Into the City: A Deterministic Operator–Trace and SSB Framework for the Generalized Riemann Hypothesis

We extend the fully deterministic operator–trace framework from The Path of Light (Paltoo
et al., 2025) to the Generalized Riemann Hypothesis (GRH).
For any Dirichlet character χ modulo q, we construct a canonical self-adjoint Hamiltonian
Hcan,χ and a trace functional Tcan,χ that deterministically encode all arithmetic information of
the primes. Standing/Sitting Band (SSB) intervals propagate primes without material clumping
across residue classes, ensuring exact correspondence between the spectrum of Hcan,χ and
nontrivial zeros of L(s, χ) on the critical line Re(s) = 1/2.
All logical steps are fully deterministic, trace-class convergence is established, and diagonal
dominance enforces spectrum on the critical line. Optional Gaussian test functions illustrate
positivity but are not essential, due to structural rigidity of the operator-trace construction.