Lagrangian Split Tori in S2 × S2 and Billiards

The paper proves the equivalence between Hamiltonian isotopy of split tori T(x,y), T(x′,y′) in Xα and the appearance of (±x′,±y′) as an admissible bouncing point on the good billiard trajectory from (x,y) (Theorem 1.3), and refines it with an explicit algebraic criterion (Theorem 2.5) . The construction (billiards ⇒ isotopy) uses symmetric probes (Theorem 2.1, Lemma 2.2) ; the obstruction (isotopy ⇒ algebra) uses Chekanov-type invariants (Theorem 2.6, plus a relative homology constraint) and yields y=y′ and x=±x′+2k1+2k2y (eqs. (30),(31), Lemma 2.7) ; the algebra ⇒ billiards direction is handled by the unfolding argument (Lemma 2.4) . The candidate solution mirrors the high-level structure but makes a critical misstep: it asserts that 1∈Γ(p) for the subgroup Γ generated by facet distances and uses this to deduce y′=±y from Γ(x,y)=Γ(x′,y′). In fact, from the generators {r−y, r+y, 1+r−x, 1+r+x} one only directly obtains 2(1+r)∈Γ, not necessarily 1+r; the leap to 1∈Γ (and hence to ⟨1,y⟩=⟨1,y′⟩) is unjustified. The paper instead derives y=y′ from the first Chekanov invariant d=α−y on the relevant region Q, and then obtains the x-relation via a relative homology argument, avoiding this gap . The model’s conclusion matches the paper, but the given proof contains this flaw, so as written it is not correct.