Cosmic Expansion and Thermodynamic Time from a Tachyonic Scalar Field

We present a self-consistent field-theoretic model in which late-time cosmic acceleration and an intrinsic notion of temporal progression arise from a single tachyonic scalar field with bounded kinetic energy. The framework retains the Dirac-Born-Infeld (DBI) form of the Lagrangian but enforces φ̇² < c², ensuring real-valued energy density and causal propagation. We derive the effective temperature T* from the Kubo-Martin-Schwinger (KMS) boundary condition and demonstrate equivalence between geometric and thermodynamic definitions of field proper time. The field entropy functional Sφ is constructed from configuration space geometry, and numerical integration confirms monotonic entropy growth, validating the Second Law within this framework. The model respects energy conservation, reproduces ΛCDM behavior at late times (w ≈ -1.000), and offers a testable extension within the class of k-essence cosmologies. This work provides an explicit field-theoretic realization of the Connes-Rovelli thermal time hypothesis, with potential applications to quantum cosmology and the problem of time in quantum gravity.