Mechanism Classification of Critical Positive Operator Families

This paper studies the classification problem for positive finite-dimensional operator families whose coercivity degenerates in a limiting process. Building on the framework of critical positive operator families, it isolates three mechanisms of coercivity loss: Uniform Contraction, Correlation Collapse, and Boundary Drift. The classification is formulated through mechanism invariants (an amplitude invariant, a normalized Riesz collapse invariant, and a boundary concentration invariant), a critical scale profile separating local, bulk, and spectral degeneration, stability properties under bounded congruence and subscale perturbation, and a partial classification theorem identifying three verifiable, sufficient locations of coercivity loss. The full exhaustion problem — whether these three mechanisms account for every possible loss of coercivity — is stated as an open problem and its difficulty is discussed explicitly, rather than left unaddressed. The paper includes model examples and counterexamples separating the mechanisms: the Brownian covariance kernel as a stable negative control, the Hilbert matrix as the model example for Correlation Collapse, hyperbolic Cauchy modes for Uniform Contraction (with a fully self-contained positivity proof via the classical Eisenstein summation identity), and a fully self-contained digamma boundary kernel for Boundary Drift, including a new closed-form evaluation of its normalization constant and an unconditional theorem for its exact asymptotic sign crossing. This is the third paper in a series on critical positive operator families, following a general framework paper and a paper on the Cauchy generator class; it organizes and classifies the mechanisms introduced in the earlier work rather than introducing a new object.