Theorem: Complete-expansion decision

Under Assumption ass:expansion:

1. $t_{\delta }-S_{\delta }\to T$ strongly in ${\mathcal{T}}^{\prime }$, with the compact-local rate (eq:remainder);

2. the raw family converges strongly exactly when all coefficients in (eq:singular) vanish;

3. if a complete nonsummable coefficient survives, there are one fixed compact joint test ${\Phi }_{\ast }$, an index $({\lambda }_{\ast },k_{\ast })$, and $c\ne 0$ such that

   $$t_{\delta }({\Phi }_{\ast })=c{\delta }^{{\lambda }_{\ast }}(\log \delta {)}^{k_{\ast }}(1+o(1)),$$

   where either ${\lambda }_{\ast }<0$, or ${\lambda }_{\ast }=0,k_{\ast }\ge 1$. Hence $|t_{\delta }({\Phi }_{\ast })|\to \infty$;

4. the nonsummable coefficients and $T$ are unique in this finite real power–logarithm class and intrinsic to the actual positive-scale family.

Proof. Every bounded test set has common compact support and bounded $\Vert \cdot {\Vert }_{m_L,L}$. Thus (eq:remainder) proves (i), and proves raw convergence when all nonsummable coefficients vanish.

Otherwise choose the smallest exponent having a nonzero coefficient, treating the degree-zero logarithms as exponent zero. Within that exponent choose the largest logarithmic degree with a nonzero coefficient. Its distribution is detected by one fixed compact test. Higher exponents are negligible relative to its scalar factor, since positive powers dominate every fixed power of a logarithm. Lower logarithmic degrees at the same exponent are also negligible. The finite term and remainder are negligible. This proves (eq:dominant), and excludes raw convergence even in the weak distributional sense.

For uniqueness, subtract two expansions and combine their finite inventories. A nonzero nonsummable difference would have an unbounded fixed pairing, while the difference of the finite terms and remainders has a finite limit. Thus all nonsummable differences vanish. Taking the limit identifies the finite distributions.