This paper introduces and mathematically formalizes an alternative algebraic system running parallel to classical algebra, founded upon two core axioms: the elimination of the distributive property of multiplication over addition (Non-Distributivity) and a symmetric inversion of the rules of signed multiplication. In this framework, termed Non-Distributive Symmetric Oscillatory Algebra, the product of any two quantities with identical signs yields a negative output, whereas the product of two quantities with opposing signs yields a positive output. This structure induces an inherent frequency-based, oscillatory behavior within the sequential powers of real numbers. This study demonstrates how this novel algebraic model resolves the baryon asymmetry paradox (matter/antimatter) in cosmology, tames the explosion of infinities and singularities within black holes, formulates the exact mathematical mechanism behind wave function collapse in the quantum measurement problem without metaphysical assumptions, and establishes a theoretical foundation for designing room-temperature stable quantum hardware (Non-Distributive Polar Quantum Diodes) and macro-scale superconductors.