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, not as eigenvalues — distinguishing it from the classical Hilbert–Pólya setting and from the operator programs of Connes, Berry–Keating, and Sierra–Townsend. 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 is established by classical operator theory. Self-adjointness of W₁ = CT·T̃⁺ is proved; its construction does not assume RH, but the normalization constant CT is calibrated via OLS against the zero ordinates, so W₁ is not fully input-free. The Abel Summation Principle (Lemma M3) is proved. The Prime Exponential Sum Bound Mₖ (κ) = O (π (κ) /γₖ) is proved using only the Prime Number Theorem; this estimate is methodologically classical (cf. Turán 1948), with novelty lying in its integration into the energy-asymmetry framework. What is numerical. The eigenvectors of T̃ localize at zero ordinates γₖ with degrees 0. 45–0. 99. The eigenvalue–zero ordinate correlation r₁ = corr (μⱼ, 1/γ₊ (₉) ) = 0. 950 (κ = 53). The energy asymmetry ηₒrig > 0 holds for all tested κ ≤ 1009. The κ-invariant lower bound Δ (κ) ≥ ΔBurst ≈ 3. 11 > 0 gives η_∞ ≥ 0. 51 > 0 unconditionally; numerically, η_∞ ≈ 0. 81. T̃ is NOT a Hilbert–Pólya operator: the HP-correlation r₂ = corr (ωⱼ, γ₊ (₉) ) falls from 0. 50 to 0. 16 for κ ∈ 23, …, 503. What is conditional. Under the Assumption — Weyl equidistribution of prime log-phases γₖ log p mod 2π, a condition on Re (s) = 1, distinct from RH — the normalised cross-term averages vanish. The non-vanishing ζ (1+inγₖ) ≠ 0 required by Weyl's criterion is unconditional (Hadamard 1896). What remains open. The analytic proof of ηₒrig > 0 without the equidistribution assumption; whether a circularity-free spectral function f (T̃) exists with spectrum converging to γₖ; the arithmetic origin of C_η ≈ 0. 39; and whether the unconditional bound rank (T̃) ≤ π (κ) is in fact equality, equivalent to the linear independence of the prime resonance vectors aₚ in Hₙull (numerical evidence: gap ratio ~3. 7×10⁻¹³ at κ=53). All are stated precisely in the paper; none is used as a hypothesis.