Algebraic Phase Theory I: Radical Phase Geometry and Structural Boundaries

TL;DR

This paper introduces Algebraic Phase Theory (APT), an axiomatic framework for analyzing phase-based data, revealing intrinsic algebraic structures, defect invariants, and structural boundaries through finite filtrations, exemplified in quadratic phase operators over finite rings.

Contribution

It develops a novel axiomatic framework for phase analysis, establishing a phase extraction theorem and identifying minimal radical quadratic phases with intrinsic defect and boundary phenomena.

Findings

Finite filtration of quadratic phases over finite rings.

Nilpotent interactions produce finite depth filtrations.

Radical quadratic phases are minimal examples with intrinsic defect and boundary phenomena.

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.