Object Mathematics (o-math) H2: A Dynamic Semantic Framework for Mathematical Theories

This work introduces o-math, a formal semantic framework for treating mathematical theories as dynamic objects rather than fixed axiomatic systems.The framework is based on a structured semantic state, explicitly constrained admissible extensions, and a strict separation between syntactic rewriting and semantic state evolution.Semantic notions such as matching, validity, and evaluation admissibility are treated uniformly as predicates in the internal logic of the semantic layer. All results are stated relative to explicit structural contracts and admissibility conditions. Invariance, conservativity, and non-interference properties are therefore established as conditional guarantees rather than absolute metatheorems.Categorical and topos-theoretic constructions are used solely as semantic tools, with no additional axioms beyond those required for the stated results. The framework is intended as a reference semantic interface for modular, extensible, and evolving mathematical theories, making explicit which properties depend on structural assumptions and which arise from optional choices of rewriting, evaluation, or truth layers.