Let \(N=p^d\) and retain the constructed full character, canonical beta extension, and compatible central completion. Both families give an actual irreducible admissible supercuspidal candidate of \({\mathrm{GL}}_N(F)\) with torsion number 1.
For odd \(p\),
\[
\begin{aligned}
{\operatorname{Sw}}(\pi)&=p^{d-1}(p+1)h_d,\\
\dim_{{\mathbb F}_p}(J^1/H^1)&=N-1,\quad\dim\eta=p^{(N-1)/2},\\
a(\pi^\vee\times\pi)&=p^{2d}-1+\sum_{i=1}^d(p^{2i}-p^{2i-2})h_i.
\end{aligned}
\]
For \(p=2\),
\[
\begin{aligned}
{\operatorname{Sw}}(\pi)&=3\,2^{d-1}h_d,\\
\dim_{{\mathbb F}_2}(J^1/H^1)&=N-2,\quad\dim\eta=2^{N/2-1},\\
a(\pi^\vee\times\pi)&=N^2-1+\sum_{i=1}^d3\,4^{i-1}h_i.
\end{aligned}
\]
Scope
These identities belong to the same constructed candidate. Matching numerical profiles does not identify a source Weil atom, an endo-class, or H5 pair factors.