Lemma 10 ([Gap sums at a fixed period]). For fixed 1≤r<R₀ , uniformly in h and j∈[K/3,2K/3] , Σ_r,h (j)≤C_r(Km)^-1/2 λ^r-1 , Σ_r,h^bd (j) ≤C_r,s₀ (Km)^-1/2 m^-1/2 λ^r-2 (r≥2).

The left gaps 0,…,h-1 sum to j-12 , and the right gaps h,…,r sum to K-j+12 . Each total is order K . Select a largest gap on each side; both are at least c_rK , so their two factors cost C_r(Km)^ -1 . Given j and the other r-1 gaps, the two total constraints determine the selected gaps. Each remaining free sum costs ∑_ γ>0; γ≤K (mγ)^ -1/2 ≤C_*√K/m=C_*λ on either the integer or half-integer lattice. Grouping by the at most C_r choices of selected gaps and multiplying by √Km proves the first bound. For the restricted sum, an internal gap at most |I| cannot be either selected largest gap for all sufficiently large K . Union over its at most r-1 possible indices. Its sum is at most ∑_ γ=1 ^ |I| (mγ)^ -1/2 ≤C_ s_0 m^ -1/2 instead of C_*λ . The other r-2 free sums are unchanged, proving the second bound.