Semantic Gravitation III: Quantum-Compatible Semantic Fields — A Hilbert-Space Embedding of Quantum States, Observables, and Dynamics

Semantic Gravitation is an open research programme that investigates whether meaning can be treated as a dynamical object with its own geometry, stability structure, and flow laws. This third paper develops the quantum-compatible layer of the programme. It shows how the standard formalism of quantum mechanics—quantum states, observables, expectation values, and dynamics—can be embedded faithfully and without modification into the semantic Hilbert-space framework introduced in Papers I and II. Quantum states, represented as density operators on a quantum Hilbert space HQ, are embedded as a distinguished sector of the global semantic state space S via an explicit linear isometric construction. Expectation values are reproduced exactly by representing quantum observables as continuous linear functionals on the embedded sector, and both unitary and open-system quantum dynamics (including GKLS evolutions) are shown to induce compatible dynamics within the semantic framework. Conceptually, the paper demonstrates that standard quantum theory can be realised as a structurally faithful subspace of a broader semantic state space, coexisting with gradient-flow and free-energy dynamics on probability measures. This establishes the mathematical foundation for coupling quantum dynamics to semantic field theories and complex-system models developed in subsequent parts of the programme.