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

==== 1. Typo in Theorem statement. “for all n≥1n\ge 1n≥1 and ε∈(0,12]\varepsilon\in(0,\tfrac12]ε∈(0,21],s” → trailing “s” should be removed. (§III.F) ====
# Lemma \ref{lem:info-W2}, Gaussian integral constant. You write “∫0∞K′(y)2/K(y) dy=1\int_0^\infty K'(y)^2/K(y)\,dy=1∫0∞K′(y)2/K(y)dy=1” but use π/τ\pi/\tauπ/τ downstream. The correct value is ∫0∞K′(y)2K(y) dy=∫0∞y2K(y) dy=12.\int_0^\infty \frac{K'(y)^2}{K(y)}\,dy=\int_0^\infty y^2 K(y)\,dy=\frac{1}{2}.∫0∞K(y)K′(y)2dy=∫0∞y2K(y)dy=21. With K(0)=1/2πK(0)=1/\sqrt{2\pi}K(0)=1/2π, this yields I(τ)=π/τI(\tau)=\pi/\tauI(τ)=π/τ as claimed. (§III.G)
# Language about “edge strips of width τ\tauτ”. It is true that νθ,τ\nu_{\theta,\tau}νθ,τ assigns mass τ\tauτ to each side beyond θ±(12−τ)\theta\pm(\tfrac12-\tau)θ±(21−τ), but the tails extend to +∞+\infty+∞. When discussing supports of derivatives, say “outside the bulk” rather than “on strips of width τ\tauτ”.
# “Strictly decreasing C1C^1C1 score”. For the present fff, ψτ\psi_\tauψτ is continuous and piecewise linear in uuu with a derivative jump at the junctions; it is not C1C^1C1. Replace by “continuous, nonincreasing, and piecewise C1C^1C1” (or adopt Fix B so it is C1C^1C1). (§III.H)