The partition function of tricritical PET gravity was calculated in section \[sec2\]. $$\label{ZcPET}\hskip 1cm
 Z_{\rm cPET}=Z_{\rm Ein}\cdot\frac{1}{\det(-\nabla^2-2)_2^{TT}}$$ Using formula (\[zdelta\]) we find $$\label{?PET?}\hskip 1cm
 Z_{\rm cPET} =Z_{\rm Ein}\cdot\bigg(\prod_{m\geq0}^\infty|1-q^{m+2}|^{-2-2m}\bigg)^2 
                =Z_{\rm Ein}\cdot\bigg(\prod_{m\geq2}^\infty\prod_{n\geq0}^\infty\frac{1}{|1-q^{m+n}|^2}\bigg)^2 \,.$$ For $q=\bar{q}$ this agrees with (\[Zpet\]). Just as in (\[?ZNMG?\]) we summed over one factor square root of $\det(-\nabla^2-2)_2^{TT}$ for each pole of the retarded correlators. To obtain (\[?PET?\]) we need three such factors and indeed one finds triple poles at a tricritical point [@Kim:2012rz].

Now that we identified where the separate terms in (\[ZcNMG\]) and (\[ZcPET\]) come from we can address the issue of truncating the theory.

The calculations in (\[?ZNMG?\]) and (\[?PET?\]) suggest that the conjecture of [@Denef:2009kn] holds for NMG and PET gravity at the (tri-)critical point. Based on this finding, we go on to make use of the quasi-normal mode method and excise certain modes from the spectrum by ignoring their contribution to the partition function. For example, in critical NMG an intriguing idea would be not to sum over the spin-two modes at all because they correspond either to null states or negative energy states. Of course this would alter formula (\[zdelta\]) in a drastic way: $$\hskip 1cm
 Z_{\Delta_2}=1 \,.$$ In the following we 