Semantic Clarification and Symmetric Correction of Exponent Labels — On 0 as an Exponent Converter Rather Than an Exponent Value

This paper identifies a foundational calibration gap in the conventional exponent labeling system: the label 0 simultaneously serves as the multiplicative identity baseline (x⁰=1) and the additive identity value (0+x=x), creating a semantic overload that manifests as singularities in zeroth-root operations, confusion between positive and negative zeroth roots, and the existential contradiction of the 0-dimensional point. We propose a symmetric correction: 0 is reclassified as an exponent converter rather than an exponent value, with +1 and −1 as the positive and negative unit exponents (both equal to x⁰=1 in value but opposite in direction). The correction is proven fully compatible with all existing computational results while resolving six boundary problems near the origin. We further derive a coupled superposition formula +k₁×+k₂×...×+kₙ = +(Σkᵢ−n+1), an algebraic consequence invisible under conventional labels. Implications for calculus, dimension theory, measure theory, linear algebra, group theory, category theory, topology, and differential geometry are discussed.