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. The normalised Gram matrix Gₙorm is σ-invariant (Proposition 3. 5). What is numerical. The energy asymmetry ηₒrig = Δ/Eₛtr ∈ 0. 650, 0. 700 for benchmark parameters (κ = 53, ε = 0. 05, N = 100), σ ∈ 0. 1, 0. 95. The spectral bound Bₘax ≈ 0. 0735 at reference parameters; Bₘax 0 for the canonical weight vector; the spectral bound for canonical weights; the analytical cancellation bound; the analytic explanation of Bₘax; the renormalized shell energy. All five are stated precisely as open problems; none is used as a hypothesis.