Physics of Difference IV: Emergence of Scalar Accessibility Structure from Relational Distance Organization

We investigate the minimal structural conditions under which effective scalar accessibility organization may emerge within a relational framework based solely on distinguishable states and admissible realizations of comparison. Building upon the comparison accessibility and emergent distance structures developed in the preceding papers of the series, we analyze how local variation of relational accessibility may admit effective scalar representation despite underlying comparison multiplicity. A central feature of the framework is that admissible comparison between distinguishable states is generally non-unique. Multiple admissible realizations of comparison may therefore exist between the same pair of states, inducing microscopic path-dependent accessibility organization. Nevertheless, sufficiently organized large-scale accessibility structure may admit effective reduction through collective accessibility organization. By introducing scalar accessibility representation as an aggregated characterization of local accessibility organization, we show that accessibility gradients naturally induce preferred realizations of comparison and effective directional accessibility organization. This results in effective attractive relational behavior arising from accessibility minimization itself, without introducing prior notions of spacetime geometry, metric manifolds, force laws, field equations, probability, or thermodynamic assumptions. Importantly, the scalar accessibility structure developed here is not interpreted as fundamental. Rather, it emerges as an effectively integrable large-scale sector of a more general comparison accessibility framework involving underlying comparison multiplicity. The present work does not derive gravitation or relativistic dynamics in full generality. Instead, it identifies structural conditions under which scalar-potential-like organization becomes compatible with relational accessibility organization itself within minimal physical description.