Admissible Mathematics and Physical Reality: Detectability, Closure, and the Exclusion of Hidden-Defect Theories

This paper develops a general criterion of admissibility for mathematical and physical frameworks, formalized through the notion of Complete Detectable Spacetime Geometry (CDSG). The central problem addressed is the absence of a structural condition distinguishing frameworks that are merely formally coherent from those that are admissible as final descriptions. To resolve this, detectable structures are defined via configuration spaces, probe algebras, and response maps, and hidden defects are introduced as irreducible response-invisible distinctions. A no-go theorem is established showing that hidden defects cannot persist in any admissible framework, together with an exclusion theorem stating that frameworks whose essential ontology requires such defects are fundamentally inadmissible. Hidden defects are further classified into three master classes-spectral, logical, and coordination-and refined through five structural invariants into a unified fifteen-fold manifestation scheme, yielding a common structural description of obstructions across analytic number theory, formal logic, computational complexity, and physical theory. Within this framework, admissibility classes of mathematical systems are defined, and it is shown that formal consistency alone is insufficient for foundational legitimacy. A dimension-selection constraint for spacetime-level realizations is derived, demonstrating that complete admissibility requires a 3 + 1-dimensional structure in the weak structural sense. For physical theories, a criterion of admissibility is established requiring the exclusion of irreducibly hidden ontic sectors and stability under admissible evolution, extension, and composition. The framework extends classical criteria such as consistency and falsifiability by introducing admissibility as a structural condition on the ontology of a theory itself. A first nontrivial realization of this program is provided by the detectability-based treatment of the Riemann Hypothesis, which appears as a special case of the CDSG principle. The results suggest that CDSG may serve as a unifying criterion across mathematics and physics, distinguishing structurally complete frameworks from those that remain effective but ontologically incomplete.