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: m^2= 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.