Entropy Without Observers: A Relational Reformulation Bridging Thermodynamics, Information Theory, and the Free Energy Principle

The concept of entropy exists in two seemingly distinct forms: thermodynamic entropy(Boltzmann-Gibbs) and information-theoretic entropy (Shannon). Despite formal similar-ities, their unification remains incomplete, obscured by epistemological concepts such as“observer,” “information,” and “erasure.” This paper proposes a relational reformulationthat strips away these observer-dependent labels, revealing both entropies as descriptions ofthe same underlying structure: geometric patterns in state space. We reinterpret Landauer’sprinciple as a consequence of phase space volume conservation under degeneracy-inducingmappings, requiring no reference to “knowledge” or “information.” Building on this founda-tion, we outline a pathway to connect this relational framework with Friston’s Free EnergyPrinciple, potentially grounding the FEP in physics rather than epistemology. This re-formulation has implications for theories of cognition, particularly for understanding bothbiological neural systems and large language models.