Planck-Scale Periodicity

The Topological Lagrangian Model necessitates a geometric extension of the standard spacetime metric to account for the intrinsic vibration of the Unified Coherence Field.Standard Minkowski geometries sufficiently describe the extrinsic trajectory of particles, yet they lack the degrees of freedom required to model the intrinsic regeneration of the quantum state vector.To resolve this, we formally define the physical substrate as a five-dimensional product manifold M = M K4 × R τ [1].While the spatial component M K4 describes the macroscopic, non-orientable geometry, the component R τ represents an orthogonal Internal Time dimension.This dimension operates independently of the coordinate time t, governing the microscopic regeneration of the field state at the fundamental granularity of the universe.