Continuity of the superpotentials and slices of tropical currents

The uploaded paper proves exactly that for positively weighted tropical cycles C1,C2 in R^d, the tropical-current wedge equals the current of the stable intersection, and the result is balanced of dimension p+q−d (Theorem 5.13) . The wedge product is defined via toric compactifications and is independent of the choice of smooth projective fan, agreeing with Bedford–Taylor in bidegree (1,1) (Propositions 5.9–5.10 and Remark 5.12) . Multiplicities are computed locally by intersecting fibres and counting with lattice index, using the 0-dimensional case (Katz’s index) and slicing to reduce higher-dimensional intersections , while balancedness follows from closedness of the wedge together with the balanced⇔closed correspondence for tropical currents (Theorem 4.3) . The candidate solution reaches the same conclusion by a direct local/fan-displacement argument: (i) a linear-subspace wedge formula TH1∧TH2 equals the index times TH1∩H2; (ii) cellwise computation in the transverse case using Definition 4.2 ; (iii) reduction to stars and generic displacement, matching Definition 3.3 of stable intersection ; and (iv) balancing via wedge-closedness. This is parallel in spirit to the paper but uses different tools (no Monge–Ampère; less superpotential machinery). Minor points that could be clarified in the model write-up include a justification of the averaging/Fubini step in the linear lemma and a brief mention that the wedge is defined as (T⊗S)∧Δ, which explains the bidimension p+q−d . Overall, both are correct; the paper’s proof is via superpotential theory and slicing limits (including the translation continuity step (e^{εb})∗TC1 ∧ TC2 → TC1 ∧ TC2) , whereas the model gives a direct combinatorial–current computation.