Semantic Gravitation IV: Semantic Fields on Spacetime Manifolds — An Effective Field Theory for Coherence

Semantic Gravitation is an open research programme that explores whether “meaning” can be treated as a dynamical object with its own geometry, stability structure, and flow laws—analogous to how classical and quantum field theories treat dynamical fields, potentials, and flows. This fourth paper develops the spacetime field-theoretic layer of the programme. It extends the Hilbert-space and free-energy formalisms of Papers I and II to a covariant classical semantic field theory on a fixed globally hyperbolic spacetime (M, g). A semantic field is modelled as a map: M S into a separable Hilbert space of semantic states, with dynamics derived from an action functional of kinetic-minus-potential type, where local semantic structure is encoded by a family of semantic potentials Wₓ. The paper derives the associated stress–energy tensor, energy and flux balance relations, and establishes local well-posedness of the Cauchy problem under standard regularity and growth assumptions. Spatially homogeneous solutions recover the Hilbert-space gradient systems of Paper I, while the probabilistic and thermodynamic perspective connects to the free-energy framework developed in Paper II.