The Non-Stationary Landauer Efficiency: Memory Growth, Informational Nostalgia, and the Thermodynamics of Learning Systems

Memory is not a record but an active thermodynamic process that continuously pays for itself. A bacterium holds the memory of a ligand concentration through receptor methylation; a mammalian embryo erases its epigenetic past through massive reprogramming; a neural network in continual-learning mode expends energy updating its weights; biospheres over aeonic windows hold in their genomes a chronicle of an environment that no longer exists. In all these systems memory is maintained against an environment that drifts faster than the pace of its updating, and a growing fraction of the stored bits no longer predicts anything. We call this fraction informational nostalgia (a technical term, not a psychological referent) and extend the Landauer efficiency to the non-stationary regime under a single methodological condition: every bit must be paid for by the system's own free energy (the self-payment postulate). Two results are proved. Lemma 1: holding each bit against thermodynamic erosion requires strictly positive power with an explicit lower bound. Lemma 2: for any finite refresh rate and a slow environmental drift of Ornstein-Uhlenbeck type, the nostalgic fraction asymptotically does not fall below an explicit positive constant - memory staleness is inevitable. Two consequences follow: efficiency vanishes above a critical level of nostalgia, and there exists an optimal rate of complete memory reset, of an order matching the timescale of epigenetic reprogramming in early mammalian embryogenesis and seasonal diapause in Daphnia. The framework binds bet-hedging, active inference, and catastrophic forgetting within a common thermodynamic budget; a pre-registered prediction for E. coli chemotaxis makes it falsifiable on a standard laboratory protocol. The stationary case - zero memory growth and refresh rate - is developed in a companion work (Andriishin 2026, DOI 10.5281/zenodo.20262947) on the Landauer efficiency of self-modeling and is recovered here as a strict limiting case.