Persistence Fingerprint Analysis (PFA): A Cross-Scale Mathematical Morphology Framework

Persistence Fingerprint Analysis (PFA) is a comparative structural framework developed within the Paton System for identifying recurring persistence topology across mathematically distinct systems. Rather than comparing systems through ontology, terminology, or disciplinary boundaries, PFA reduces systems into persistence fingerprints based on continuation structure, weighting behaviour, instability response, confinement geometry, collapse dynamics, reconstruction pathways, and admissibility topology. The framework investigates whether mathematically distinct systems may preserve recurring persistence architecture despite differing interpretation and scale. PFA introduces persistence fingerprint extraction, topological normalisation, morphological fit testing, cross-scale topology comparison, and stability density analysis. The framework is applied across quantum systems, field systems, recursive cognitive systems, dynamical systems, organisational systems, and unresolved micro-scale persistence terrain. PFA does not propose ontological equivalence or replacement of existing theories. Instead, it functions as a comparative persistence morphology framework for analysing recurring transformational structure across mathematically distinct domains.