Are text in the derivation files look like this 'b := (1/L)∫ϕ⁴ (0 < b ≤ a ≤ 1); Φ := ϕ²̂; g := ϕ²⋆ϕ² (autocorrelation); A_ϕ := ϕ⋆ϕ; w := 1. Facts (checked): Φ(0) = aL; pair-correlation kernel R_kernel = |ϕ̂|² ≥ 0 by construction; ϕ²̂ = A_ϕ̂... (ϕ² = ϕ·ϕ ⟹ ϕ²̂ = (ϕ̂∗ϕ̂)/2π conventions aside); Φ² = ĝ on ℝ; ∫Φ² = 2πg(0) = 2πbL (Fourier inversion at 0: g(0) = ∫ϕ⁴ = bL ✔); sandwich (5.2): (L−2w−|y|)₊ ≤ g(y) ≤ A_ϕ(y) ≤ (L−|y|)₊ ✔ (from 1_[−L/2+w, L/2−w] ≤ ϕ² ≤ ϕ ≤ 1_[−L/2,L/2]). Decay envelope (5.3): max(|ϕ̂(r)|, |Φ(r)|) ≤ ϑ(r) := min(L, 2/|r|, C_χ/(wr²)), with (5.4) ∫₀^∞ ϑ = 4 + 2 log(C_χL/4w) ≪ log L; ∫ϑ²|r| ≪ log L; ∫ϑ² ≤ 8L. [Structure: L¹-norm ≪ log L is the source of the log log T in the final rate.]' ??