Spectral Rigidity and Sobolev Norm Bounds for the Riemann Zeta Function via Fano Graphon Refinements

We propose a geometric confinement mechanism for the non-trivial zeros of the Riemann zeta function $\mathbf{\zeta(s)}$ by embedding its local arithmetic symmetries into the Szegedy-Lovasz graphon limit of the projective plane $PG(2,q)$ over finite fields $\mathbb{F}_q$. By explicitly calculating the exact singular spectrum of the iterated tensor refinements of the incidence matrix over $PG(2,q)$, we demonstrate a strict separation between the trivial Perron-Frobenius eigenvalue and the universal non-trivial spectral radius $\lambda = \sqrt{q}$. Formulating a Strengthened Trace Equivalence Conjecture between the continuous logarithmic phase space of $\mathbf{\zeta(s)}$ and the cylindrical state space of the graphon limit, we construct a proto-unitary isomorphism $U_0$ on a dense subspace and characterize the scale-absorption functional map $\Phi$ via Sobolev trace bounds. We show that within this framework, the amplitude of continuous spectral modes is rigidly pinned to the radial geometry $R(x) = \sqrt{x}$, yielding the stationary spectral condition $\mathrm{Re}(s) = 1/2$ along the critical line. A comprehensive cylinder stress test validates the analytical predictions for all perturbations, and a uniqueness theorem based on the characteristic polynomial of the Fano plane demonstrates that the projective structure is the unique geometric realization of the TEC-Weil framework.