Discernibility, Persistence, and the Origin of Curvature

This preprint develops a necessity-based structural derivation showing which concepts are forced once discernibility is taken as a minimal condition for description. Rather than postulating spacetime, dynamics, probability, or physical law, the paper reconstructs—step by step—the minimal structure required for distinctions to persist under repeated variation. Starting from discernibility as a primitive, the analysis derives persistence, iterability, irreversible loss, entropy as accumulated degradation of distinctions, path dependence, and finally curvature as the obstruction to extending local preservation rules consistently across composed sequences. All notions are introduced only when their absence would render earlier distinctions unstable or incoherent. The main text is purely structural and makes no empirical claims. Familiar physical and informational quantities (Shannon entropy, energy conservation, mass, force, geometric curvature) appear only in an appendix, where their correspondence is established under explicitly stated additional assumptions. These assumptions are not used in the derivation itself. The work is intended as a contribution to foundational structure rather than reinterpretation. It provides a minimal, conceptually transparent framework clarifying the logical order in which entropy, irreversibility, energy, and curvature emerge once persistence of distinctions is required.