Lemma 5 ([A bounded flux form]). Under the topological, completeness, and asymptotic hypotheses of Definition 1 ([Admissible purely electric exterior]), the full-cut infimum satisfies 0<a≤_g(S). This assertion does not use the constraint …

Choose a small boundary disk in one component of S . A smooth embedded arc beginning transversely at that disk can be continued into the coordinate end and then along a straight coordinate ray. By shrinking a tubular neighborhood, obtain a proper embedded tube [0,∞)×B^ n-1 _δ⟶Ω whose initial face lies in the chosen boundary disk. The transverse coordinates have fixed small coordinate width sufficiently far along the ray. Choose a nonnegative function ζ∈C_c^∞(B^ n-1 _δ) with integral one. The form ζ(z)\, z^1∧⋯∧ z^ n-1 on the tube is closed. Extend it by zero across the lateral boundary; call the resulting smooth closed form β . Its comass is bounded: on the compact part this follows from smoothness, and on the end from asymptotic Euclidean comparability and the fixed transverse width. Orient the initial disk so that ∫_Sβ=1 . Every full cut Γ is homologous to S with the orientation toward the end. This can be seen by truncating both exteriors at one distant coordinate sphere and applying Stokes to their difference; portions of S shared by the two boundaries cancel with their full multiplicity. Equivalently, integration over the compact difference of the two codimension-zero sets gives the same identity. Thus 1=∫_Γβ ≤\|β\|_ comass,∞ _g(Γ). Taking the infimum proves the lower bound. The admissible choice Γ=S gives the upper bound. In particular, this argument uses the entire intrinsic boundary and never replaces it by an empty relative frontier.