Polynomial growth, scalar diagnostics, and raw divergence

The following supplied records are comparisons only Em01B,Em01C,Em02D:

• Em01-B, version 3, gives polynomial smoothing growth for its retained contraction. A bound by a negative power of a smoothing scale is not convergence through a characteristic locus. It is not the $O(\epsilon )$ estimate proved here.

• Em01-C, version 3, separates its scalar aligned-null expansion from the source-specific complete-sum characteristic estimate. Its scalar diagnostic is not the completed physical coefficient and not the present coordinate kernel.

• Em02-D, version 4, has a fixed-model separated detecting test with

  $$t_{\lambda }({\Phi }_A)=c_A{\lambda }^{-3}+O({\lambda }^{-2}),c_A\ne 0.$$

  That raw family has no weak limit. A correction supported on the excluded attachment cannot change a detecting test supported away from it.

The last nonconvergence follows because weak convergence would make $t_{\lambda }({\Phi }_A)$ bounded, contrary to the displayed asymptotic. An $O({\lambda }^{-2})$ remainder does not by itself establish a subtracted finite part. None of these distribution families is replaced or repaired by Theorem thm:convergence.