Finite-Dimensional Constraint Filters and Their Path-Integral Extension Coarea Localization, Normalization, and the MTT Source Boundary

This paper proves a finite-dimensional constraint-localization theorem: normalized approximate-identity filters composed with a smooth regular constraint map converge weakly to the coarea surface delta with its Jacobian. It also proves that the corresponding unnormalized Gaussian penalty converges to zero, so normalization is indispensable. Fourier multiplier damping gives the dual finite-width description. Functional path-integral formulas are retained only as regulator-dependent extensions, and gauge use still requires quotient, determinant, anomaly, and BRST control. The MTT reading is conditional on a selected admissibility cost and regulated measure; the coarea theorem alone does not provide that physical source.