ENTROPIC GRAVITY AND THE PERONA–MALIK EQUATION FROM THE LOGARITHMIC DERIVATIVE OF THE STEPANOV TRANSFORMATION ON THE NONCOMMUTATIVE TORUS

We derive the equation of anisotropic curvature diffusion—the Perona–Malik equation—from the entropic functional of the noncommutative super-torus T 2 θ . This establishes a rigorous geometric foundation for entropic gravity à la Bianconi.The vacuum superconnection on T 2 θ is the logarithmic derivative of the Stepanov Jacobian: Γ∗ = J−1dJ = d(ln J). We prove that ln J is the information-theoretic entropy density of the quasi-periodic Stepanov flow, and the Connes–Yang–Mills func-tional ∥∇∥^2 = Trω(R†⋆R) is the Fisher information of this flow. The condition R = 0 is equivalent to the extremisation of the Fisher information—the state of maximal entropy.We establish three principal results:(1) Entropic functional as Fisher information (Theorem 2.2): The vacuum energy functional coincides with the Fisher information of the Stepanov flow. Its minimisation (R = 0) is the condition of maximal entropy.(2) Bianconi action from dimensional reduction (Theorem 3.1): The dimensional reduction of the Connes–Yang–Mills functional from M4 × T 2 θ to the emergent 4D spacetime yields the entropic action of Bianconi, including the non-local termRln(□/Λ^2NC)R. All coefficients are determined by τ0 and θ.(3) Perona–Malik equation from entropic gradient flow (Theorem 4.1): The gradient flow of the entropic functional, with the automorphism parameter t as thetime variable, yields the Perona–Malik equation for anisotropic curvature diffusion:∂gij/∂t = ∇^2Rij − ∇i∇jR + anisotropic corrections. (1)This equation describes the evolution of spacetime geometry towards maximal entropy, providing a microscopic foundation for the thermodynamic interpretation of gravity.As a corollary, we formulate the entropic origin of Newton’s law as a conjecture: the entropic gradient ∇(ln J) combined with the Unruh temperature yields F = GMm/r^2.The proof requires solving the inhomogeneous Riccati equation with a point mass source, which we outline as a programme for future work.The theory contains no free parameters.