The Physics, Information, and Computation of Perennial Learning: Kolmogorov Complexity, Information Distance, and Port-Hamiltonian Thermodynamics

Real-world autonomous agents learn under nonstationarity, safety constraints, and finite energetic budgets. We develop a framework for perennial learning—agents that continuously refine their models while provably controlling the cost of forgetting—by unifying three classical pillars: Kolmogorov complexity, which equates scientific discovery with algorithmic compression; Landauer's principle, which assigns a minimal thermodynamic cost of kBT ln 2 per erased bit to every irreversible model update; and port-Hamiltonian (PH) dynamics, whose (J − R) ∇H decomposition separates zero-cost reversible inference from costly irreversible forgetting by construction. The Maxwell demon analogy is formalized: each learning episode is a Szilard cycle in which information acquisition, belief transport, and memory erasure must balance thermodynamically. The information-distance framework, comprising the normalized information distance (NID) and normalized compression distance (NCD), provides a computable geometry for measuring learning progress and guiding curriculum design. We separate the ideal uncomputable regularizer based on prefix complexity from the practical compressor/MDL (minimum description length) surrogate that appears in optimization and prove a calibration lemma linking the two under a mild uniform-accuracy assumption. Under explicit regularity, compact-sublevel, and non-energy-extracting assumptions, we prove a passivity speed limit for curriculum-induced contractions of the effective feasible set. Under local asymptotic normality, we reprove that Fisher information is a local posterior codelength proxy rather than an exact theorem about algorithmic entropy. A conditional sequential information-budget proposition shows that the per-stage sample requirement scales as \ (O (kₜ/_) \), where \ (kₜ\) is the number of materially changed model coordinates (not the total model complexity \ (kₜ\) ) ; the \ (k³ k\) improvement is conditional on a warm-start assumption and a chosen cold-start baseline. A double-integrator running example with a moving obstacle illustrates the architecture.