Completed results and the remaining physical application

The main mathematical result is Theorem thm:main: a conditional decision for one fixed, complete family, with intrinsic coefficient extraction and separately proved family and differentiated remainders.

The revision also supplies:

1. the finite source generator and propagated-output conversion;

2. the density derivative, with its identification condition;

3. the full seven-row, three-vertex inventory;

4. exact characteristic coordinates, density, and shifted cones;

5. a complete translation representation using only defined tensor products and compact pushforwards;

6. coefficient uniqueness, reference-dependent support, profile comparison, and normalization criteria;

7. an evaluated nonphysical diagnostic in the appendix.

For the designated physical graph, two distinct tasks remain:

1. identify the actual covariance and density-map package, without substituting diagnostic data;

2. prove a complete origin-germ certificate, direct scale-integrability bound, or another controlled complete asymptotic calculation for that package.

Neither task is discharged by relabeling an arbitrary parameter as physical. In particular, the appendix's ${\delta }^{-3},{\delta }^{-2},{\delta }^{-1}$ powers are not assigned to the physical family.

Thus this manuscript releases a self-contained conditional theorem and a completed diagnostic, but not an evaluated physical convergence or failure verdict, an authorized physical extension, or a normalized quantum coefficient.