Theorem of Complex Binarity

This work introduces and proves the Theorem of Complex Binarity, a geometric extension of binary logic. The theorem states that when the structural complexity of a system exceeds the resolution capacity of a binary distinction operator, distinguishability cannot be preserved without dimensional extension. A complex binary operator is defined by augmenting binary opposition with an orthogonal parameter encoding transition, orientation, or accumulated structural stress. The extension preserves binary projection while restoring injectivity under saturation conditions. Zero is formally redefined as a transitional singularity rather than a state. Three degradation modes are derived as geometric consequences of saturation. The theorem establishes minimal conditions under which binary logic must be extended to preserve form and relates the result to the Nyquist–Shannon sampling theorem and Ashby’s law of requisite variety.