Abstract: We present a model-theoretic framework that derives the mathematical structures of physics from operational coherence constraints. Uniquely, we employ a “nominalist air-gap,” extracting these structures from a first-order language containing no mathematical objects in its ontology. Our construction has three main components: (1) We show how any minimal coherence model M |= T_min generates a canonical effect algebra E(M ) via a constructive free completion. (2) We demonstrate that the apparent conflict between Quantum Mechanics and General Relativity can be expressed as a difference in the scope of the Riesz Decomposition Property (RDP). Global RDP (T_rich) forces the algebra to embed into an ordered real vector space, while Local RDP (T_grav) gives rise to a presheaf of local algebras over a derived context topology. (3) We analyze a Path Independence property: global RDP is naturally associated with path-independent transport in context space, whereas non-trivial holonomy obstructs any such global linear structure. Motivated by this obstruction, we formulate a Logical Preservation Principle according to which physical evolution must restrict to locally coherent sectors when global RDP would lead to inconsistency, and we interpret this restriction as objective collapse.
An observable on a quantum structure is any
σ-homomorphism of quantum structures from the Borel
σ-algebra into the quantum structure. We show that our partial information on an observable known only for all intervals of the form
(−∞,t) is sufficient to determine uniquely the whole observable defined on quantum structures like
σ-MV-algebras,
σ-effect algebras, Boolean
σ-algebras, monotone
σ-complete effect algebras with the Riesz Decomposition Property, the effect algebra of effect operators of a Hilbert space, and a system of functions, and an effect-tribe.