Coherent-Sector Reduction in Modal Triplet Theory: Feshbach Control, Projector Stability, and Local Universality

A coherent-sector truncation is controlled only after the retained subspace, full closed operator, block domains, spectral parameter, eliminated-sector resolvent, and comparison norm have all been specified. A spectral gap alone is not a complete truncation theorem. This paper gives a domain-explicit coherent-sector reduction for a self-adjoint block operator. It first resolves a basic ambiguity: an exact spectral projector of the same operator reduces that operator, so its off-diagonal blocks vanish and the truncation is exact. A nonzero Feshbach correction therefore requires a projector selected from a reference operator, symmetry, Galerkin space, or approximate coherent criterion. For a reference-selected projector, the paper states the exact Feshbach-Schur reduction at a declared spectral parameter and bounds the omitted-sector self-energy by the product of the two coupling norms and the eliminated-sector resolvent norm. It proves the inverse formula, gives a two-mode example, derives Riesz-projector stability under bounded perturbations, and establishes local operator-norm robustness with all block variations explicit. These results define a controlled local comparison class, not a global universality theorem. Resolvent closeness does not imply uniform-in-time dynamical closeness; graph-norm and form robustness require separate hypotheses; and an internal one-parameter unitary is not physical time without a model-specific clock and intertwining theorem. The result is a rigorous methods paper and an executable certificate format for future Modal Triplet Theory reductions.