Discrete History Selection and Reconfiguration Dynamics in Topological Tensor Networks

This article develops a variational extension of the discrete topological tensor-network framework established in a preceding series of papers that introduced the underlying network model, its mathematical foundations, and the local reconfiguration dynamics. The presentation is intentionally restricted to the mathematical formulation of this variational framework: physical interpretation of the structures introduced here, continuum limits, and phenomenological consequences lie outside its scope and are addressed separately in subsequent publications. We formulate a framework for the comparison and selection of complete network histories, introducing transition functionals, reconfiguration measures, a History Action, and an associated variational principle on the discrete history space. The construction presupposes no auxiliary continuum structure such as a manifold, a metric, or an external time parameter; all quantities are defined intrinsically on the discrete state space. The principal result establishes that the global variational principle admits a complete local formulation: a local selection principle, based on finite history variations, localizes the global History Action through segment action differences. We further prove additivity of the History Action, derive a sector-restricted discrete Euler--Lagrange equation as its differentiable representation, and, under a local regularity assumption, obtain locally unique discrete evolution equations within differentiable variational sectors.