Raw convergence or fixed-test failure

If all coefficients in (eq:singularsum) vanish, (eq:mainrem) gives strong convergence to $T^{\rho }$.

Conversely, choose the smallest degree with a nonzero nonsummable coefficient and, within that degree, the highest logarithmic power whose coefficient is nonzero. There is one fixed compact joint test ${\Phi }_{\ast }$ detecting it. For that test, the corresponding scalar term dominates all larger degrees and all lower logarithmic powers at the same degree.

If the degree is negative, an inverse power times a nonzero logarithmic polynomial is unbounded. If the degree is zero, the highest surviving logarithmic degree is at least one and is also unbounded. The finite part and the remainder cannot cancel this growth.

This proves all raw-family assertions of Theorem thm:main. The admissibility statement is addressed in Section sec:admissibility.