5.10 Geometric Hilbert Expansion — Euler Equations

G4 has derived the information field dynamical equation — the geometric version of the Boltzmann equation. This article (G5) performs the first order of the Hilbert expansion on that equation, rigorously deriving the compressible Euler equations. Core results: (1) the equilibrium state is the uniform distribution on ; (2) the first-order expansion yields the continuity equation, momentum equation, and energy equation; (3) the geometric pressure is determined by the algebraic action, and the adiabatic index is determined by the dimension of the Clifford algebra . The mathematical structure is completely parallel to Deng-Yu's derivation of the Euler equations from the Boltzmann equation, but all parameters are locked by spectral rigidity. **Keywords**: Hilbert expansion; Euler equations; equilibrium distribution; collision invariants; moment closure; geometric pressure; algebraic action