RH-Conditional Positivity in a Theta-Kernel Langevin Model

Abstract

A classical Brownian-bridge representation of Riemann's xi function defines an even, rapidly confining probability density and a reversible Langevin model. Our principal physical observable is the transient variance after a quadratic-potential quench; we also analyze a tilted bridge-range response and a sixth-order cumulant gap. For a specified passive harmonic trap with bounded delayed residual feedback, we derive a non-vacuous finite-time variance-increase bound under explicit calibration and sampling assumptions. An elementary global curvature estimate and de Bruijn's strip-contraction theorem give a direct proof of a strict susceptibility bound for every nonnegative quadratic tilt, without assuming the Riemann hypothesis. The inequality itself is a classical GHS consequence, valid more generally for every real quadratic tilt. Reversibility then gives explicit finite-time response and deterministic-control action bounds (not thermodynamic energy). The same curvature estimate gives monotone variance relaxation after an outward quadratic quench, with an exponential deficit bound. At the undeformed density, all alternating zero-field cumulant signs and the displayed low-order cumulant gap are also unconditional; neither is an RH test. The genuinely stronger RH-conditional structure is a positive Stieltjes representation and positive semidefiniteness of divided-difference matrices of every size. At the undeformed density that entire matrix hierarchy is equivalent to RH. We separate these exact statements from finite-sample confidence bounds, numerical discretization evidence, and theoretical physical implementations. The work supplies neither a proof of RH nor a laboratory realization.