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

Proposition 10 ([Dual identity]). For compatible periodic Voronoi data on 𝕋=2/Γ with N≥1 distinct sites, lower edge distance d0>0 , and |𝕋|=N , let h be the mean-zero potential. Let I be finite or countable, let ki≥6 be integers with …

Lemma 4. Let F be a periodic admissible electric field or current, with simple charges and period cell P of area A . For either square or ball cutoffs with fixed boundary width and uniformly bounded gradients, its energy per unit area has …

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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 .

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