The Phase-Variance Sieve A Unified Framework for Self-Filtering Dynamic States across Number Fields and Quantum Topologies

Traditional mathematical modeling and industrial engineering paradigms rely on the static equals sign ($=$) to represent state preservation and equilibrium. This approach treats balance as a passive identity, necessitating significant computational overhead to detect, calculate, and correct out-of-phase error states. This paper introduces The Phase-Variance Sieve, an analytical framework that shatters the static identity barrier by implementing an active Line of Symmetry (∥). By driving mathematical and physical topologies to a binary crossroads, we demonstrate that symmetrical configurations achieve infinite, scale-invariant resonance (fractalization), while asymmetrical configurations introduce directional phase friction that forces the state to compound structural variance and self-terminate. We verify this cross-domain invariant law via exact analytic proofs across three major frontiers: pure number fields (the critical line of the Riemann Zeta Function), superconducting quantum processing gates (Google's 105-qubit Willow chip architecture), and high-frequency wireless wave propagation (6G Terahertz communication channels).