Editing Openai/6922876a-7988-8007-9c62-5f71772af6aa (section)

==== - In Lemma \ref{lem:info-W2} you can slightly streamline the W2W_2W2 computation by the change of variables q=(1−2τ)+2τS(u)q=(1-2\tau)+2\tau S(u)q=(1−2τ)+2τS(u), which gives the “cubic constant” in one line: W22(μθ,νθ,τ)=8τ3∫0∞(S(u)−12−K(0)u)2 K(u) du,W_2^2(\mu_\theta,\nu_{\theta,\tau}) = 8\tau^3 \int_0^\infty \Big(S(u)-\tfrac12 - K(0)u\Big)^2\,K(u)\,du,W22(μθ,νθ,τ)=8τ3∫0∞(S(u)−21−K(0)u)2K(u)du, from which your expression for ccc follows after evaluating ∫u2K(u)du\int u^2K(u)du∫u2K(u)du and ∫u(2S(u)−1)K(u) du\int u(2S(u)-1)K(u)\,du∫u(2S(u)−1)K(u)du. ====
* If you prefer to state the theorem for the unclipped score and the present fff, add a succinct assumption that rules out adversarial fourth‑moment explosions, e.g. “assume QQQ has finite fourth moment uniformly on the W2W_2W2-ball” or “assume QQQ is dominated by a sub‑Gaussian envelope,” but this weakens the minimax statement. The clipping (III.B) avoids this altogether.

===== 1. Fix Lemma \ref{lem:dyn-W2} (velocity field) — §III.A. =====
# Replace Lemma \ref{lem:score-moments} by a valid version. Easiest is clipping (§III.B) or use a globally smoothed fff (§III.C).
# Acknowledge/resolve non‑uniqueness (tie‑break or global smoothing) — §III.D.
# Correct the bias-term algebra (use ε2τ⋆\varepsilon^2\tau_\starε2τ⋆, not ε2τ⋆3\varepsilon^2\tau_\star^3ε2τ⋆3) — §III.E.
# Fix constants/typos — §III.F–G, and relax (A1) as per §III.H.

With these amendments, the proof becomes sound and delivers the claimed sharp IDS rate ε2/3/n\varepsilon^{2/3}/nε2/3/n (up to constants).