Hilbert Substrate Reformulation of the Riemann Hypothesis: A Weighted Frame Inequality for the Prime–Power Lattice

We reformulate the Riemann Hypothesis as a uniform weighted frame inequality for band-limited functions sampled on the prime–power lattice. By introducing a symmetric Hilbert geometry with kernel K(\tau)=(\tfrac14+\tau^2)^{-1} and an explicit ε-regularization of the discrepancy symbol, we isolate the analytic obstruction from arithmetic structure. Phase-locked “ghost” constructions are shown to fail via a rigidity mechanism, while the genuine prime–power lattice leads to a well-posed open problem in weighted sampling theory.