the leading symbol map thus yield a morphism of extensions $$\vcenter{
\xymatrix{ 0 \ar[r] & \Psi_c^{-\infty}(M)\rtimes G \ar[r] \ar[d] & \Psi_{H,c}^0(M)\rtimes G \ar[r] \ar@{=}[d] & {\mathcal{S}}_{H,c}^0(M)\rtimes G \ar[r] \ar[d] & 0 {\vspace{2mm}}0 \ar[r] & \Psi_{H,c}^{-1}(M)\rtimes G \ar[r] & \Psi_{H,c}^0(M)\rtimes G \ar[r] & {C^{\infty}_c}(S_H^*M)\rtimes G \ar[r] & 0 }}$$ The cyclic cohomology class of the operator trace ${\mathrm{Tr}}_{[1]}$ localized at unit extends in a straightforward manner to a cyclic cohomology class $[\tau]\in {\mathrm{HP}}^0(\Psi_{H,c}^{-1}(M)\rtimes G)$. The latter is represented, for any choice of even integer $k>v+2h$, by the cyclic $k$-cocycle $\tau_k$ defined as follows: $$\tau_k(a_0,\ldots, a_k) = {\mathrm{Tr}}_{[1]} (a_0\ldots a_k)$$ for all $a_i\in \Psi_{H,c}^{-1}(M)\rtimes G$. By naturality of excision, the class $\partial[{\mathrm{Tr}}_{[1]}] \in {\mathrm{HP}}^1({\mathcal{S}}_{H,c}^0(M)\rtimes G)$ is the pullback of $\partial [\tau] \in {\mathrm{HP}}^1({C^{\infty}_c}(S_H^*M)\rtimes G)$ under the leading symbol homomorphism. Now the computation of $\partial[\tau]$ is fairly analogous to the above computation of $\partial[{\mathrm{Tr}}_{[1]}]$. We use the generalized Goodwillie theorem of Cuntz and Quillen [@CQ95], which states that the periodic cyclic cohomology of an associative algebra ${\mathcal{A}}$ is isomorphic to the periodic cyclic cohomology of its *completed tensor algebra* $${\widehat{T}}{\mathcal{A}}= \varprojlim_n T{\mathcal{A}}/(J{\mathcal{A}})^n\ ,$$ where the two-sided ideal $J{\mathcal{A}}\subset T{\mathcal{A}}$ is the kernel of the multiplication homomorphism $T{\mathcal{A}}\to{\mathcal{A}}$, $a_1\otimes\ldots\otimes a_n\mapsto a_1\ldots a_n$. We let 