A Logical-Systemic Analysis of the Zero Axiom and Its Implications for Theoretical Physics and Mathematics

This work proposes a fundamental revision of the mathematical axioms underpinning modern theoretical physics. It argues that three of contemporary science's most persistent paradoxes — the cosmological singularity, the vacuum energy catastrophe, and the probabilistic interpretation of quantum mechanics — are not independent physical problems but correlated symptoms of a single systemic failure: the mathematical reification of the concept of "absence" (zero) as a valid operational entity. The central thesis contends that treating zero as a real number capable of participating in arithmetic operations introduces a topological discontinuity in the number line that propagates through mathematical formalism, generating paradoxes at scales ranging from elementary financial transactions to black hole cosmology. The "Null Operator Axiom" is proposed as a correction principle, redefining operations with physically unrealizable premises (such as division by zero volume) as logically invalid, preserving the system's antecedent state. The work demonstrates how this axiomatic correction simultaneously resolves: (1) singularity formation in General Relativity, replacing them with finite-density objects at the Planck limit; (2) apparent quantum non-locality and superposition, reinterpreted as epistemological artifacts; (3) the Riemann Hypothesis, which emerges as a necessary consequence of the restored symmetry of the numerical system. Additionally, empirical evidence of this logical flaw is presented through the "billion-dollar mistake" in computer science — the null reference in programming languages — demonstrating that the same conceptual anomaly generates measurable systemic failures in modern digital systems. This work calls for the development of a "Mathematics of Reality", an axiomatic formalism that incorporates logical firewalls against physically unrealizable premises, respecting the fundamental distinction between quantity (semantics) and position (syntax) in numerical representation.