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

We introduce the operator $\tilde{T} := \Phi\circ\Phi^*$, the dual of the loop operator $T = \Phi^*\circ\Phi$ from Paper 3 of this series. $\tilde{T}$ acts on the finite-dimensional zero space $H_{\mathrm{null}}$ and encodes the prime-mediated coupling between zero ordinates $\gamma_k$ of the Riemann zeta function.

We prove: (1) the algebraic spectral identity $\sigma(T)\setminus\{0\} = \sigma(\tilde{T})\setminus\{0\}$; (2) self-adjointness of $W_1 = C_T\cdot\tilde{T}^+$; (3) the Abel Summation Principle (Lemma M3); (4) the Prime Exponential Sum Bound $M_k(\kappa) = O(\pi(\kappa)/\gamma_k)$, using only the Prime Number Theorem.

We establish numerically: $\tilde{T}$ is NOT a Hilbert–Pólya operator; the energy asymmetry $\eta_{\mathrm{orig}} > 0$ for all tested $\kappa \leq 1009$; and the $\kappa$-invariant lower bound $\Delta(\kappa) \geq \Delta_{\mathrm{Burst}} \approx 3.11 > 0$, giving $\eta_\infty \geq 0.51 > 0$ unconditionally. Numerically, $\eta_\infty \approx 0.81$.

Under Assumption A — a single genuinely open condition on $\mathrm{Re}(s)=1$: the equidistribution of prime log-phases $\{\gamma_k \log p \bmod 2\pi\}$ via Weyl's criterion (Part (b)) — we prove conditionally that the normalised cross-term averages vanish. Part (a) of the original Assumption A, namely $\zeta(1+in\gamma_k)\neq 0$, is settled unconditionally by Hadamard (1896) and is no longer an open condition.

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