.16. Both complete26,723-byte hosted packets match frozen local bytes. The synthetic test checkout has the exact testedtree and canonical/public parents; no repository merge. These are finite checks; this is a finite certificate plus written proof, not human/kernel acceptance.

Only research/clean-query-rank-optimum is added. No dependency, shared code, selected frontier or κ change. Changed/partial bit-new-register-model task; native caller, helper preparation, basis implementation, routing/chronology, precision/stock/restoration and complete physical recursive invoice remain unknown. All five selected assumptions remain. No broader construction/classification reserved.

Credit Nick Petalas #6800 for the method/projection inequality and clean/dirty distinction; own excluded #6808 for attainment/helper minimum; Josie Elliston #6138, Petalas #6122, Cinna Davis, Ali Romman, Matarener and retained lineage. Classical UV/VU, rank-nullity and dual-number methods are credited. LICENSE/NOTICE and immutable source pins retained. Substantial source-aware OpenAI Codex proposal/proof/implementation/review assistance; not blind independent or human review.

Claim/investigation: #6808 (comment)
Hub: #1166 (comment)

Main/source/overlap refreshed; canonical main remains 7c49fe3. No merge; maintainer edits and auto-merge disabled. A replay-manifest status error stopped before science; corrected to ready with original attempt retained and verification clock restarted. First successful packet's zero-control wording was clarified, all other typed fields unchanged. Exact receipts and remaining obligations accompany the handoff. Reproduce with the two commands in README. Public head f333cd5.