Quantum and stastical interpretation of entropic gravity

This work develops a quantum and statistical extension of the Entropy–Hessian geometric framework(https://doi.org/10.5281/zenodo.18773391), in which the spacetime metric is defined as the Hessian of a single scalar generator S,

g_µν = ∇µ∇νS.

The paper does not introduce additional dynamical fields or external quantization postulates. Instead, it investigates whether probabilistic and quantum-like structures can arise intrinsically from the scalar-generated geometry itself.

The main elements of the analysis include:

•Formulation of a conserved probability current derived from the gradient of the scalar generator

•Functional integral formulation over the scalar configuration space

•Nonlinear Hamiltonian analysis demonstrating propagation of a single scalar degree of freedom

•Proof of degeneracy of the higher-derivative structure in the Ostrogradsky sense

•Demonstration of dynamical preservation of Lorentzian signature

•Establishment of hyperbolicity and well-posed Cauchy evolution

•Self-adjointness of the linearized fluctuation operator

•Emergence of Schrödinger-type dynamics in appropriate quasi-static regimes

The causal structure is shown to arise from the Hessian metric itself, and the principal symbol of the field equations is governed by the scalar-generated geometry. No independent background metric is assumed.

The paper is presented as a structural and mathematical extension of the Entropy–Hessian framework. It does not claim a complete quantum theory of gravity but instead analyzes the internal consistency, stability properties, and statistical interpretation of the scalar-generated geometric model.

Subsequent works explore gravitational recovery, gauge structure, spectral constraints, and cosmological implications within the same framework.