Proposition 15 ([Positive-mixture transfer]). Let C⊂² be locally finite with centered disk density one. Suppose that, for every α>0 , _R→∞ 1 N_R ∑_x,y∈ C_R; x≠y e^-πα|x-y|² ≥∑_a∈A0 e^-πα|a|² . Then every smooth completely monotone …

Fix ε>0 and set F_ε(t)=g(ε+t) for t≥0 . It satisfies Lemma 14 ([Positive mixtures]), with a finite representing measure ρ_ε of mass g(ε) . Choose any real sequence R_n→∞ , discarding finitely many terms so that N_ R_n >0 , and define A_n(v)=1 N_ R_n ∑_ x,y∈ C_ R_n ; x≠y v^ |x-y|^2 (0≤v≤1). Every exponent is positive, so A_n is continuous and nonnegative on [0,1] , with 0^t=0 for t>0 . Define the nonnegative extended-valued function L(v)=∑_ a∈A0 v^ |a|^2 (0≤v≤1). For 0<v<1 , the Gaussian hypothesis with α=-(v)/π gives _n A_n(v)≥L(v). A full-radius lower-limit inequality holds along every sequence tending to infinity. The same comparison holds at the endpoints: A_n(0)=L(0)=0 , while A_n(1)=N_ R_n -1→∞=L(1) . The finite-sum representation, Fatou's lemma, and Tonelli's theorem now give , _n1 N_ R_n ∑_ x,y∈ C_ R_n ; x≠y F_ε(|x-y|^2); , =_n∫A_n(v)\,ρ_ε(v); , ≥∫_n A_n(v)\,ρ_ε(v) ≥∫L(v)\,ρ_ε(v); , =∑_ a∈A0 F_ε(|a|^2). Every integrand and summand is nonnegative, so these steps remain valid when the last sum is infinite. Since the sequence R_n was arbitrary, this is the full-radius lower bound for F_ε . Complete monotonicity gives g'≤0 , hence g(t)≥F_ε(t) for t>0 . Termwise comparison therefore yields E_g( C)≥ ∑_ a∈A0 g(ε+|a|^2) (ε>0). As ε0 , the summands increase to g(|a|^2) . Monotone convergence in the lattice sum proves. No limit in ε has been interchanged with the configuration lower limit, and no unshifted representing measure has been used.