5.9 Information Field Dynamical Equation (Geometric Boltzmann Analogy)

G1–G3 have proved that higher-order cumulants () of the information field decay exponentially to zero under steady-state causal circulation. This article (G4) derives, from this conclusion, the closed dynamical equation of the information field — a geometric version of the Boltzmann equation. Core steps: (1) establish the information field BBGKY hierarchy; (2) use the cumulant truncation theorem to prove that the hierarchy truncates at second order after ; (3) derive the single-body dynamical equation (discrete); (4) obtain the information field Boltzmann equation in the continuum limit. This equation is structurally parallel to Deng-Yu's hard-sphere Boltzmann equation, but the microscopic mechanism differs — collisions arise from cross-sector coupling of the Birkhoff mixing matrix, rather than elastic collisions between particles. **Keywords**: BBGKY hierarchy; cumulant truncation; information field dynamical equation; Boltzmann equation; cross-sector coupling; Clifford sector transitions