Lemma 9 ([Finite differences of a rank kernel]). Fix 1≤r<R₀ , an admissible central count, and a tagged tuple in K periods with coarse gaps γ_i . For disjoint sets D, V⊆1,…,r , put v'=| D|+2| V| . Then \|∏_h∈ D D_h ∏_h∈ V V_h W_j \|_…

Let S_i increase gap i 's length in the numerator of . Since T_h=S_ h-1 S_h^ -1 , D_h, =12(T_h-1)(1+T_h^ -1 ),, V_h, =12(T_h-1)^2T_h^ -1 ,; T_h-1, =(S_ h-1 -1)S_h^ -1 +(S_h^ -1 -1). Use also S_i^ -1 -1=-(S_i-1)S_i^ -1 . Expanding assigns exactly v' first length differences to gaps, with C_r terms and shifts of their lengths by O_r(1) . For an assigned term write b_i for the order on gap i ; thus ∑_i b_i=v' . Write (∇f)_g=f_ g+1 -f_g for the first length difference. At any integer length g≥1 , the order- b difference (∇^b f)_g has Fourier transform θ^b(e^ it -1)^b(1-θ+θe^ it )^g. On [-π,π] the last factor has modulus at most (-c_*δ^2g t^2) , since θ(1-θ)δ^2 . Parseval and Fourier inversion therefore give, respectively, \|(∇^b f)_g\|_2^2 ≤C_bδ^ -(2b+1) g^ -b-1/2 , \|(∇^b f)_g\|_∞^2 ≤C_bδ^ -(2b+2) g^ -b-1 . All shifted lengths are comparable to g_i for large m , because r is fixed and g_i m . Choose a largest coarse gap e , so its shifted length g_e_r Km . In the squared rank sum, solve for z_e and use the squared supremum there, with squared ℓ^2 sums on the other r gaps. For one assigned term F , the displayed bounds and f_ Km (l)^ -2 δ^2Km give \| F\|_2^2 , ≤C_rδ^ -(2v'+r) Km\, g_e^ -1-b_e ∏_ i≠e g_i^ -1/2-b_i ; , ≤C_rδ^ -(2v'+r) √Km\,m^ -v' ∏_ i=0 ^r(mγ_i)^ -1/2 . The second line uses g_i mγ_i , g_i m , and the extra factor g_e^ -1/2 _r(Km)^ -1/2 . Combining the C_r assigned terms proves.