-   APS:

    \[eq:coeffs-first\]$$\begin{aligned}
          f_\pm(v) &= [B(v\pm 4)]^{-\frac{1}{2}}\tilde{f}(v\pm 2)[B(v)]^{-\frac{1}{2}} , \\
          f_o(v) &= [B(v)]^{-1} [f_+(v)+f_-(v)] ,
        \end{aligned}$$

    where [@rep-diff]

    $$\begin{aligned}
          \tilde{f}(v) &= (3\pi G/8) |v| \big| |v+1| - |v-1| \big| \\
          B(v) &= (27/8) |v| \big| |v+1|^{1/3} - |v-1|^{1/3} \big|^3
        \end{aligned}$$

-   sLQC:

    \[eq:fslqc\]$$\begin{aligned}
          f_\pm(v) &= (3\pi G/4) \sqrt{v(v\pm 4)}  (v\pm 2) , \\  
          f_o(v) &= (3\pi G/2) v^2 ,
        \end{aligned}$$

-   MMO:

    \[eq:fmmo\]$$\begin{aligned}
          f_{\pm}(v) &= C g(v\pm 4)s_{\pm}(v\pm 2)g^2(v\pm 2)s_{\pm}(v)g(v), \notag \\
          f_o(v) &= C g^2(v) [ g^2(v-2)s^2_-(v) + g^2(v+2) s^2_+(v) ] , \tag{\ref{eq:fmmo}}
        \end{aligned}$$

    where

    \[eq:Theta-form-last\]$$\begin{aligned}
          g(v) &= \big| |1+1/v|^{1/3} - |1-1/v|^{1/3} \big|^{-1/2} , \\
          s_{\pm}(v) &= \sgn(v\pm 2) + \sgn(v) , \\
          C &= \pi G/12 .
        \end{aligned}$$

The operator $\Theta$ is denoted, respectively, by $\Theta_{\operatorname*{APS}}$, $\Theta_{sLQC}$ and $\Theta_{\operatorname*{MMO}}$ and is well defined in all the listed prescriptions in particular for $\varepsilon=0$ (see the detailed discussion in [@klp-aspects] for APS and [@mmo] for MMO).

$\bullet$ *Building a physical space out of states annihilated by the constraint.* Since the operator is essentially self-adjoint [@kl-flat], this step can be realized via systematic procedure of the *group averaging* [@gave; @gave2]. On the other hand its form selects another natural way of finding the solutions [@aps-imp; @GaSch] which in this case is 