Proofs for Section 2 We give two lemmas before proving Theorem 1. Lemma A.1. For any T, U ∈ S and any fluctuation submodel dP$_{t}$ = (1 + th$_{1}$ + t $^{2}$h$_{2}$ ) dP$_{0}$ , we have that, for all ˜ t in a neighborhood of zero, Proof of A.1. We have that where the derivative is passed under the integral in view of (S2). The result follows by the chain rule. For each T, U ∈ S , define We have omitted the dependence of D TU on P$_{0}$ in the notation. We first give a key lemma about the parameter Φ $^{TU}$. Lemma A.2 (First-order canonical gradient of Φ $^{TU}$) . Let T and U be members of S . Then Φ TU has canonical gradient D TU at P$_{0}$ . 41 D $^{TU}$( o ) ≜ − 2Φ $^{TU}$( P$_{0}$ ) +$^{∫}$ $^{{}$2 [ U$_{0}$ ( o$_{1}$ ) − T$_{0}$ ( o )] D T 0 ( o ) + 1 $^{}}$e − [ T$_{0}$ ( o ) − U$_{0}$ ( o$_{1}$ )] $^{2}$dP$_{0}$ ( o$_{1}$ ) +$^{∫}$ $^{{}$2 [ T$_{0}$ ( o$_{1}$ ) − U$_{0}$ ( o )] D U 0 ( o ) + 1 $^{}}$e − [ T$_{0}$ ( o$_{1}$ ) − U$_{0}$ ( o )] $^{2}$dP$_{0}$ ( o$_{1}$ ) . ˙ Φ TU ˜ t = d dt ∫ ∫ e − [ T$_{t}$ ( x T 1 ) − U$_{t}$ ( x U 2 )] 2 { 2 ∏ j =1 $^{[}$1 + th$_{1}$ ( o$_{j}$ ) + t $^{2}$h$_{2}$ ( o$_{j}$ ) ] } dP$_{0}$ ( o$_{2}$ ) dP$_{0}$ ( o$_{1}$ )$^{∣}$ ∣ ∣ t = ˜ t = ∫ ∫ d dt e − [ T$_{t}$ ( x T 1 ) − U$_{t}$ ( x U 2 )] 2 { 2 ∏ j =1 $^{[}$1 + th$_{1}$ ( o$_{j}$ ) + t $^{2}$h$_{2}$ ( o$_{j}$ ) ] } ∣ ∣ ∣ t = ˜ t dP$_{0}$ ( o$_{2}$ ) dP$_{0}$ ( o$_{1}$ ) , ˙ Φ TU ˜ t = ∫ [ ∫ e − [ T$_{˜}$ $_{t}$( x T 1 ) − U$_{˜}$ $_{t}$( x U 2 )] $^{2}$dP$_{˜}$ $_{t}$( x T $_{1}$)$^{]}$ $^{[}$h$_{1}$ ( o$_{2}$ ) + 2 ˜ th$_{2}$ ( o$_{2}$ ) $^{]}$dP$_{0}$ ( o$_{2}$ ) +$^{∫}$ [ ∫ e − [ T$_{˜}$ $_{t}$( x T 1 ) − U$_{˜}$ $_{t}$( x U 2 )] $^{2}$dP$_{˜}$ $_{t}$( x U 2 )$^{]}$ $^{[}$h$_{1}$ ( o$_{1}$ ) + 2 ˜ th$_{2}$ ( o$_{1}$ ) $^{]}$dP$_{0}$ ( o$_{1}$ ) − 2$^{∫ ∫}$ $^{[}$T$_{˜}$ $_{t}$( x T $_{1}$) − U$_{˜}$ $_{t}$( x U 2 ) ] [ d dt T$_{t}$ ( x T $_{1}$)$^{∣}$ ∣ ∣ t = ˜ t − d dt U$_{t}$ ( x U 2 )$^{∣}$ ∣ ∣ t = ˜ t ] e − [ T$_{˜}$ $_{t}$( x T 1 ) − U$_{˜}$ $_{t}$( x U 2 )] $^{2}$dP$_{˜}$ $_{t}$( x U 2 ) dP$_{˜}$ $_{t}$( x T $_{1}$) .