From Polynomial Gravity to Fano Spectral Geometry: Exploratory Notes on Arithmetic Selection, Discrete Curvature, and the Spectral Metric of the Fano Plane

Version 3. This version supersedes the previous draft. It incorporates the diagonal-gauge audit of the stored Fano generation-corner matrices Dc1. npy, Dc2. npy, and Dc3. npy. The audit shows that all three stored matrices are diagonally gauge-equivalent to their entrywise moduli. Consequently, for these stored data, the complex phases do not modify the Connes Lipschitz constraints. This corrects the interpretation of Version 2: the previously reported “phase-corrected” Fano distance hierarchy cannot yet be interpreted as a genuine phase effect of the stored spectral data. The reported hierarchy 4·1 + 6·C₂ + 2·S₄ + 5·deg18 + S₂₂ + 3 open = 21 is retained as the output of the complex SDP/Newton–PSLQ pipeline, but now as a computational diagnostic pending audit of matrix generation, scaling conventions, realification, and SDPA input construction. The accompanying reproducibility bundle includes the bilingual paper, the stored Fano generation-corner matrices, the SDP-GMP input/output files, Newton–PSLQ and Sage elimination scripts, the previously reported complex-distance diagnostics, and the phase-cocycle / diagonal-gauge audit scripts. The incidence structure gives the skeleton; the next task is to determine whether the observed arithmetic hierarchy belongs to the spectral metric itself, or to a mismatch between computational conventions.