[apoursh]

Change and mathematical scope

Every real 104-by-104 matrix of rank at most seven approximating the complete P104 Gram block has maximum coefficient error at least 1/4, and 1/4 is attained. This resolves Nick Petalas's PR6183 first open question and goes beyond arbitrary carrier filters: the candidate may have any row and column spaces. Symmetry, positivity and contractivity are unnecessary. No new κ is claimed.

Choose eight columns indexed by orthogonal D8 roots. A rank-seven candidate has a nonzero null vector on those columns. A real two-coordinate absolute-value identity bounds its coefficient norm by twice the top-two-coordinate norm, while the D8 rows force a residual of half that norm. This yields the 1/4 floor. PR6183's omitted-root projector supplies attainment and retains its credit.

Adds only research/p104-rank-seven-error-floor/. The P104-to-D8 identification is still a cited premise from Matarener's PR6166, not a new physical-port audit. Both predecessors are archived as partially reviewed; archival inclusion is not scientific acceptance. The proof concerns the full real Gram matrix, not masked accuracy, arbitrary complex approximants, nonlinear consumers, or the complete supplier. Corrections, restoration and native/caller/precision costs remain open; the selected result and five framework assumptions are unchanged.

Validation

CPython 3.12 standard-library normal and optimized registered replays pass. Exact checks cover 6,560 nonzero ternary directions, 128 rational directions, sixteen attaining projectors and all 173,056 matrix-error entries; three carrier fixtures; a nonsymmetric rank-seven candidate outside both carrier spaces; four boundary/normalization controls and three malformed-input rejections. The universal claim rests on the written proof. Separate agent mathematical and code audits found no blocking issue; no human or formal-kernel review is claimed.

The separately written Fraction checker passes 147 vector fixtures and 194,688 full matrix entries, including an oblique projector with error 3/8 and a candidate outside carrier factorization with error 1/2. Both checkers are registered; the three-command replay takes about 4.4 seconds locally.

At commit be3fc854aa380b3d81e134ce9a4282468e4fa999, committed bytes, executable modes and source ancestry pass for all ten package/input files. Official export integrity and both direct normal/optimized modes of both checkers pass in a fresh read-only Git-free export with empty PATH; all eleven exported files retain their hashes. A source-byte corruption is rejected before replay. Hosted CI is not yet complete at submission.

Reproduce from the repository root with python3.12 -B scripts/research_package.py --package research/p104-rank-seven-error-floor. The package declares all three pinned predecessor inputs and imports no predecessor executable code. The selected binding-only orientation was reused after confirming all 1,227 recorded inputs were unchanged; no selected scientific cold replay was run.

Coordination and credit

Canonical upstream was pulled first and refreshed at a5bff3bf133e482541c09eba4e773884bdb2b1b7, priority revision 8. The canonical claim records the initial carrier-only scope and its strengthening. The original coverage-law-change card, scope key 2bf6355a8dd01ca324fb, is retained with an explicit scope change; this is an algebraic approximation decision, not supplier admission. PR6166's cited coordinate identification is pinned at 096938513138868f0ed27639061f1a5054900f96.

Bounded live all-state PR searches and the newest visible handoffs found no competing result. The local overlap helper covered only 28/8,623 notes and had no remote cache. Nearest work is PR6183's contractive theorem, PR6181's exact spectral conjugacy obstruction and PR6173's code-support invariant; their different quantities are not compared numerically. The completion check found no newer PR than #6190 and confirmed PR6183's unchanged head eaf77790466f34b86ce72080e3ec690e26dfacab.

New lower-bound proof and verifier: apoursh with substantial OpenAI Codex assistance. Nick Petalas retains the preceding question and attaining projection; Matarener retains the D8 identification and omitted lines. Rohan Arun, Chafik Boukhalfa and the retained P104/root/frame/native/helper/compiler/BIT lineage retain their source credit. Apache-2.0.

Open next questions

The package's next questions concern a fully paid side correction and arbitrary complex rank-seven approximants. Start respectively with one residual in an actual scalar word or an exact complex witness/dual certificate. Physical costs and the complex analogue of the real norm inequality are missing. No follow-up search is reserved.