A Finite-Cutoff Hilbert-Space Model for Prime–Zero Energy Structure

This paper is the third in a series developing an operator-theoretic framework for studying the distribution of the non-trivial zeros of the Riemann zeta function. Given a prime cutoff κ and truncated zero ordinates γ₁, …, γN, we define two finite-dimensional real Hilbert spaces Hₛtr (over primes) and Hₙull (over zeros), connected by a linear map Φ built from Gaussian-damped prime resonance vectors. The self-adjoint loop operator T = Φ*∘Φ is the Gram matrix of these vectors — explicitly constructed, not postulated. No hypothetical input is used: the Riemann Hypothesis, the GUE conjecture, and the Hilbert–Pólya postulate are explicitly avoided throughout. What is proved. The algebraic variance identity Δ = Eₛtr − Eₛpec is established by direct expansion. The loop operator T = Φ*∘Φ is proved self-adjoint and positive semi-definite, with σ-invariant normalized Gram structure. What is numerical. The energy asymmetry ηₒrig = Δ/Eₛtr ∈ 0. 598, 0. 704 for benchmark parameters (κ = 53, ε = 0. 05, N = 100), strictly monotone decreasing in σ. The uniform spectral bound Bₘax ≤ 0. 106 0 for the canonical weight vector; the spectral bound for canonical weights; the analytical cancellation mechanism underlying Bₘax < 1. All are stated precisely in the paper; none is used as a hypothesis.