Physics of Difference VIII: Necessity of Additive Representation under Reduction Obstruction

The present work investigates the representational consequences of realization-dependent accessibility organization under globally incomplete endpoint reduction. The preceding papers of the Physics of Difference series established that admissible realizations connecting identical endpoint accessibility relations may nevertheless remain compositionally distinguishable within realization organization itself. In particular, realization-dependent distinctions may remain globally persistent whenever endpoint reduction fails to extend to a globally compatible compositional reduction structure.Under such conditions, endpoint accessibility relations no longer provide a structurally complete representation of realization organization. The present analysis therefore investigates what representational organization becomes necessary once realization multiplicity remains irreducible under endpoint accessibility reduction.The analysis shows that any representation framework preserving realization organization faithfully must simultaneously support realization distinguishability, compositional compatibility, multiplicity-preserving aggregation, decomposability, and recombinational stability. These constraints naturally motivate additive decomposable representation.Importantly, additive representation is not introduced as an independent quantum-theoretic postulate. Rather, it emerges structurally as a multiplicity-preserving response to reduction obstruction itself.The resulting framework admits several quantum-like organizational features. In particular, realization composition induces residual compositional organization, manifest accessibility restriction generates projection-like representational structure, and compatibility-preserving aggregation admits naturally emerging quadratic organizational forms.At the same time, the framework remains intentionally weaker than conventional quantum mechanics. The present analysis does not assume or derive Hilbert-space structure, Schrödinger dynamics, unitary evolution, or Born-rule probability interpretation.The framework should therefore be interpreted not as a reconstruction of quantum mechanics itself, but as the identification of minimal representational conditions under which quantum-like organizational structure begins to emerge from irreducible realization multiplicity under compositional accessibility organization.