A Scalar Weighted Metric and Gradient Flow Structure on Riemannian Manifolds

This work develops a scalar–weighted geometric framework on smooth Riemannian manifolds based on three positive scalar fields S,C,F representing structural, coherence and fractal informational components. A symmetric tensor constructed from the gradients of these fields induces a non-degenerate deformation of the background metric, yielding a well-posed weighted Riemannian geometry. An associated variational functional and its gradient flow are introduced, and global existence, uniqueness and convergence to stationary configurations are established under standard geometric assumptions. A discrete approximation on weighted Hamming graphs is shown to converge to the continuous formulation. The resulting framework provides a rigorous geometric backbone for informational models based on scalar field triplets.This version includes a supplementary spectral appendix related to RH interpretation. The main framework remains unchanged.