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

We introduce the operator T̃ := Φ∘Φ*, the dual of the loop operator T = Φ*∘Φ from Paper 3 of this series. T̃ acts on the finite-dimensional zero space H_null and encodes the prime-mediated coupling between zero ordinates γ_k of the Riemann zeta function.

We prove: (1) the algebraic spectral identity σ(T){0} = σ(T̃){0}, established by classical operator theory and confirmed numerically to machine precision; (2) self-adjointness of W₁ = C_T·T̃⁺; (3) the Abel Summation Principle (Lemma M3); (4) the Prime Exponential Sum Bound M_k(κ) = O(π(κ)/γ_k), using only the Prime Number Theorem (no Riemann Hypothesis).

We establish numerically: T̃ is NOT a Hilbert–Pólya operator (the correlation r₂ falls from 0.50 to 0.16 as κ grows); the energy asymmetry η_orig > 0 for all tested κ ≤ 1009; and the κ-invariant lower bound Δ(κ) ≥ Δ_Burst ≈ 3.11 > 0, which gives η_∞ ≥ 0.51 > 0 unconditionally. Numerically, η_∞ ≈ 0.81.

Under Assumption A — two conditions on Re(s)=1, distinct from the Riemann Hypothesis which concerns Re(s)=1/2: (a) ζ(1+inγ_k) ≠ 0 for all n ≥ 1, and (b) joint equidistribution of prime log-phases — we prove conditionally that the normalised cross-term averages vanish.

No Riemann Hypothesis, GUE conjecture, Montgomery pair correlation conjecture, or Hilbert–Pólya postulate is used anywhere.

Part of a series on the curvature of the Riemann zeta function. Verification code: https://github.com/utehrani/analysislab-nt