Algebraic Phase Theory I: Radical Phase Geometry and Structural Boundaries
Joe Gildea
TL;DR
Algebraic Phase Theory (APT) addresses phase interactions that arise over finite nilpotent rings by extracting intrinsic algebraic invariants. It defines defect as the first nonvanishing additive derivative and encodes it into a canonical, finite defect filtration of the emergent algebraic Phase, which terminates at a structural boundary determined by the analytic degree. The radical quadratic model over $R=\mathbb{F}_2[u]/(u^2)$ provides the minimal nontrivial instance, where defect appears at quadratic depth and terminates at depth $2$, establishing a sharp boundary for extensions. The main contributions are the axiomatic framework (Axioms I–V), a canonical phase extraction procedure from admissible data, and a structural boundary theorem showing that beyond quadratic depth any finite-termination extension must introduce higher defect strata.
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.