NO BOUNDED GEOMETRY WANDERING DOMAINS FOR SUFFICIENTLY REGULAR AUTOMORPHISMS

The paper proves Theorem 1.7 by introducing the transboundary modulus on T^n relative to the minimal set Λ, constructing the horizontal path family Γ_h, and showing mod_Λ(Γ_h) < ∞ using bounded geometry of wandering domains. Existence and uniqueness of an extremal mass distribution are established, and a conformal invariance lemma shows mod_Λ(f(Γ_h)) = mod_Λ(Γ_h) for quasiconformal f when Λ has measure zero. Since f is isotopic to the identity, f(Γ_h) = Γ_h. Uniqueness then forces constant weights on infinitely many domains in a single wandering orbit; finiteness of mass implies those weights vanish, contradicting admissibility, hence no such f exists. This argument is complete and rigorous in the paper's Section 4 and Lemma 3.5 (conformal invariance) together with the existence/uniqueness of extremals for the transboundary modulus. See the statement of Theorem 1.7 and its proof outline in the paper and the details around Lemma 3.5 and the extremal mass distribution uniqueness. The candidate solution proposes a different, annulus-barrier modulus strategy relying on Loewner space properties and a serial law for classical n-modulus. However, it leaves the pivotal separation/barrier construction unproven: there is no rigorous selection of disjoint wandering annuli that any curve joining two parallel (n−1)-tori must cross. Without this, the claimed “infinite in series” lower bound on modulus is unsupported. Hence the model’s approach is incomplete.