For a packet of orbit length \(r\) and closing scalar \(a\),
\[
\det(I-u\Phi\mid V^{I_K})=
\begin{cases}1-au^r,&\text{if inertia acts trivially},\\1,&\text{otherwise}.\end{cases}
\]
Thus tame tensor local factors multiply \((1-a_iu^{r_i})^{-1}\) over precisely the refined packets whose inertia exponent vanishes.
An inertia-trivial companion cycle has a Weil-fixed vector exactly when \(a_i=1\), and then its fixed space is one dimensional. Therefore
\[
\dim(\tau_\xi\otimes\tau_\eta)^{W_K}
=\#\{i:c_i=0,\ a_i=1\}.
\]
For an irreducible nontrivial tame atom of dimension \(n>1\), wild inertia acts trivially and inertia has no fixed vectors. Hence
\[
{\operatorname{Sw}}(\tau)=0,\qquad \operatorname{Art}(\tau)=n.
\]
Scope
These exact tame formulas do not establish the full wild pair-factor identities required by H5.