The Universal Master Equation on Exceptional Geometries, A Stochastic Template for ReversibleDissipative Dynamics

We propose the Universal Master Equation (UME) as a standard normal form for dynamics on Riemannian and quaternionic-Kähler manifolds. The UME provides a coordinateinvariant, modular decomposition into four irreducible driversconservative (J∇H), dissipative (−G∇H), selection (−G∇Φ), and uctuation (Σ ◦ dW)plus an optional harmonic term from Hodge decomposition when H1 (M) ̸= 0. The selection functional Φ enforces viability constraints (we term this autopoietic structure, following MaturanaVarela, though no biological claim is intended). The UME is not a uniqueness theorem about nature but a universal intermediate representation for well-posed dynamicsa compilation target with optional modules for continuous diusions, discrete Markov chains, and memory. We present: (1) the Theorem of Dynamical Completeness via Hodge decomposition on compact Riemannian-symplectic manifolds; (2) an instantiation on the 112-dimensional Wolf space M112 = E8(−24)/(E7 ×SU(2)), where topological sectoring via H2 (B, Z3) ∼= Z3 and HirzebruchRiemannRoch index computations yield a mechanism for three fermion generations (χ(W) = 1 per sector); and (3) a discrete extension showing that Markov chains admit a parallel four-term decomposition, with Boolean logic emerging as the topological limit of smooth geometry. The framework interfaces with GENERIC/metriplectic systems, Wasserstein gradient flows, diffusion models, and the Free Energy Principle, though we emphasize structural parallels rather than identities. Applications to prompt engineering and conformance standards are discussed as methodological illustrations.