Gaussian-Filtered Contact Loops and Finite One-Loop Tadpoles Exact Euclidean Benchmark and the Local-Limit Counterterm

This paper studies an exact Euclidean scalar one-loop benchmark with a declared Gaussian momentum filter. The filtered tadpole is finite in every finite dimension for positive mass and filter scale. We derive its exact Schwinger representation, its four-dimensional exponential-integral form, and its local-limit asymptotics. The divergent local theory is recovered as the filter scale tends to zero, so ordinary counterterms remain the appropriate sharp-limit subtraction. The calculation does not derive the external filter from an internal MTT fixed-point operator and does not prove all-loop finiteness or physical Lorentzian consistency. MTT promotion requires a selected internal-to-external source map; gauge and BRST identities, reflection positivity, causality, and unitarity require separate certificates.