Finite-Time Scattering Windows and Asymptotic Delta Limits Standard Fourier Mathematics and the MTT Source Boundary

This paper proves standard finite-time and finite-support scattering limits. A constant perturbation acting for a finite duration produces a sinc-shaped transition kernel, and its normalized square converges distributionally to the energy-conservation delta as the duration tends to infinity. Smooth windows give the corresponding approximate identities through Fourier transformation and Plancherel normalization, while finite spacetime support yields finite energy-momentum kernels whose infinite-support limits are the usual scattering deltas. The MTT interpretation is conditional: these kernels become MTT bookkeeping outputs only if selected dynamics derives the preparation, switching, detector, or finite-time profile. Finite experimental duration must not be identified with a fundamental admissibility width. Exact conservation continues to follow from the symmetry assumptions of the closed theory.