Towards Scalar-Field Actions in General Relativity from a Maximum-Entropy Displacement Ensemble

We present a maximum-entropy (MaxEnt) derivation of spacetime geometry starting from a quantum thermal ensemble of local displacement fluctuations. The sole constraint imposed is the expectation value of a quadratic line-element observable. Maximization of entropy yields a Gaussian displacement kernel whose second moments encode an emergent metric structure. Beginning in a locally inertial (flattened Minkowski) frame, we show how curved spacetime geometry and field-space metrics arise through pushforward of the same MaxEnt measure, performed entirely inside the defining integrals. We demonstrate the equivalence of this formulation with the quantum thermal (Matsubara) density-matrix description, without assuming a prior Hilbert-space structure. The resulting geometry is expectation-valued and information-theoretic in origin. This framework provides a unified statistical foundation for spacetime geometry consistent with information geometry, quantum statistical mechanics, and covariant field theory.