ETE IX — Discrete Thermodynamics, Coarse–Graining, and the Arrow of Time

We show how thermodynamics emerges in the Elective Space–Time (ETE) framework without
introducing continuous variables or a priori probabilities. A microstate at a front time t is
purely discrete: for each node X ∈ Ft we keep an internal frame RX ∈ W(E8) and a finite
integer state u(X) ∈ Z8. Macroscopic variables are integer counts in windows (zigzag density,
class histograms in the center of the group algebra Z[W(E8)], and norms of u). We define
entropies by discrete counts; inheritance with central mixing is bistochastic and yields a discrete
H–theorem. Coarse–graining (LMR in statistical mode) produces diffusion–type effective laws
and an operational notion of temperature. The arrow of time is the monotone growth of these
entropies under front expansion. We also include refinement suggestions: a reference check for
the H–theorem, the effect of discrete gravity μ(X) on entropy production, and fluctuation scaling
across dimensions.