The final inequality above holds by the FKG inequality (Fortuin et al., 1971). It follows that where the final equality holds under H$_{0}$ by a Taylor expansion of z ↦→ z (2 z 2 − 3) e − z 2 and analogous calculations to those used in (A.10). We note that the second equality above uses a different F and a different big-O term than the line above, and that the big-O term can be upper bounded by C ‖ M TU P ‖ 4 4 ,P$_{0}$ for a constant C > 0. Plugging (A.8), (A.9), and (A.11) into (A.7), applying the triangle inequality, and using the bounds on B gives the result. We give a lemma before proving Theorem 3. Lemma A.6. Let K$_{P}$ ≜ ‖ L RS P ‖$_{1}$$_{,P}$$_{0}$ + ‖ M RS P ‖ 2 2 ,P$_{0}$ for all P ∈ M . If H$_{0}$ holds, then for all P ∈ M , where O ′ ⊆ O is some P$_{0}$ probability 1 set. More generally, for all P$_{0}$ ∈ M , Proof of A.6. For T,U ∈ S , we have that Above we have omitted the dependence of Γ TU on ( o$_{1}$,o$_{2}$ ), T and D T P on o$_{1}$ , and U and D U P 51 Γ TU P = $^{[}$1 + 2( T$_{P}$ − U$_{P}$ ) D U P $^{]}$e − [ T$_{P}$ − U$_{P}$ ] 2 − 2 $^{[}$( T$_{P}$ − U$_{P}$ ) + $^{(}$2( T$_{P}$ − U$_{P}$ ) 2 − 1 $^{)}$D U P $^{]}$D T $_{P}$e − [ T$_{P}$ − U$_{P}$ ] $^{2}$. ∣ $_{∣}$P 2 $_{0}$Γ$_{P}$ − ψ$_{0}$ ∣ ≲ K$_{P}$. sup o$_{1}$ ∈O ′ | P$_{0}$ Γ$_{P}$ ( o$_{1}$, · ) | ≲ K$_{P}$, E$_{P}$ 2 0 [ $^{(}$[ T − T$_{0}$ ] 2 + [ U − U$_{0}$ ] 2 $^{) (}$2 B 2 − 1 $^{)}$e − B 2 $^{]}$+ 2 3 E$_{P}$$_{0}$ $^{[}$B $^{(}$2 B 2 − 3 $^{)}$( B$_{0}$ − B ) $^{3}$e − B 2 ] = 4 3 E$_{P}$ 2 0 [ $^{(}$[ T − T$_{0}$ ] 3 − [ U − U$_{0}$ ] 3 $^{)}$B (2 B 2 − 3) e − B 2 $^{]}$+ F ( P,P$_{0}$,T,U ) + O ( ‖ M TU P ‖ 4 4 $_{,P$_{0}$}$) = F ( P,P$_{0}$,T,U ) + O ( ‖ M TU P ‖ 4 4 $_{,P$_{0}$}$) , (A.11)