Momentum-Conservation Deltas from Translation Symmetry and Finite Windows Exact Ward–Noether Closure and the MTT Bookkeeping Boundary

This paper proves that exact momentum-conservation deltas are distributional limits of finite interaction-window Fourier kernels. Gaussian windows give explicit finite-width kernels, while box and finite-time windows yield sinc or Dirichlet-type forms. The physical authority for exact vertex conservation is the Ward-Noether consequence of exact translation symmetry, and the exact delta must be preserved when that symmetry holds. Finite windows instead describe switching, finite volume or time, detector resolution, or an explicitly non-translation-invariant effective interaction; they are not a universal MTT softening of conservation. The circle-sector bookkeeping interpretation is conditional on a selected MTT source for the relevant finite window.