Possest–PQF: An Operatorial Theory of Filtration, Intensity, and Topological Complexity

This work introduces the Possest–PQF (Persistent Quantification Framework), a novel operatorial approach to complexity, accessibility, and novelty. PQF formalizes accessibility as algebraic dynamics generated by the interplay of two operators: a filtration operator (Δ𝔽) and a recursive operator (δ*). The framework demonstrates how non-commutativity between these operators gives rise to measurable structural novelty, detectable through bottleneck distances in persistent homology, directed shifts in topological entropy, and order-sensitive loss differences in algorithmic processes. We compare PQF with established paradigms — Shannon information theory, Kolmogorov complexity, dynamical systems, statistical complexity, topological data analysis, and AI evaluation metrics — highlighting its role as a foundational reorientation from external metrics to endogenous accessibility dynamics. Worked examples (toy barcodes, minimal SGD experiment) and a categorical operator diagram illustrate both the theoretical rigor and the applied diagnostic power of PQF. This establishes PQF as a universal diagnostic for novelty, bridging mathematics, topology, and machine learning.Persistent Homology, Topological Data Analysis, Complexity, Novelty Detection, Accessibility Operators, PQF, Shannon Information, Kolmogorov Complexity, Dynamical Systems, Statistical Complexity, Deep Learning, Operator Algebra, Topological Entropy