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

We present a structural operator-theoretic framework for analyzing the spectral rigidity of the Riemann zeta function ζ(s), centered on the Strengthened Trace Equivalence Conjecture (TEC) and a geometric confinement mechanism for its non-trivial zeros. The framework embeds the local arithmetic symmetries of ζ(s) into the Szegedy-Lov\'asz graphon limit of the projective plane $PG(2,q)$ over finite fields Fq. 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 λ = √q. The core of the work is the formulation of the Strengthened Trace Equivalence Conjecture, establishing a structural isomorphism between the continuous logarithmic phase space of ζ(s) and the cylindrical state space of the graphon limit. We construct a proto-unitary isomorphism U₀ on a dense subspace and characterize the scale-absorption functional map Φ via Sobolev trace bounds. This construction yields the operator identity U Φ(D) U⁻¹ = B A, where D is the Hilbert-P\'olya operator, A is the radial growth operator, and B is a bounded operator encoding the projective geometry. Within this framework, the amplitude of continuous spectral modes is shown to be rigidly pinned to the radial geometry R(x) = √x, imposing the stationary condition Re(s) = 1/2 along the critical line. The equivalence between TEC and Weil's explicit formula is developed through an operator-theoretic derivation addressing operator-norm convergence, M\"obius inversion of primitive cycle densities, global Mellin-Fourier construction of the unitary U, and Sobolev multiplier bounds for Φ. A comprehensive computational validation suite is provided, comprising five independent numerical tests that are strictly falsifiable: radial drift test, cycle-prime correspondence, operator norm convergence, TEC-Weil trace identity, and dynamic ETA-sweep. All tests pass within machine precision. A cylinder stress test evaluates the analytical predictions against four distinct perturbations of the cylindrical framework, assessing the structural stability of the critical line. An algebraic uniqueness theorem based on the characteristic polynomial of the Fano plane further identifies the projective structure as a unique geometric realization of the TEC-Weil framework. The accompanying Python script TEC_WEIL_complete_validation_suite.py implements all validation tests and generates diagnostic figures, ensuring full reproducibility. This work formulates a complete operator-theoretic model for ζ(s), framing spectral confinement as a structural property of metric preservation in the limit space of dense graph refinements over finite fields.