Relational Entropy, Finite Resolution, and Hierarchical Constraints in Informational Dynamics

This is a preprint version of a manuscript submitted to Foundations of Physics.The paper is currently under review and has not yet been peer reviewed. Abstract: We propose a relational interpretation of entropy in which entropy is not a property of physical objects, but a measure of loss of distinguishability in relational structures under finite descriptive resolution. We adopt the working hypothesis that relational structures, possibly of very high informational complexity, need not have a temporal origin and may be ontologically prior to time, matter, and energy. In this framework, entropy-related irreversibility emerges as a necessary con- sequence of unavoidable coarse-graining induced by finite resolution of relational updates. We formalize this idea using projection maps on relational state spaces and show that coarse-graining necessarily reduces Shannon entropy of the described vari- able, while the loss of distinguishability (conditional entropy) necessarily increases or remains constant. We further introduce the notion of hierarchical informational constraints that restrict the space of admissible relational trajectories without mod- ifying local dynamics, thereby providing a structural mechanism for the existence of long-lived low-entropy structures. We argue that the evolution of relational descriptions is generically non-linear due to the presence of mathematical attractors and metastable regimes in relational state space. This leads to plateaus, quasi-stationary structures, and the emergence of ordered patterns such as crystalline, quasi-periodic, and fractal forms. Finally, we discuss the implications of this framework for irreversibility, emergence, and the effectiveness of mathematics in physics.