Differential Entangled Topology: towards a rigorous programme with computational methods and dynamic applications

Abstract

Differential Entangled Topology (DET) proposes a mathematical framework for analysing evolving topological structure in dynamical systems. We present a formalisation of DET on smooth manifolds, define entanglement functionals derived from geodesic interactions and clique complexes, and introduce computable surrogates for topological entropy based on families of Vietoris-Rips complexes across thresholds.

Background: The methodology connects differential topology, stochastic calculus, and topological data analysis (TDA), motivated by a broader research programme to develop DET beyond its original context in consciousness modelling to complex networks, information-theoretic characterisations, and quantum-inspired systems.

Methods: A stochastic-deterministic flow on the sphere S² illustrates how entanglement indices and Betti curves evolve as control parameters change, using computational methods from topological data analysis.

Results: The results show that changes in noise and rotational shear elevate an entanglement index and shift Betti curves in ways consistent with qualitative phase transitions, providing a roadmap for proof-theoretic progress and scalable computation.