Deterministic Arithmetic–Spectral Correspondence and the Necessity of the Critical Line

We construct a deterministic correspondence between prime-generated arithmeticdata and the spectrum of a canonical self-adjoint Hamiltonian acting on L2(R), builtfrom logarithmic translation operators. The correspondence is forward-causal, rigidityenforcing,and independent of analytic continuation, functional equations, or probabilisticmodels.Within this framework, Dirichlet generating functions arise as trace functionals undera restricted class of admissible test functions preserving prime support, logarithmicembedding, and canonical orientation. Using a deterministic rigidity inequality derivedfrom weighted prime cancellation, we show that the existence of such an arithmetic–spectral correspondence forces all admissible zeros to lie on the critical line ℜ(s) = 12 .Consequently, the Riemann Hypothesis is not an optional analytic refinement buta necessary structural condition: if the Riemann zeta function is to arise from deterministicprime causality through a self-adjoint spectral mechanism, then its nontrivialzeros must lie on the critical line.