Geometric and arithmetic characterization of [Formula: see text]-module flatness with applications to tensor products.

This paper establishes a comprehensive framework for studying flatness properties and tensor products of [Formula: see text]-modules across algebraic, geometric, and arithmetic contexts. We develop new criteria characterizing flatness through Lagrangian geometry, homological algebra, and irregular Hodge theory, revealing deep connections between these perspectives. The work introduces a geometric obstruction theory for globalizing pointwise flat modules and proves fundamental results about the monoidal structure of the derived tensor product category. Applications include compatibility theorems for Beilinson-Bernstein localization and arithmetic characterizations of flatness in characteristic p. The methods combine microlocal analysis, irregular Riemann-Hilbert correspondence, and p-adic techniques to yield new insights into the interplay between local and global properties of differential systems.