Dirac Delta Limits, Spectral Kernels, and the MTT Finite-Kernel Diagnostic Exact Approximate Identities, Constraint Tubes, and Scope Boundaries

Dirac delta distributions play several mathematically different roles in physics. They are identity kernels, point sources, hard constraints, conservation distributions, formal densities of continuous observables, and local gauge-slice selectors. This paper separates those roles before asking which admit finite-resolution representatives. On a compact elliptic background, spectral projector kernels converge to the diagonal delta distribution with an explicit Sobolev truncation error. Heat kernels provide a different, nonprojective approximate identity with a quantitative Sobolev error. For a smooth submersion between Euclidean spaces, normalized Gaussian tubes converge by the coarea formula to the correctly Jacobian-weighted constraint distribution on the zero level set. Compression of the canonical commutation relations by an orthogonal projection replaces the full inner-product kernel by the projected kernel exactly, but only in that declared compressed theory. These results support a restricted Modal Triplet Theory diagnostic: when a delta occurs in a downstream model, one should ask whether selected upstream geometry emits a finite projector, smoothing kernel, source profile, detector response, or constraint tube, and in which topology the sharp limit is controlled. They do not imply that every delta is a hidden projection, that a spectral gap fixes a physical width, or that finite smearing by itself solves gauge fixing, measurement, ultraviolet renormalization, or outcome selection.