Uniform area derivatives imply width comparison

Integrate the derivative upper bound on a common short interval using the mean-value inequality. The extra term given by the initial area gap divided by the interval length consumes at most that gap. The area-energy inequality bounds the initial slice supremum by the finite energy cap. Taking the supremum of the evolved slices gives exactly the short-time area comparison required by WidthComparisonData. The common interval and cutoff are essential hypotheses; pointwise variation of a single surface is insufficient.