Zero-Equation: Unified Criterion for Real-Virtual Conjugate Symmetry and Necessary and Sufficient Condition for Symmetry Breaking

In conventional physics, Noether’s theorem establishes a one-to-one correspondence between continuous symmetries and conservation laws, yet it fails to interpret the breaking of discrete symmetries such as spacetime inversion and matter inversion. Experimental observations including parity non-conservation in weak interactions and CP violation in kaon decays have long lacked a unified criterion derived from first principles, remaining a critical unsolved puzzle within the Standard Model of particle physics.Based on the real-virtual dual-spacetime ontology of discrete spacetime theory, this paper puts forward the Zero-Equation centered on the rule that the aggregate physical quantity over real and virtual domains equals zero, extending symmetry definition from a sole real realm to real-virtual conjugate symmetry. We rigorously prove that breakdown of the Zero-Equation serves as the necessary and sufficient condition for all discrete symmetry breaking; spacetime and matter inversion symmetries are preserved if and only if A_real​+iA_virtual​=0.We further redefine conjugate transformations for discrete symmetries and formulate a generalized conservation law. The framework uniformly accounts for all observed symmetry-breaking effects in the Standard Model, attributing parity non-conservation and CP violation to unbalanced physical quantities across real and virtual domains. Complementing Noether’s theorem, it separately governs conservation rules for discrete and continuous symmetries, offering an innovative mathematical-physical viewpoint toward unifying fundamental interactions.