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,…,γ_N, we define two finite-dimensional real Hilbert spaces H_str (over primes) and H_null (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 = E_str − E_spec is established by direct expansion. The loop operator T = Φ*∘Φ is proved self-adjoint and positive semi-definite. The normalised Gram matrix G_norm is σ-invariant (Proposition 3.5). What is numerical. The energy asymmetry η_orig = Δ/E_str ∈ [0.650, 0.700] for benchmark parameters (κ = 53, ε = 0.05, N = 100), σ ∈ [0.1, 0.95]. The spectral bound B_max ≈ 0.0735 at reference parameters; B_max < 0.382 across the tested grid. What remains open. The analytic proof of η_orig > 0 for the canonical weight vector; the spectral bound for canonical weights; the analytical cancellation bound; the analytic explanation of B_max; the renormalized shell energy. All five are stated precisely as open problems; none is used as a hypothesis.