Prove
Proposition 11 ([Preparation at rest]). Suppose the original data have a…Lemma 9 ([The decaying Neumann problem and strictification]). Assume g-δ=O_2(r^…Proposition 34 ([Prepared numerical bound]). For the prepared data, for every…Proposition
Apply the rest preparation and strict preparation in Proposition 11 ([Preparation at rest]) and Lemma 9 ([The decaying Neumann problem and strictification]), retaining the two closed forms. The magnetic support stays in the compact original region. There the strictly positive margin absorbs the arbitrarily small rescaling errors. The gluing and bending annuli can be taken outside that support, so on those annuli and on the end the electric estimates are unchanged. This yields prepared data with charge Q , energies E_ *,j →m , and full-cut infima a_ *,j ≥(1-o_j(1))a . For each fixed s , apply Proposition 34 ([Prepared numerical bound]) and then pass to this limit. The right side is increasing in the area variable, so the lower comparison of infima is sufficient. This proves. The elementary optimization is given in Proposition 13 ([Reduction of the numerical theorem]). The argument does not require the component electric fluxes to have one sign.