Abstract
An agent that operates sequentially must incorporate new experience without forgetting old experience, under a fixed memory budget. We propose a framework in which memory is not a parameter vector but a stochastic process: a Bridge Diffusion on a replay interval [0,1], whose terminal marginal encodes the present and whose intermediate marginals encode the past. New experience is incorporated via a three-step \emph{Compress--Add--Smooth} (CAS) recursion. We test the framework on the class of models with marginal probability densities modeled via Gaussian mixtures of fixed number of components~K in d dimensions; temporal complexity is controlled by a fixed number~L of piecewise-linear protocol segments whose nodes store Gaussian-mixture states. The entire recursion costs O(LKd^2) flops per day -- no backpropagation, no stored data, no neural networks -- making it viable for controller-light hardware.
Forgetting in this framework arises not from parameter interference but from lossy temporal compression: the re-approximation of a finer protocol by a coarser one under a fixed segment budget. We find that the retention half-life scales linearly as a_{1/2}\approx c\,L with a constant c>1 that depends on the dynamics but not on the mixture complexity~K, the dimension~d, or the geometry of the target family. The constant~c admits an information-theoretic interpretation analogous to the Shannon channel capacity. The stochastic process underlying the bridge provides temporally coherent ``movie'' replay -- compressed narratives of the agent's history, demonstrated visually on an MNIST latent-space illustration. The framework provides a fully analytical ``Ising model'' of continual learning in which the mechanism, rate, and form of forgetting can be studied with mathematical precision.
