Structural Constraints of Cohesive ∞-Toposes on Physical Modeling: A Meta-Theoretical Analysis with Predictive Extension via Topological Completeness

We present a comprehensive analysis of an axiomatic framework proposing that physical theories be formulated within cohesive ∞-toposes. Through systematic examination of seven foundational axioms, we demonstrate that this framework provides rigorous mathematical foundations for gauge theories, quantum field theories, and gravity, while remaining fundamentally distinct from empirical physical theories. Our analysis reveals that the framework functions as a universal test engine: given hypothetical or empirical data, it computes the minimal mathematical structures required and enforces consistency constraints. In this extended version, we introduce an Axiom of Topological Completeness (Axiom 8), which serves as the selection principle previously identified as missing. We demonstrate that particles emerge as data required to patch coherence gaps caused by topological obstructions in spacetime manifolds. Applying this principle, we derive structural predictions for neutrino physics (Majorana nature), fermion generations (Ng = 3), mass hierarchy constraints, and the necessity of a Higgs-like scalar sector. The framework thus evolves from a universal consistency engine into a predictive theory for matter content and structural parameters.