DBI Tachyonic Dark Energy and Thermal Time from Infrared Entropy Production

We construct a stable Dirac–Born–Infeld (DBI) tachyonic scalar field model that provides a microphysical basis for late-time cosmic acceleration, field–proper time, and a thermodynamic arrow of time. The DBI structure imposes a speed limit, keeps the energy density real and non-phantom, and yields de Sitter–like late-time attractors with w → -1. At the perturbative level we derive the DBI mode functions in the de Sitter limit and show that super-horizon field fluctuations obey a characteristic scaling with sound speed c_s < 1. Coarse-graining on sound-horizon scales leads to a stochastic description with diffusion coefficient D(H, c_s) = H^3 / (4 π^2 c_s^3) and a fluctuation–dissipation temperature T_FD = 3 H^4 / (8 π^2 c_s^3) in the Itô convention, fixed directly by the mode functions. From the coarse-grained probability distribution we construct an infrared Gibbs entropy S_IR[P] obeying an H-theorem dS_IR/dt ≥ 0, which defines a strictly monotonic thermal time parameter τ_T proportional to S_IR. This yields a concrete semi-classical realization of a state-induced temporal flow in the spirit of the Connes–Rovelli thermal-time hypothesis. Numerically we show that strong Hubble friction drives the field to freeze on a plateau, reproducing ΛCDM at the background level with |w(z < 3) + 1| ≲ 10^-3 and c_s^2(z < 3) ≈ 1. We also construct a controlled dynamical benchmark with percent-level deviations in H(z) and fσ_8(z), and a simple mock MCMC analysis illustrating how such deviations can be constrained with precision cosmological data.