Retained hidden excess generates memory in price-limited markets

What the math gives to ML

The paper provides a constructive mechanism by which hard clipping creates memory: the clipped-off latent excess is stored and partially injected into the next time step, even when the external shocks are independent. Its transferable asset is a bounded recurrent unit with an explicit latent overshoot state, rather than a memoryless saturating activation. This can be implemented as a recurrent or state-space neural layer whose output is constrained while retained saturation excess drives future hidden states. The mechanism makes sharp predictions: post-saturation responses have the same sign, scale linearly with the clipping width, repeated same-side saturation approaches a nonzero probability, and opposite-side saturation is power-law suppressed.