Gaussian-Filtered Euclidean Propagators and UV Behaviour Exact Domination under a Declared External Filter

This paper studies Euclidean propagators obtained after declaring an external Gaussian momentum filter. On a single declared operator domain, functional calculus defines a unique compressed covariance. We prove distributional recovery of the ordinary propagator as the filter width tends to zero, finite coincidence values at positive width, and Gaussian ultraviolet domination. Every scalar Euclidean graph whose internal lines are explicitly assigned this filter is absolutely ultraviolet finite under the stated quadratic-control hypothesis. This is a conditional model theorem, not a source theorem: an internal MTT gap does not select external momentum damping without an internal-to-external intertwiner. Reflection positivity, gauge invariance, causal support, and Lorentzian unitarity remain independent requirements.