Proposition 16. Assume for the functions and constants just constructed. Let 𝕋=²/Γ be a flat torus of area N , where N≥1 is an integer, and let v₁,…,v_N be distinct sites. Suppose their full periodic lift gives compatible periodic …

Proposition 12 ([From sector bounds to energy bounds]). Let d₀>0 , let I be finite or countable, let k_i≥6 be integers with repetitions permitted, and let B,D_i∈C¹((0,∞)) satisfy and with this lower distance d₀ . For every sector with …

Lemma 11 ([Signed endpoint parameters]). Let K<0 and d₀>0 . For each d≥d₀ , let G(d,·) be an even locally integrable real function on , and put g_d(x)=∫₀^xG(d,s)\, s. Suppose that, for every d≥d₀ , 2g_d(x)≥K (x≥0), ∫₀^∞0,-G(d,s)\, …

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The radial normalization and compact summability established above supply the radial hypotheses of Proposition 12 ([From sector bounds to energy bounds]) with d_0=b . Evenness of G and supply the hypotheses of Lemma 11 ([Signed endpoint parameters]) at this same lower distance. Combining its conclusion with gives J(S)≥K+λA(S)+μθ(S) for every sector with d≥b , including the signed endpoint cases. The compatible-torus conclusion now gives W_ 𝕋(h)≥N(6K+λ+2πμ) =NW(E_). The equality is the exact triangular calibration , not a numerical estimate of the triangular energy.

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Proposition 16. Assume for the functions and constants just constructed. Let 𝕋=2/Γ be a flat torus of area N , where N≥1 is an integer, and let v1,…,vN be distinct sites. Suppose their full periodic lift gives compatible periodic …

Proposition 12 ([From sector bounds to energy bounds]). Let d0>0 , let I be finite or countable, let ki≥6 be integers with repetitions permitted, and let B,Di∈C1((0,∞)) satisfy and with this lower distance d0 . For every sector with …

Lemma 11 ([Signed endpoint parameters]). Let K<0 and d0>0 . For each d≥d0 , let G(d,·) be an even locally integrable real function on , and put gd(x)=∫0xG(d,s)\, s. Suppose that, for every d≥d0 , 2gd(x)≥K (x≥0), ∫0∞0,-G(d,s)\, …

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Proposition 16. Assume for the functions and constants just constructed. Let 𝕋=²/Γ be a flat torus of area N , where N≥1 is an integer, and let v₁,…,v_N be distinct sites. Suppose their full periodic lift gives compatible periodic …

Proposition 12 ([From sector bounds to energy bounds]). Let d₀>0 , let I be finite or countable, let k_i≥6 be integers with repetitions permitted, and let B,D_i∈C¹((0,∞)) satisfy and with this lower distance d₀ . For every sector with …

Lemma 11 ([Signed endpoint parameters]). Let K<0 and d₀>0 . For each d≥d₀ , let G(d,·) be an even locally integrable real function on , and put g_d(x)=∫₀^xG(d,s)\, s. Suppose that, for every d≥d₀ , 2g_d(x)≥K (x≥0), ∫₀^∞0,-G(d,s)\, …