The Memory Engine: Self-Organized Coherence from Internal Feedback

We present a continuous-space realization of the Coupled Memory Graph Process (CMGP), a minimal non-Markovian framework in which coherence emerges through internal feedback. A single Brownian particle evolves on a viscoelastic substrate that records its trajectory as a scalar memory field and exerts local forces via the gradient $\nabla$ of accumulated imprints. This autonomous, closed-loop dynamics generates structured, phase-locked motion without external forcing. The system is governed by coupled integro-differential equations: the memory field evolves as a spatiotemporal convolution of the particle's path, while its velocity responds to the gradient of this evolving field. Simulations reveal a sharp transition from unstructured diffusion to coherent burst-trap cycles, controlled by substrate stiffness and marked by multimodal speed distributions, directional locking, and spectral entrainment. This coherence point aligns across three axes: (i) saturation of memory energy, (ii) peak transfer entropy, and (iii) a bifurcation in transverse stability. We interpret this as the emergence of a \textit{memory engine} -- a self-organizing mechanism converting stored memory into predictive motion -- illustrating that coherence arises not from tuning, but from coupling.