Discrete vs continuum gravitational diagrams in the soft synchronous gauge

Due to the non-renormalizability of gravity, the perturbative expansion has sense, say, for its discrete simplicial (Regge calculus) version. A finite-difference form of gravity action has diffeomorphism symmetry at leading order over metric variations from site to site, and we add a term bilinear in $n^\lambda(g_{\lambda\mu}-g_{\lambda \mu}^{(0)})$, $n^\lambda=[1,-\varepsilon(\Delta^{(s)\alpha}\Delta^{(s)}_\alpha)^{-1}\Delta^{(s)\beta}]$, to "softly" fix the synchronous gauge $g_{0\lambda}=g_{0\lambda}^{(0)}=-\delta_{0\lambda}$ at $\varepsilon\to0$, thus removing singularities at $p_0=0$. For the symmetric derivative $\Delta^{(s)}_\lambda$, the propagator has a graviton pole at $\sin^2p_0=\sum^3_{\alpha=1}\sin^2p_\alpha$ or, at small $p_\alpha$, at $p_0$ close to 0 or $\pm \pi$. This pole doubling compared to the continuum does not arise from $\sin^2(p_0/2)=\sum^3_{\alpha=1}\sin^2(p_\alpha/2)$ obtained from the action $\check{S}_{\rm g}$ with the usual derivative $\Delta_\lambda = \exp (ip_\lambda)-1$ instead of $\Delta^{(s)}_\lambda=i\sin p_\lambda$ in some terms, including in the k part of some term, and $\Delta^{(s)}_\lambda$ in the 1-k part of that term. Given the propagator $\check{G}(n,\overline{n})$, we form a principal value type propagator $[\check{G}(n,n)+\check{G}(\overline{n},\overline{n})]/2$ by analytically continuing from real $n=\overline{n}$. Singularities are roughly resolved as $p_0^{-j}\Rightarrow[(p_0+i\varepsilon)^{-j}+(p_0-i\varepsilon)^{-j}]/2$ leading to separate diagram finiteness at $\varepsilon\to0$. We find that k=1 is needed for this prescription to work properly and match the continuum case. The gauge-fixing term needed for this propagator and its finiteness are considered, the ghost contribution is found to vanish at $\varepsilon\to0$. We use these results in arXiv:2601.03228. Calculations are illustrated by the electromagnetic (Yang-Mills) case.