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.