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 = CT·T̃⁺; its normalization constant CT 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 Mₖ (κ) ≤ (1/√ (1+γₖ²) + 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 (μⱼ, 1/γ₊ (₉) ) = 0. 950 (κ=53). Energy asymmetry ηᵣen > 0 on the tested grid κ ∈ 23, 53, 101, 199, 503, 1009; values ≈ 0. 81. T̃ is NOT a Hilbert–Pólya operator: r₂ = corr (ωⱼ, γ₊ (₉) ) falls from 0. 50 to 0. 16. The κ-invariant burst term ΔBurst = 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 (Eᵣem) — remainder control |ΔCross + ΔStream| ≤ ρ·ΔBurst for some ρ 0 is proved by triangle inequality. What remains open. The analytic proof of ηᵣen > 0 without Assumption (Eᵣem) ; whether a circularity-free spectral function f (T̃) exists with spectrum converging to γₖ; the arithmetic origin of C_η ≈ 0. 39; and whether rank (T̃) = π (κ) (linear independence of aₚ in Hₙull).