Abstract
We introduce a Gaussian spectral-packet detector for horizontal displacement of the non-trivial zeros of the Riemann zeta function and connect it quantitatively to the prime side of the explicit formula. For distinct zeros \(\rho=\beta_\rho+i\gamma_\rho\) with multiplicity \(m_\rho\), let \(S_\lambda(t)=\sum_\rho m_\rho e^{-\lambda(z_\rho-t)^2},\qquad z_\rho=\gamma_\rho-i(\beta_\rho-\tfrac12),\) and for a compact interval \(I\), \(Q_I=\max\left\{0,\sup_\rho\left[(\beta_\rho-\tfrac12)^2-\mathrm{dist}(\gamma_\rho,I)^2\right]\right\}\). We prove that the logarithmic exponential rate of \(\sup_{t\in I}|S_\lambda(t)|\) is exactly \(Q_I\): a limsup for arbitrary compact \(I\), and a genuine limit when \(I\) has positive length. The interval lower bound uses an \(O(1/\lambda)\) micro-shift and a finite-dimensional norm-separation argument, preventing dominant-cluster phase cancellation from destroying the exponential rate. This yields an exact RH-equivalent subexponential-growth criterion. Using an even shifted Gaussian and the Guinand–Weil formula in a fixed \(2\pi\) Fourier convention, we derive \(S_\lambda=B_\lambda+G_\lambda-P_\lambda\) and prove an explicit uniform prime-tail estimate. In particular, a finite arithmetic reconstruction preserves the same exponential rate when \(\log N(\lambda)\ge(2+\varepsilon)\lambda\). We then examine the pre-asymptotic regime computationally: finite-prime peak reconstruction, blind synthetic off-critical-pair recovery, multi-pair interference, and nuisance-projected waveform identifiability. These computations test finite-scale identifiability only; they do not prove RH or identify an actual off-critical zero.