This paper argues that physical singularities are incompatible with minimal consistency requirements of physical theory, including operational time, unitary quantum evolution, bounded field dynamics, and ultraviolet closure. Singularities are reinterpreted not as physical entities or events, but as diagnostic indicators of theoretical incompleteness. The argument is framework-independent and does not rely on speculative assumptions about Planck-scale physics. Rupture–Containment Cosmology is referenced only as an existence proof that singularity-free completion is achievable, without being required to follow the argument.