The Minimal Unit of Description: A General Theory of Phase Collapse

This paper develops a quotient-theoretic theory of phase collapse. The central idea is that a phase collapse is not primarily a discontinuity of a scalar observable, the appearance of an order parameter, or the failure of a chosen effective model. Such events may be signatures of collapse, but they are not its structural definition. The defining event is a change in the minimal unit of admissible description. The primitive object is a state space observed through a bounded admissible readout. Such a readout identifies primitive distinctions and induces a collapse quotient. Every description admissible for the readout must factor through this quotient, and therefore cannot recover distinctions that the readout has already killed. Once an observable layer is placed on the quotient, the resulting descriptive quotient determines the minimal descriptive units. Along compatible monotone readout families, phase collapse occurs when previously distinct descriptive units merge. The paper organizes this structure through a fibered architecture consisting of a carrier, a reduction or orbit law, an effective algebra, a typicality or selector coordinate, an observable layer, and a representative. This separates the quotient-level collapse event from the numerical or asymptotic quantities that may represent it. The theory also introduces canonicality levels for representatives. Level 0 is pre-representative. Level 1 is measure-relative in the broad sense of depending on an externally chosen typicality law. Level 2a is selected by an intrinsic reference class. Level 2b is selected by an enlarged datum such as a source law, distortion function, calibration distribution, response-test family, or validation protocol. Level 3 consists of typicality-independent representatives. The framework is applied to the Euclidean Gauss subclass, rate-distortion quotients, defect-cocycle representatives, engineering readout metrics, comparison cases from classical phase transition theory, and the joint spectral radius as a stable nontrivial Level 3 representative. The resulting theory provides a structural language for asking when a readable description has changed its unit, when a representative is induced rather than imposed, and how much data are required before that representative becomes measure-relative, intrinsic, datum-relative, or typicality-independent.