Semantic Gravitation I: Gradient Fields of Semantic Dynamics on Hilbert Spaces

Semantic Gravitation is a new open research programme that explores whether “meaning” can be treated as a dynamical object with its own geometry and stability structure—analogous to how physics treats states, flows, and potentials. The aim is not to claim a finished theory, but to propose a precise formal scaffold that can be inspected, tested, and either refined or rejected. This first paper develops the local mathematical core of the programme: semantic states are represented as points in separable Hilbert spaces and evolve under (i) deterministic gradient flows driven by a potential and (ii) stochastic Langevin-type dynamics. On the level of probability distributions, these dynamics induce Fokker–Planck equations with an associated free-energy functional acting as a Lyapunov function. The focus is structural and analytic rather than ontological. Under explicit assumptions, the paper establishes existence, stability, and qualitative behaviour of the dynamics, and illustrates the framework through finite-dimensional and field-like examples. It provides the Hilbert-space foundation for subsequent parts of the Semantic Gravitation series.