We introduce Topological Threshold Dynamics (TTD), a unified measurement framework in which any cognitive, physical, or informational system at time t is represented as a directed weighted graph G (t) = (V, E, w), and its structural state is characterized by the directed persistent homology of that graph via the path-homology complex of Grigor'yan, Lin, Muranov, and Yau. Phase transitions are detected by a threshold dynamic tau defined over the eigenstructure of the directed path-Laplacian operator Lₚ, which tracks topological change in the allowed p-chain spaces of the directed complex. The developmental delta between two time points is the bottleneck distance dB between persistence diagrams, stable under perturbation by the Cohen-Steiner theorem. The framework is domain-invariant: the same formal apparatus measures cognitive growth in multi-instance learning, phase transitions in financial markets, sensory decay in neurophysiology, manifold deformation in computer vision, structural anomalies in spacetime geometry, conformational change in molecular biology, bifurcation dynamics in chaos theory, and equilibrium topology in game-theoretic environments. We introduce the NELE algorithm as the topological state oracle that scans, memorializes, and evaluates topological change across the multi-instance ensemble, outputting developmental deltas as finite-difference ratios of bottleneck distance and inter-instance divergence matrices. Perturbation amplification is grounded in the Ollivier-Ricci curvature of the directed graph and in the non-normality of the directed transition operator, locating sensitivity to initial conditions in the curvature geometry rather than in the contractive Laplacian spectrum. Regression specifications for machine learning are grounded in the Morse-Smale complex of the discretized gradient vector field rather than in level-set topology alone. TTD implies improved optimization algorithms, new observational targets for astrophysics, and a general theory of phase-transition detection across physical and informational scales.
Eric Hoppe (Mon,) studied this question.