The Prime Resonance Ecosystem: A Spectral Framework for the Riemann Hypothesis

We present a spectral framework for the Riemann zeta function based on prime

shift operators. The Hilbert space L2(R+, dx/x), unitary operators Upf(x) = f(px),

and the Euler product operator E = Q p(I − Up)−1 define a lossless resonance network.

A rigged Hilbert space S ⊂ H ⊂ S′ with S = S1/2 1/2 (a nuclear Gelfand– Shilov space) is introduced. The functional equation forces zeros to appear in symmetric pairs. The corresponding log-time resonant state would be ϕ(u) = 2 cosh((σ−1/2)u)eiγu. We prove that cosh(αu) /∈ S′ unless α = 0. This reduces the

Riemann Hypothesis to constructing a two-dimensional extension of E whose kernel

projects continuously onto S′. Two precise gaps are identified. The Hilbert–PÅLolya

conjecture is reformulated but not proved. No claim of proof is made. A research

program is offered.

Keywords: Riemann Hypothesis, Hilbert–PÅLolya conjecture, prime shift operators,

rigged Hilbert spaces, Gelfand–Shilov spaces, spectral theory.