The Non-Zero Measurability Theorem: Quantization of Observable Existence in Srinivas Bounded Mathematics

We establish a fundamental theorem within Srinivas Bounded Mathematics (SBM): all measurable quantities are doubly bounded. For any measurable quantity Q in context Π, we prove εAR (Π) ≤ |Q| ≤ ð (Π), where εAR > 0 is the lower (AR) bound and ð 0 for photons, gluons, gravitons), no singularities (r ≥ εAR > 0), no UV divergences (momentum bounded), and the Yang–Mills mass gap Δ > 0 follows as a one-line corollary. We establish the mass gap as independent of subdivision axioms—analogous to the Continuum Hypothesis being independent of ZFC. The Physical Incompleteness Theorem proves that empirical input is logically necessary for any formal system describing reality. We further derive the Bekenstein bound, the Principle of Least Action, and modified Wightman axioms as natural consequences of the doubly bounded framework.