The Einstein equation from Gauss–Codazzi embedding geometry: exact coupling constant and Planck-scale corrections for a Hessian-potential ambient space

We consider a 4-dimensional Lorentzian submanifold V embedded in a higher-dimensional Riemannian ambient space whose metric is the Hessian of a scalar potential Φ. Such Hessian metrics arise in information geometry, thermodynamic geometry, and the geometry of quantum state spaces. We perform the complete Gauss–Codazzi analysis and show that, in the sharp-embedding limit and under vacuum isotropy (Assumption 4), the intrinsic geometry of V satisfies the Einstein equation G_μν + Λg_μν = κT_μν with three results: (i) the coupling constant is expressed in terms of the mean vacuum embedding curvature, κ = 2/Ā²_vac; (ii) Newton’s constant is G = c⁴/(4πĀ²_vac); (iii) Planck-scale corrections are given by the projected ambient curvature tensor H_μν, which is symmetric and divergence-free. The Planck length is identified as the inverse vacuum embedding curvature. Parametric predictions include a generalised uncertainty principle with β = O(1), modified photon dispersion, and gravitational decoherence; numerical coefficients are model-dependent. We compare with Regge–Teitelboim embedding gravity, Randall–Sundrum braneworlds, Jacobson’s thermodynamic derivation, Verlinde’s entropic gravity, and loop quantum gravity.