Estimating Phase from Observed Trajectories Using the Temporal 1-Form

The paper’s Theorem 1 states exactly the three claims at issue: (1) the Temporal 1-Form dφ is the unique continuous closed 1-form on X with ⟨dφ,f⟩≡ω; (2) for any circle-valued phase ϕ, one has ϕ∘P=ϕ and dφ=ϕ∗(dθ)=−i(dϕ)/ϕ; (3) conversely, given x0, ϕ(x)=exp(i∫γ dφ) is well defined and unique (Lemma 2, Lemma 3, and Theorem 1) . The paper proves uniqueness of dφ by reducing to uniqueness of circle-valued phases modulo rotations and the rotation-invariance of dθ, together with Lemma 3’s reconstruction argument . The candidate solution proves the same three items but via a different, more geometric argument: uniqueness is established by analyzing η:=β−P∗α on flow rectangles, showing η annihilates both f and ker DP, hence η≡0. This differs from the paper’s cohomological/phase-based route but reaches the same conclusions. The candidate’s Step 2 (ϕ∘P=ϕ and dφ=ϕ∗dθ=−i dϕ/ϕ) and Step 3 (reconstruction from path integrals with winding-number reasoning) match the paper’s statements and proofs in substance . Minor caveat: the candidate’s rectangle argument implicitly uses uniform contraction to the limit cycle along fibers; the paper avoids this by leaning on deformation retraction and uniqueness of phases (Lemma 2), so the paper’s route is a bit cleaner on technical hypotheses. Overall, both are correct; the proofs are different in style.