Algebraic Phase Theory I: Radical Phase Geometry and Structural Boundaries

Abstract: We develop Algebraic Phase Theory (APT), an axiomatic framework for extracting intrinsic algebraic structure from phase based analytic data. From minimal admissible phase input we prove a general phase extraction theorem that yields algebraic Phases equipped with functorial defect invariants and a uniquely determined canonical filtration. Finite termination of this filtration forces a structural boundary: any extension compatible with defect control creates new complexity strata. These mechanisms are verified in the minimal nontrivial setting of quadratic phase multiplication operators over finite rings with nontrivial Jacobson radical. In this case nilpotent interactions produce a finite filtration of quadratic depth, and no higher degree extension is compatible with the axioms. This identifies the radical quadratic Phase as the minimal example in which defect, filtration, and boundary phenomena occur intrinsically.