Logic of ER III - Physics Underdetermination: Constructive Toy Models for a Dynamic Ontology in Spacetime and Hilbert Space

We challenge the common view that relativity and quantum formalisms compel a static block-universe ontology. Instead, we show that Existential Realism (ER) – can be formulated as an ontological extension of the specific physical models studied here, leaving empirical predictions unchanged so long as the added ER structure remains empirically idle. First, we clarify that the mathematical formalism of physics is neutral: one can impose a preferred foliation or an existence predicate without changing the equations. Second, we explicitly construct ER-compatible models for Minkowski spacetime (special relativity) and a finite-dimensional quantum system. In these Minkowski-spacetime and finite-dimensional quantum toy models, we show that the standard relativistic dynamical laws, Born-rule calculations, and unitary evolution between measurements are preserved, while the ER additions remain extra ontological structure. More precisely, we have constructed models that are empirically equivalent and dynamically identical while differing in their ontological assignments, thereby demonstrating underdetermination in a model-theoretic sense. We conclude that relativity and quantum theory, in the model presented here, do not logically force eternalism, leaving ER as a viable alternative ontology.