Problem, contribution, and logical dependencies

The fixed-kernel problem is

The internal distribution is kept exact. Only the grouped crossed array is smoothed.

The supplied Em01-B and Em01-C excerpts describe positive-scale regularization, moderate growth, characteristic coordinates, and an exact translation representation Em01B,Em01C. The supplied project proposal attributes a homogeneous-origin route to Em01-D-v2. These are statements of project provenance: no unprovided predecessor proof is needed for the results below. All defining formulas and analytic arguments used here are stated in this manuscript.

The scale-integrability and homogeneous-origin methods are infrastructure, not claimed as new analytic discoveries. The contribution is their coefficient-level organization, the operational verification procedures, and the evaluated complete specialization.

The principal deliverables are:

1. the complete-expansion decision theorem (Theorem thm:decision);

2. the exact-family verification protocol and its local-to-global requirements (Sections sec:routes and sec:protocol);

3. the evaluated expansion, fixed detecting test, and conormal coefficient bounds (Section sec:application).