Chronon Field Theory in 1+1D: A Solvable Model for Emergent Mass, Charge, and Geometry

We study a dimensionally reduced version of Chronon Field Theory in 1+1 dimensions, offer ing a solvable framework in which spacetime geometry, gauge structure, and quantized matter all emerge from a single underlying field. The model is built from a unit-norm timelike vector field whose internal phase θ (x µ) encodes both curvature and U (1) holonomy. Within this setup, we construct exact topological soliton solutions that behave as charged, massive excitations. These solitons exhibit three interrelated but distinct manifestations of mass: gradient mass from spatial inhomogeneity, holonomy mass from topological winding, and coherence mass from long-range phase rigidity. We demonstrate that all three forms of mass reduce to a single Lorentz-invariant, geometric quantity: m²= η^µν ∂µθ ∂νθ, which we interpret as the root definition of mass in Chronon Field Theory. This covariant norm of the internal phase gradient not only unifies emergent mass phenomena but also suggests a new fundamental origin of mass applicable to broader field-theoretic contexts, including real-world physics. In addition, we introduce a physically meaningful notion of soliton size, derived from the energy localization profile, which provides a geometric length scale associated with particle struc ture. Linearized perturbations around soliton backgrounds give rise to photon- and graviton-like modes, and soliton composites exhibit discrete mass spectra governed by coherence conditions. The 1+1D model thus provides an analytically tractable setting for exploring how matter, charge analogues, particle size, and now mass itself can emerge from internal phase geometry and topology in Chronon-based field theories.