Companion Note to the Entropic Gravity Framework (V5): Mathematical Justification of the Non-Minimal Weight 𝐹 ( 𝜅 ) = 𝑒^− 𝜅

This companion note to Entropic Scalar–Tensor–Quantum Gravity: Informational Field κ(x,t) and Unified Extension of Einstein Gravity (V5) [DOI: 10.5281/zenodo.16971040] provides a rigorous mathematical justification of the exponential weight F(κ)=e^−κ used in the scalar–tensor action. By identifying the entropic field κ(x,t) with the weight function of Perelman’s weighted manifolds, the construction is shown to be canonically rooted in Bakry–Émery geometry rather than introduced ad hoc. The document includes: the full variational derivation of the field equations, the role of the weighted measure in conservation laws, the Perelman–κ \ κ entropy functional, and implications for FLRW cosmology and strong-curvature regimes. This establishes that the entropic weight e^-κ is both mathematically mandated and physically fertile, reinforcing the internal consistency of the entropic gravity programme.