An Energy Criterion for RH: Heat-Flow, Thermal Sieve, and Mesoscopic Reduction

We give an energy formulation of RH on the line t = log x and introduce aheat-flow yielding exact variational monotonicity for mesoscopic times τ ∈ [0, αT^2].This reduces RH to a single mesoscopic bound (normalized by local zero density)at τ = αT^2. We prove a thermal large sieve (Gaussian window, explicit constantK0 < 1.001) that controls the real-frequency spectrum. We conclude that thenormalized mesoscopic bound is equivalent to RH: under RH the normalized energyis uniformly O(1), while any zero with ℜρ > 1/2 forces divergence as t0 varies.