Lemma 15. The calibrated constants satisfy W(E_)=6K+λ+2πμ.

Take 𝕋=^2/Λ_ and N=1 . Its regular Voronoi hexagon supplies six compatible sectors with d=p and α=β=π/6 . The radial normalization and summability proved above supply the hypotheses of Proposition 10 ([Dual identity]) with d_0=p , and the torus has area one. This application uses no square-torus minimizer or incidence theorem. On each sector p≤t≤R_0 , u_0(r)=γ(r) and u_m(r)=(-1)^mc_mr^ k_m for r≤t . The edge trace and mode cancellation established above therefore give H(r,φ)=γ(r)+∑_ m≥1 c_mr^ k_m (k_mφ) =h_-C_. All six normals differ by multiples of π/3 , which leaves these cosines unchanged. Hence ∇H=∇h_ and the nonnegative square in is zero. Each sector has area 1/6 and angle π/3 ; its right side in is 2g_p(x_0)=K . Summing the six contributions gives the cell energy 6K+λ+2πμ . Lemma 4 identifies this cell energy with W(E_) , proving .