Weighted force propagation in the de Bruijn–Newman zero dynamics

Rodgers and Tao proved that the de Bruijn–Newman constant is nonnegative. A delicatepart of their argument propagates a time-averaged renormalized inverse-square energyestimate to time zero by repeatedly choosing good boundary gaps. We give an alternativepropagation mechanism. The raw logarithmic force of the zero configuration satisfies anexact heat equation on a time-dependent complete graph with conductances 2/(xi − xj)2.A Cauchy weight in the physical variable obeys a compatible barrier estimate, leading to aglobal weighted-force inequality without hard spatial boundaries or microscopic lower-gapassumptions. A finite Bregman-force identity connects the Rodgers–Tao renormalized energyto the force, while an infinite-line completion gives the required time-zero nearest-gapcoercivity. Together with the integrated energy estimate of Rodgers and Tao, this recoverstheir local-equilibrium input for the final contradiction.