Lemma 30 ([Pair energy after local loss]). There is a sequence ε_M→0 such that, for every M≥2 and every q≥30M , ∑_b=1^2q-1 B_b-1^*B_b-1 ≤∑_b>M B_b-1^*B_b-1 +ε_MI F_q.

The following two elementary estimates specialize the general pair localization bounds to the cutoff used here. In particular, the exponential estimate on distant rows will let us preserve coefficient one on their energy. Both follow directly from the coefficients in . For b≤q/2 and every fixed integer r≥0 , ∑_ i<j; i+j=b |v_ b-1 (i,j)|\,|i-b/2|^r ≤C_r b^ r/2+1/4 . For b>10M and q≥30M , ∑_ i<j, i+j=b; i<3M |v_ b-1 (i,j)| ≤Cb^ 3/2 e^ -c_1b . Constants may change after decreasing c_1>0 . Here is a direct verification. Set p_ q,b (i)= qi q b-i 2qb . The exact pair formula gives |v_ b-1 (i,b-i)|^2 =(2(2q-1)(2i-b)^2)/(b(2q-b))p_ q,b (i), p_ q,b (i+1) p_ q,b (i) =((q-i)(b-i))/((i+1)(q-b+i+1)). When b≤q/2 , these ratios, their inverses, and symmetry about b/2 show p_ q,b (i)≤Cb^ -1/2 -c_2(i-b/2)^2/b. For completeness, on the upper half i≥(b-1)/2 , the first factor in the ratio is at most one and the second is at most -(2i+1-b)/(b+1) . Summing logarithms from a middle index gives the Gaussian upper bound relative to the middle value. In the range |i-b/2|≤√b , the absolute value of that logarithm is at most C(1+|i-b/2|)/b . Summing these lower bounds and using ∑_i p_ q,b (i)=1 bounds the middle value by C/√b . Insertion in, followed by summing Gaussian moments, proves. To prove, there is nothing to check if b>q+3M , because the sum is empty. Otherwise b≤q+3M≤11q/10 , so 2(2q-1)/[b(2q-b)]≤C/b in . For every allowed i≤2b/5 and b≥20 , the ratio in is at least 4/3 . Starting with i<3M<3b/10 and taking at least b/10-O(1) steps toward 2b/5 gives p_ q,b (i)≤Ce^ -c b , since each probability is at most one. The square root of is therefore at most C√b\,e^ -c_1b , and there are at most b terms. This proves . The modified row vanishes when b≤M . For M<b≤10M , put x=b/(2M) and t=T(x) . The smooth even function z↦T(x+z)T(x-z) has zero first derivative at zero. By , every derivative of positive order there is bounded by C_r√1-t . Taylor expansion through degree 41 , with a bounded remainder of order 42 , and yield B_ b-1 =t^2 B_ b-1 +E_b, \|E_b\|≤CM^ -3/4 √1-t+CM^ -20 . We used \|c_jc_i\|≤1 . More explicitly, the degree- r contribution is bounded by C_rM^ -r b^ r/2+1/4 √1-t, 2≤r≤41, and the remainder is at most CM^ -42 b^ 21+1/4 . Let δ=M^ -8 and s=1-t+δ . For any vector φ , Cauchy's inequality gives \|(t^2B_ b-1 +E_b)φ\|^2 ≤(t^4+s)\|B_ b-1 φ\|^2 +(1+t^4/s)\|E_bφ\|^2. Since t^4+s≤1+δ , bounds the second term by CM^ -3/2 \|φ\|^2 . Summing over the O(M) central rows costs CM^ -1/2 \|φ\|^2 . The additional coefficient δ also disappears in this error: with r=0 implies ∑_ M<b≤10M \|B_ b-1 \|^2≤CM^ 3/2 . Finally, for b>10M the coefficients of B_ b-1 -B_ b-1 are supported where i<3M or j<3M . The increasing ordering reduces this to i<3M . Equation bounds this difference in operator norm by Cb^ 3/2 e^ -c_1b , while normalization of the pair vector gives \|B_ b-1 \|≤√b . The norm of the difference of the two row squares is therefore summable over b>10M , with sum at most Ce^ -c_3M . Combining the three ranges proves the assertion, with ε_M≤C(M^ -1/2 +M^ -13/2 +e^ -c_3M ) .