Lemma 45 ([Derivatives of the localized observable]). The parameter β>0 can be chosen so that there are a₀>0 and 0<δ<0.01 with, for every disorder matrix J , 1 n2 ∑_e|∂_e^4Φ(J)| , ≤n^-a₀ , ; 1 n2 ∑_e|∂_e^kΦ(J)| , ≤n^δT^ (k-4)/2 , 5≤k<K, …

On the support of a term containing a differentiated history, the static support conditions hold and ∑_qℓ_q≤2n^ -10 . This follows from the supports of Γ and its derivatives, and of χ and its derivatives. Split each ζ_q into its positive and negative parts. The base masses of these two checked integrals are at most n^ O(β) . Indeed the post-check kernel caps at the ends of the one or two legs dominate the joint law of the two observation spins by (C'N(cT))μ^ ⊗2 . For a stationary query the first spin already has law μ ; for a uniform-start query the first leg cap holds from each initial spin. Choose the first dyadic d_*≥cT . Then d_*≤2cT , r_2(d_*)≤r_2(cT) , and a_2(d_*)≥c_1n^ 3β . The pair cutoff therefore gives 𝔼_q[|ζ_q|∏_Iw_I] ≤n^ 1/3 r_2(cT) +n^ 1/3 (C'N(cT)-c n^ 3β ) ≤C n^ 2β +n^ -A' for arbitrary fixed A' . We used N(cT)=O(n^β) . Apply the ordinary chain rule to χ(n^ 10 ∑ℓ_q) and to F(Y) . A term has at most K positive-order history coefficients, with total order at most the differentiating order; repeated histories are represented by independent copies. The coefficients from derivatives of F are evaluated at the base Y(J) , hence are common across e , and their total absolute sums are bounded by . For loss factors, expand . The mass factors in Lemma 44 ([Averaged derivatives of histories]), summed over all choices of queries and first-failure nodes and multiplied by their n^ 10 factors, give at most (n^ 10 ∑_ q,I ℓ_ q,I )^r≤2^r when there are r loss factors. Trace mass factors cost at most n^ O_K(β) . The additive precisions in that lemma are chosen after the polynomial number of nodes and queries and after the factors n^ 10K , so their total is negligible. If a static cutoff is also differentiated, its pointwise derivative bound from Lemma 30 ([Static support]) multiplies these estimates. If only the static cutoff is differentiated, that bound is O(n^ -b ) for some b=b(β)>0 , and its remaining outer factor is bounded. Consequently the fourth averaged derivative is at most n^ -a+C'_Kβ ( n)^ C' +O(n^ -b ) , higher averaged derivatives are at most n^ C'_Kβ ( n)^ C' T^ (k-4)/2 , and the supremum bound is at most n^ C'_Kβ ( n)^ C' T^ j/2 . These estimates hold everywhere: outside the indicated supports the corresponding product-rule terms vanish. Choose β after K so that C'_Kβ<a/2 and C'_Kβ<0.005 , while satisfying the preceding sections' smallness requirements. Absorb logarithms by a still smaller fixed power. This gives-- for some a_0>0 and δ<0.01 .