Physical Realization Relative to V₀: A Common-Reference Framework for Identity, Transformability, and Derived World Structure

Abstract

We investigate a foundational formal architecture in which physical realization is represented as a relation to a single common nonphysical referent V₀, rather than being taken as an intrinsically primitive condition of an entity. The framework postulates R(X) iff X R₀ V₀ and assigns the realization relation a character ρ_X = (κ_X, λ_X), where κ_X is an identity-bearing invariant component and λ_X is a transformable realization component. Physical specification is placed downstream through an open physicalization relation Φ, while spatiality, interaction, and physical-world connectedness are treated as distinct subsequent structures. Physical worlds are defined as connected components of a physical-link graph rather than as primitive spatial containers. Within the stated postulates, the construction provides a sufficient identity criterion based on realization continuity and κ-invariance and separates physical realization from spatial realization and physical connectedness. Explicit finite witnesses and countermodels show that neither a universal spatial structure nor a universal metric time is required for the formal core to be instantiated. To test the specific role of the common-reference hypothesis, we construct an intrinsic competitor T_Q, in which Q_X = (κ_X, λ_X) is assigned directly to each realized entity. Most downstream structure can be reproduced in T_Q. However, any reduction required to preserve the universal common-reference criterion, factorization of realization character, and continuity through the same relational architecture must introduce a common-reference functional equivalent of (V₀, R₀). This is a conditional structural non-eliminability result rather than a demonstration that V₀ is physically necessary. The mathematical form of R₀, the physicalization interface Φ, realization dynamics, and an empirical discriminator between the common-reference and intrinsic architectures remain open. The framework therefore reaches structural closure within its stated postulates while leaving dynamical, physical, and empirical closure