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 ε_AR(Π) ≤ |Q| ≤ ð(Π), where ε_AR > 0 is the lower (AR) bound and ð < ∞ is the upper bound. Both bounds follow from a single axiom—Prohibition of Infinite Subdivision—via the Peano arithmetic relationship between division and multiplication: prohibiting infinite subdivision necessarily prohibits infinite growth. The system is inherently symmetric. The theorem eliminates all true zeros and infinities from measurable physics: no massless particles (m ≥ ε_AR > 0 for photons, gluons, gravitons), no singularities (r ≥ ε_AR > 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.