Reclassifying Zero: Boundary, Interior, and Dual Roles Across Mathematics, Physics, and Computation

This paper propose a formal framework for understanding zero not as possessing a fixed ontological status, but as a structural element whose mathematical character as either an interior point or boundary depends on the algebraic, topological, or physical context in which it appears. Rather than merely cataloging examples, This paper develop a rigorous taxonomy of zero’s contextual roles based on structure-preserving properties of operations and mappings. This classification unifies disparate phenomena across mathematics and physics from algebraic singularities and topological discontinuities to thermodynamic phase transitions and quantum field vacua and provides a systematic diagnostic for predicting qualitative changes in dynamical systems. This paper extend the framework to machine learning, where zero’s boundary character manifests in gradient pathologies, sparsity regularization, and neural network stability. Our analysis reveals that many ML pathologies stem from unrecognized boundary behavior at zero, suggesting design principles for more robust architectures.