Coercivity structure of positive-type memory: exact gaps, critical horizons, and singular limits

What the math gives to ML

The paper gives a precise spectral diagnosis of why positive memory is dissipative but cannot by itself provide uniform instantaneous coercivity: its frequency symbol decays to zero at high frequencies. This is directly useful for neural optimizers and recurrent or state-space modules that replace instantaneous updates with exponentially weighted history, because it predicts sluggish response to rapidly changing signals. The practical transfer is a coercivity-aware memory update that combines a completely monotone memory branch with an explicit instantaneous branch, while choosing the memory horizon from an empirical finite-horizon coercivity profile rather than assuming that longer memory always improves stability.