When Particle Counts Cannot Respond: A Finite-Time Law for Mode-Mass Locking in Stein Flows

A finite set of deterministic particles can cover every mode of a target and still assign the wrong mass to each mode. We study when a change in target mixture weight can alter an integer particle count within a fixed time. For separated Gaussian mixtures, we bound the response of finite-particle Stein variational gradient descent (SVGD) by the time-integrated sensitivity of the local score to mixture log-odds. This conditional, one-sided locking certificate motivates an intermediate smoothing window: too little Gaussian smoothing leaves responsibilities saturated, while too much suppresses the score. A threshold calibrated on half of a 378-cell phase grid predicts material count response on 179 held-out cells with 89.6% balanced accuracy. On fresh seeds, dwelling in the predicted window lowers one-dimensional mode-count error by 53.4% relative to the tested standard, force-annealed, wide-kernel, and importance-weighted $\beta$-SVGD variants. The same schedule gives a smaller gain on a symmetric two-dimensional ring and none on an irregular four-mode layout. The result is a diagnostic law for finite-time mode-mass locking and a controlled account of which annealing operations can cross it.