Lemma 11 ([Scores at separated anchors]). Let U be a cell union and let D be any collection of anchors mutually separated by more than r , all within distance Ds of U . There is a coarse-cell cover S of the (D+R_A)s neighborhood of U , …

Lift the anchors periodically to ℝ^3 . Their distinct lifts remain separated by more than r once L>2r . For a point y and a_t=√1+t , packing disjoint balls of radius r/2 bounds the number of anchors in a ball of radius u by C(1+u/r)^3 . Summing in shells of width (r,a_t) , with the Gaussian weight, gives ∑_ a∈ D e^ -d(y,a)^2/[1000(1+t)] ≤C(1+(1+t)^ 3/2 r^ -3 ). For the torus expression, select a nearest lift of each anchor; it is bounded by the full lifted Gaussian sum just estimated. Only y in S can contribute after the distance indicator in (15), because the anchors are within Ds of U . Multiplication by w(t) now proves (21), since w(t)(1+t)^ 3/2 ≤C . This proof is deterministic for every set of separated anchors and therefore permits adaptive choices. The integral of the scalar weight on the right of (21) is at most C(1+δr^ -3 )≤B_0 . Divide by B_0 and regard the result as an average of the variables n_S(ω(t)) , padding with a zero variable if the total averaging mass is less than one. Jensen's inequality for the exponential, stationarity of the one-time closed law, and (11) at λ=1/2 give 𝔼_ cl [ B_0^ -1 C∫_0^ δ/2 (w(t)+r^ -3 )n_S(ω(t))\,dt ] ≤𝔼_Pe^ n_S ≤𝔼_Pe^ n_ φ_S ≤e^ C_vρ|S| . Time reversal gives the other half of the assertion.