Physics of Difference III: Emergence of Distance from Comparison Accessibility

We investigate the minimal structural conditions under which quantitative relational organization admits a distance-like representation within physical description. Starting solely from distinguishable states and admissible realizations of comparison, we analyze how quantitative accessibility structure may emerge without introducing prior notions of geometry, metric structure, spacetime, continuity, probability, or dynamics.Within the present framework, comparison between states is realized through finite compositional paths of admissible transformations. By assigning nonnegative structural cost to admissible comparison realizations, we show that minimal comparison cost naturally induces a generalized distance structure on the state space. This structure arises directly from compositional accessibility organization and does not require metric axioms or geometric embedding.The resulting framework admits nonnegative, triangle-type, asymmetric, and path-dependent relational organization compatible with generalized distance representation. We further introduce accessibility weights derived from emergent comparison distance and show that their multiplicative structure follows naturally from additive path composition. These weights admit an effective coupling interpretation while remaining secondary to the underlying comparison organization itself.Importantly, the present framework is formulated initially in discrete form. Continuous geometric organization is not assumed a priori, but is instead interpreted as a possible large-scale representation of sufficiently organized relational accessibility structure.The present work does not introduce fundamentally new mathematical constructions. Rather, its contribution lies in the conceptual reinterpretation of distance, accessibility weight, and effective coupling structure as emergent representations of admissible comparison organization within a minimal relational framework of physical description.