The Nostalgic Undertow: The Inevitability of Memory Obsolescence in Non-Stationary Landauer Regimes

A system that learns about a changing world must store what it has learned, and — by Landauer's principle — storing memory costs energy. But the world keeps changing, so a stored memory gradually stops predicting anything useful while it still costs energy to keep. We call this growing fraction of useless but still-paid-for memory informational nostalgia (a technical term, not a psychological one), and the irreducible rate at which it drags the efficiency of memory toward zero the undertow. The central result is that the undertow is inevitable for slowly drifting, mixing environments, no matter which of three escape routes a system attempts. Tracking the world more accurately does not help: even the best possible filter runs into a hard limit on how closely it can follow a moving target (Theorem 1). Refreshing memory faster, discarding stale entries, always leaves an irreducible residue of useless memory whose size is the same for every way the world can drift (Theorem 2). And growing the memory does not escape either: no polynomial growth rate is fast enough, and the only rate that would work, exponential growth, costs so much energy that efficiency collapses anyway (Theorem 3). The deeper finding is that these three escape routes are not independent — they fail for the same two underlying reasons — so memory obsolescence is a single structural impossibility, not three separate facts: there is no clever architecture that gets around it. Each theorem comes with a falsifiable prediction, all confirmed by reproducible, deterministic Markov-chain simulations; the results are proved for mixing, slowly drifting environments in a linear-Gaussian (adiabatic) approximation. This is the third paper in a series on the Landauer efficiency of memory; the stationary case is developed in two companion works and recovered here as a limiting case. This open-access deposit contains the manuscript and supplementary material with the full proofs (English submission version and Russian primary version), the bibliography, the figures, the LaTeX sources, and the self-contained, deterministic simulation code.