Phase-sensitive tipping: How cyclic ecosystems respond to contemporary climate

The paper states and proves the exact biconditional in Proposition 4.1: for any p1, p2 on the parameter path, an instantaneous jump from p1 to p2 gives irreversible P-tipping from Γ(p1) if and only if Γ(p1) is not contained in the basin of attraction of Γ(p2) . The proof reduces directly to the definition of the basin of attraction for the frozen system, B(Γ,p) = {x0 : d(x(t,x0;p), Γ(p)) → 0 as t → ∞} . The candidate solution reproduces this argument: both directions follow immediately from the basin definition, and the note on partial basin instability matches the paper’s role for that assumption—to guarantee the existence of both tipping and non-tipping phases (phase sensitivity), not to establish the logical equivalence itself . Hence, both are correct and essentially identical in proof structure.