A Dual Operator for Prime–Zero Coupling and a Conditional Proof of Energy Asymmetry

This paper is the fourth in a series developing an operator-theoretic framework for studying the distribution of the non-trivial zeros of the Riemann zeta function. It introduces the Tehrani operator T̃ := Φ∘Φ*, the dual of the loop operator T = Φ*∘Φ from Paper 3, acting on a finite-dimensional zero space. The operator encodes prime-mediated coupling between zero ordinates as eigenvector localization. No hypothetical input is used: the Riemann Hypothesis, the GUE conjecture, the Montgomery pair correlation conjecture, and the Hilbert–Pólya postulate are explicitly avoided throughout. What is proved. The algebraic spectral identity σ(T){0} = σ(T̃){0} by classical operator theory. Self-adjointness of W₁ = C_T·T̃⁺; its normalization constant C_T is calibrated via OLS against the zero ordinates, so W₁ is not fully input-free. The Abel Summation Principle (Lemma M3). The Prime Exponential Endpoint Estimate M_k(κ) ≤ (1/√(1+γ_k²) + o(1))·π(κ) using only the Prime Number Theorem (cf. Turán's program). What is numerical. Eigenvectors of T̃ localize at zero ordinates with degrees 0.45–0.99. Correlation r₁ = corr(μ_j, 1/γₖ₍ⱼ₎) = 0.950 (κ=53). Energy asymmetry η_ren > 0 on the tested grid κ ∈ {23,53,101,199,503,1009}; values ≈ 0.81. T̃ is NOT a Hilbert–Pólya operator: r₂ = corr(ω_j, γₖ₍ⱼ₎) falls from 0.50 to 0.16. The κ-invariant burst term Δ_Burst = 3.105 for κ ≥ 7. rank(T̃) ≤ π(κ) unconditionally; gap ratio μ₁₇/μ₁₆ ≈ 3.7×10⁻¹³ at κ=53 is consistent with rank equality. What is conditional. Under Assumption (E_rem) — remainder control |Δ_Cross + Δ_Stream| ≤ ρ·Δ_Burst for some ρ < 1, numerically ρ_max = 0.583 on the tested grid — η_ren(κ) ≥ (1−ρ)·Δ_Burst/E_str(κ) > 0 is proved by triangle inequality. What remains open. The analytic proof of η_ren > 0 without Assumption (E_rem); whether a circularity-free spectral function f(T̃) exists with spectrum converging to {γ_k}; the arithmetic origin of C_η ≈ 0.39; and whether rank(T̃) = π(κ) (linear independence of {a_p} in H_null).