Memory-driven Topological Defects and Unconventional Long-Range Order

What the math gives to ML

The paper’s transferable mechanism is that delayed self-interaction can create qualitatively new collective behavior: memory feedback suppresses weak-noise fluctuations, supports true long-range order, and produces a critical relaxation law with dynamical exponent \(z=4\). The most direct neural analogue is a recurrent or state-space layer whose hidden state is coupled to an explicit slowly evolving memory state rather than relying only on instantaneous recurrence. A useful falsifiable target is quartic low-frequency relaxation, \(\tau(k)\propto k^{-4}\), together with a measurable stability boundary as the memory gain and delay are increased. Because the supplied excerpt does not specify the microscopic memory kernel, the construction below makes that kernel explicit and treats the paper’s \(z=4\) result as the design signature to test.