Limiting measures and energy growth for sequences of solutions to Taubes’s Seiberg–Witten equations

The paper proves Theorem 1.5 using a new maximum principle that works even when En = o(rn^{1/2}), establishes non-emptiness, X-invariance, θ-independence of limiting sets Z_n^θ, and, under bounded energy, finiteness of periodic orbits, all without invoking local vortex analysis or ECH. These results and their proofs are explicit in Sections 3–4, including Lemma 3.3 and the Hausdorff-limit argument (Theorem 1.5 and its proof) . By contrast, the model’s solution depends crucially on Taubes’s small-scale “vortex model,” Gaussian decay from a 1D core, and an SW→ECH dictionary in a purported stable Hamiltonian setting. The paper explicitly warns that in the unbounded-energy regime such local vortex approximations are not valid and replaces them with a maximum-principle approach . Moreover, the model assumes a stable Hamiltonian structure and applies ECH-style arguments without justification in this general exact divergence-free context. Hence, while the model’s conclusions mirror the paper’s statements, its proof strategy is not justified under the stated hypotheses.