Finite Coherent Filters in Modal Triplet Theory: Operator Types, Exact Kernels, and Sectoral Boundaries

Finite coherent filtering is a useful common language for several Modal Triplet Theory constructions, but it does not by itself derive every physical sector in which a filter can be written. This paper gives the operator statement a precise domain. It distinguishes an orthogonal projector, a smooth spectral filter, and a positive measurement effect; proves positivity and contraction of the sandwiched filter without assuming commutativity; states the additional hypothesis needed for a modal eigenvalue formula; and derives the Euclidean heat kernel and its sharp delta limit. The current finite q79 Hessian supplies an exact six-mode illustration with a rank-two coherent kernel and rank-four complement. A separate no-go theorem proves that filtering only internal modes cannot suppress arbitrarily high four-dimensional momenta. Gauge, gravitational, Lorentzian, and measurement applications therefore require their own compatible operators, constraints, and source maps. The result is a rigorous operator toolkit and a clear list of transfer obligations, not a universal derivation of quantum field theory or quantum gravity.