Klein Quartics as Eigenschemes of the Fano Adjacency Tensor

For the Cooper-Dutle normalized adjacency tensor of the Fano plane, we explicitly identify the entire projective eigenscheme at each root of lambda^2+lambda+2 with the bicanonical Klein quartic. A Fourier transformation turns the seven incidence equations into an adjugate equation and a quadric. A displayed integer identity excludes all off-hyperplane components, while a local adjugate argument removes infinitesimal thickening. The proof gives an explicit isomorphism over Q(zeta_7), including scheme structure, and a portable exact rational checker. The earlier anonymous preprint DOI 10.5281/zenodo.22850084 already reported degree-eight genus-three eigencurves and a full computational spectrum; those assertions are credited. The contribution here is their Klein-quartic identification and direct certificate-based proof. The full characteristic polynomial and resultant multiplicities are not recomputed, and AMR-030-0030 is not counted as a newly closed whole-spectrum question. AI-assisted, self-audited preprint; no independent review, formal verification or absolute-priority certification is claimed.