[apoursh]

Change and mathematical scope

Complex coefficients beat the sharp real rank-seven quarter-error floor for the fixed P104 Gram matrix. For K=RR^T/2, the explicit vector
z=(5-5i,3+9i,5-5i,7,5-5i,-1-4i,-7,-7)
gives D=I-zz*/404 and C=RDR^T/2 with complex rank seven and exact maximum complex entry error 25/101 < 1/4. A second retained witness achieves 1205/4912, with gap 23/4912 below 1/4. Neither witness is claimed optimal.

D is a Hermitian positive contractive orthogonal projector. C satisfies 0<=C<=K, not an ambient C<=I claim. Its real part has rank eight, so this answers the complex-field question in #6191 without contradicting that real theorem or giving a seven-real-mode implementation.

Scoped exact finite construction and written proof; no new κ or selected-frontier change. Odd denominators remain, and no Gaussian-dyadic/native implementation, caller installation, residual correction, ownership/restoration or complete paid/target invoice is supplied.

Task and overlap

Canonical main refreshed first and before submission at a5bff3bf133e482541c09eba4e773884bdb2b1b7, priority revision 8. Task coverage-law-change, scope key 2bf6355a8dd01ca324fb, scope changed to this algebraic approximation question; the original supplier acceptance gate remains unmet.

Canonical advance claim. Nearest predecessors:

• Prove sharp 1/4 error for every real rank-seven P104 approximation (no κ) #6191, be3fc854aa380b3d81e134ce9a4282468e4fa999: sharp real theorem and released complex question; still pending.
• Nick Petalas's Prove a sharp 1/4 error floor for contractive P104 mode deletion (no κ) #6183, eaf77790466f34b86ce72080e3ec690e26dfacab: contractive question, real projection and retained root data.
• Matarener's P104 carrier code: the length-8 decoding code is unique (E8 root code, min weight 49) #6166, 096938513138868f0ed27639061f1a5054900f96: cited physical P104-to-D8 identification and omitted lines. That identification is not newly audited.

The local helper covered 30/8623 notes, with no usable remote cache. Live checks covered the newest 20 PRs, newest 15 handoffs and all-state P104/complex/rank, 6191 and Gram/contractive searches; the relevant search and source heads were refreshed before submission. No competing complex witness appeared in those bounded results. #6194 studies a different paid frame-graph projection objective. No exhaustive novelty claim.

Validation

Publication head: 369ad15647ad0df0b0ef6f4fcb060509b67f1823. CPython 3.12.14, macOS arm64, standard library only.

python3.12 -B scripts/research_package.py --package research/p104-complex-rank-seven
python3.12 -B scripts/check_research_commit.py --package research/p104-complex-rank-seven --check-source-history --check-modes

Both separately written checkers pass normal and optimized registered replays. Each checks all 21,632 residual entries across the two full matrices. The primary checks nine hypothesis controls and all three pinned predecessor inputs; the independent Fraction-based checker uses a different root order and independently checks complex rank-seven and real-part rank-eight minors, plus four projector controls. Three primary source-corruption controls and the independent root-data corruption control reject.

Committed bytes/modes, declared source ancestry and official export integrity pass. All four direct commands pass in a fresh read-only, Git-free export with empty PATH. Normal/optimized outputs agree and all eleven exported files retain their bytes and modes. A separate AI proof audit found no blockers.

The rank/error claims combine written projection/rank arguments with complete exact finite arithmetic, not floating-point optimization or a formal proof kernel. Discovery search is not needed for replay. No further optimization was run this round.

Not run: selected scientific cold replay, physical/native/caller replay, Lean, Docker or independent human review. Binding-only orientation was reused after confirming 1227 bound inputs unchanged; this is not a fresh scientific replay. Hosted CI is pending at submission.

Credit

Prepared by apoursh with substantial OpenAI Codex assistance in earlier witness discovery, separate exact checkers, mathematical audit, proof and packaging. Nick Petalas retains #6183's contractive question and real attaining projection; Matarener retains the coordinate identification and omitted lines. Rohan Arun, Chafik Boukhalfa and the retained P104/root/frame/native/helper/compiler/BIT lineage retain their source credit. Hermitian projection and Gram-rank facts are classical. Apache-2.0; predecessor files are unchanged.

Open next questions

README and structured questions identify two next gates: one exact Gaussian-dyadic projector that beats 1/4, and one actual caller whose complete residual-correction/restoration bill is favorable. The present witnesses supply neither. Lower coefficient error alone is not a paid-work saving.