Abstract
A local completion of a coarse amplitude does not by itself supply a local low-energy selector. We assemble a conditional theory of common preparation maps, induced norms, effective dynamics, and matched quadratic effects for a supplied positive one-dimensional chain. An exact noncommutative graph identity gives a second-order whole-window dynamics bound without isolating individual levels. Finite inverse and metric polynomials localize this common interface, while their approximation order remains distinct from the dynamics order. Applying the same block identity after an orthogonal static decomposition gives a higher-order common map: fifth-order raw dynamics and compressed metric defect require an inverse-square lower gap and a cubic contrast certificate. A coefficient-controlled estimate supplies the cubic factor on the first effective window; transferring it to the second window still requires a quantitative exterior-gap certificate. Exact low-mode projectors can remain globally supported. Positive Bernstein filters replace exact selection with pass, transition, and stop sectors, but a classical polynomial derivative bound excludes shrinking physical support for a fixed physical transition in the stated nonwrapping Jacobi setting. At fixed off-spectrum poles, explicit Chebyshev resolvent truncations have bounded physical support. Pole approach, normalization, and positivity restoration incur separate costs. An affine repair restores exact positivity without enlarging a support upper bound; square restoration has a different support and transmission budget. The contribution is the organized, source-bound combination of these estimates and obstructions, not a new general graph-subspace mechanism or rational-filter family. Carriers, couplings, preparation, action scale, and the physical interpretation of the supplied readout remain external.
Where it sits in the release
The preparation-and-selection paper of the branch, and the last of the fourteen. Carriers, couplings, preparation, action scale, and the physical interpretation of the supplied readout remain external.
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