Finite Coherent Overlap Vertices and Their Local Contact Limit Conditional EFT Interpretation and the Renormalization Boundary

This paper proves that a point-local interaction vertex is the zero-width limit of a finite coherent overlap vertex. For smooth fields, bounded approximate-identity kernels give convergence of general finite-overlap interactions to their local contact limits. On compact spectral models, finite projection yields finite vertex tensors and finite coincident covariance within the retained sector. The MTT interpretation is conditional: if a selected physical sector supplies such a kernel and an intertwiner to external fields, the local vertex may be read as its sharp limit. At nonzero width the construction is generally a nonlocal effective interaction, not automatically a local ultraviolet completion. A local parent mediator or equivalent localization theorem would be needed to change that classification. The paper does not infer all-loop renormalizability, ultraviolet completion, Lorentzian causality, unitarity, reflection positivity, or gauge and BRST consistency from finite width alone.