Free multiplet, algebra, and the inherited symbols

The independent multiplet is $(\gamma ,\varphi ,c,\bar{c},b)$ with the free rule $s_0\gamma =\mathcal{R}c$, $s_0\varphi =s_{0}c=s_{0}b=0$, $s_0\bar{c}=ib$. The gauge fermion of XI-B is

and elimination of $b=D\gamma$ gives $S_{\mathrm{gf}}=\int D_{\nu }\gamma D^{\nu }\gamma$ with Euler derivative $E_{\mathrm{gf}}[\gamma ]=2D^{\ast }D\gamma$ XIB,XIA. The free shell ${\mathcal{C}}_{\mathrm{sol}}$ is $-\square \mathcal{P}\gamma =0$, $b=D\gamma$, $P_m\varphi =0$, $\square c=\square \bar{c}=0$.

XI-E's Route 1 constructed the polynomial on-shell Hadamard algebra ${\mathcal{E}}_H=({{\mathcal{F}}_{\mathrm{mc},\mathrm{pol}}}^{\mathrm{sol}}[[\hslash ,\kappa ]],{\star }_H)$ with block data $H_{\gamma \gamma }=-\mathcal{P}{h}_0$, $H_{\gamma b}=D_y{H}_{\gamma \gamma }$, $H_{\varphi \varphi }=-h_m$, $H_{c\bar{c}}=-i{h}_0$, graded antisymmetric part equal to XI-B's causal matrix, and no state selected. Its domain is: