The Quantization Threshold: Spectral Explosions and Non-Renormalizability as Mathematical Proofs for Bidirectional Constraint Closure

Abstract The historical failure to mathematically quantize continuous, nonlinear, and gravitational systems is widely interpreted as a limitation of current field theories. This paper argues that these mathematical breakdowns—specifically spectral explosions, quantum anomalies, and non-renormalizability—are not defects, but rigorous mathematical proofs confirming that constraint is the foundational ontological primitive. By formally defining the Bidirectional Constraint Closure (BCC) and Constraint-Relaxation Energy Model (CREM) frameworks, we demonstrate that attempting to quantize an unquantizable system is structurally equivalent to violating invariant topological boundaries. To ensure rigorous external validation, this paper integrates standard literature from quantum gravity and operator algebra, establishing a mathematically grounded, constraint-first ontology that reads independently of prior internal archival literature.