Physics of Difference I: Minimal Structure of Physical Description

Abstract We present a minimal reconstruction of physical description based on a single assumption: the existence of distinguishable states. Without presupposing space, time, or dynamical laws, we formulate physical description in terms of states and their updates, and examine the structural requirements for consistency. Within this minimal framework, we show that state updates are generically non-injective, and that their composition forms a semigroup. From these properties, a preorder structure naturally arises, providing a notion of temporal ordering. At the same time, entropy emerges as a measure of the compression of distinguishability, corresponding to an intrinsic and irreversible loss of information. A key result is that this loss of information follows directly from the existence of difference itself: distinct states cannot, in general, be mapped in a way that preserves all distinguishing features. Irreversibility is therefore not an artifact of coarse-graining or observational limitations, but a structural consequence of the requirements for a consistent physical description. These results suggest that time and entropy are not independent physical inputs but arise jointly from the same underlying structure. This work provides a minimal foundation for further extensions toward geometric, field-theoretic, and quantum descriptions.