M20 Operational Separation SC Circuits, Branch Addresses, and the Architecture of P<>NP

This monograph introduces the SC circuit model, a rank-aware computational framework derived from the Symmetric Core hierarchy. Within this model, the complexity classes P[SC] and NP[SC] are defined and formally separated via the Lifting Cost and Algebraic Inexpressibility results. The work does not claim P≠NP in the classical sense; instead, it isolates a single precise gap — the Model Equivalence Conjecture — between SC computation and classical Turing computation. The result provides a structurally clean reformulation of the P vs NP problem and clarifies the geometric and algebraic obstructions underlying computational hardness.