${\mathcal{A}}= {C^{\infty}_c}(S^*_HM)\rtimes G$ and choose any linear splitting $\sigma:{\mathcal{A}}\to \Psi^0_{H,c}(M)\rtimes G$ of the leading symbol homomorphism. $\sigma$ is multiplicative up to the ideal $\Psi_{H,c}^{-1}(M)\rtimes G$. By the universal property of the tensor algebra, $\sigma$ extends to an homomorphism $\sigma_*:T{\mathcal{A}}\to \Psi^0_H(M)\rtimes G$ respecting the ideals, whence a morphism of extensions $$\xymatrix{ 0 \ar[r] & J{\mathcal{A}}\ar[r] \ar[d]_{\sigma_*} & T{\mathcal{A}}\ar[r] \ar[d]_{\sigma_*} & {\mathcal{A}}\ar[r] \ar@{=}[d] \ar@{.>}[dl]^{\sigma} & 0 {\vspace{2mm}}0 \ar[r] & \Psi_{H,c}^{-1}(M)\rtimes G \ar[r] & \Psi_{H,c}^0(M)\rtimes G \ar[r] & {C^{\infty}_c}(S_H^*M)\rtimes G \ar[r] & 0 }$$ Observe that the cocycle $\phi$, viewed as a cyclic 1-cocycle over $\Psi_{H,c}^0(M)\rtimes G$, vanishes on the large powers of the ideal $\Psi_{H,c}^{-1}(M)\rtimes G$ because it involves the Connes-Moscovici residue. Hence the composite $\phi\circ\sigma_*$, which is a bilinear form on $T{\mathcal{A}}$, extends to a cyclic 1-cocycle over ${\widehat{T}}{\mathcal{A}}$. By [@Per2012] Corollary 2.6, this cocycle is precisely a representative of the periodic cyclic cohomology class $\partial [\tau] \in {\mathrm{HP}}^1({C^{\infty}_c}(S^*_HM)\rtimes G)$. Therefore we obtain

\[pcocycle\] Let $0\to \Psi_H^{-1}(M)\rtimes G \to \Psi_H^0(M)\rtimes G \to {C^{\infty}}(S_H^*M)\rtimes G \to 0$ be the fundamental extension in the Heisenberg pseudodifferential calculus. Then the image $\partial[\tau]$ of the canonical trace localized at units under the excision map $\partial : {\mathrm{HP}}^0(\Psi_{H,c}^{-1}(M)\rtimes G) \to {\mathrm{HP}}^1({C^{\infty}_c}(S_H^*M)\rtimes G)$ is represented by a 