Coherent Green Functions from Declared Projection Kernels Elliptic Results and the MTT Source Boundary

This paper develops a worked finite-source sequel to the observation that a Dirac delta can arise as the zero-width limit of a bounded projection kernel. For a positive self-adjoint elliptic operator, finite spectral projectors and heat-kernel filters produce smooth finite-width sources, smooth responses, finite diagonal values, and distributional convergence to the ordinary Green kernel only in the singular limit. A worked circle model displays both the band-limited and heat-filtered constructions explicitly and recovers the point-source equation as the bandwidth tends to infinity or the heat scale tends to zero. Within Modal Triplet Theory, these results provide a conditional downstream interface rather than a universal replacement rule. A selected physical realization must still supply the elliptic operator, projector or filter, its width, and the map from an internal or Euclidean spectral problem to the claimed external source. It must also prove that covariance, gauge constraints, locality or causal support, and boundary conditions survive. Different finite kernels can share the same sharp limit while giving different finite predictions. The rigorous result is therefore the elliptic response theorem and its singular limit; physical kernel selection remains open.