Lemma 18 ([Range of the fitted curvature]). Let an arc have positive base height h , peak height t>h , and width r . Its fitted curvature satisfies_h≤u≤t φ''(u) ≤κ(h,r)≤ _h≤u≤t φ''(u). If φ'' is nonconstant on [h,t] , both inequalities …

Write m=_ [h,t] φ'' and b=_ [h,t] φ'' . If the fitted κ were less than m , the actual curvature would be strictly larger than this constant reference curvature along the whole characteristic. Lemma 7 ([Curvature comparison]) would then make the actual final L strictly larger than the reference L with the same final angle, contradicting the fit. A fitted curvature greater than b gives the reversed contradiction. This proves Equation. If φ'' is nonconstant and κ=m , the forcing in Equation is nonpositive and strictly negative on a positive interior width interval. Its integral representation gives E≤0 everywhere and E<0 on a nonempty interval. Equation makes the reference label nonincreasing and nonconstant. Relative to its final constant label, the actual earlier angles are therefore no smaller and are strictly larger on a nonempty open interval. As in Equation, their L integral is strictly larger, a contradiction. The case κ=b reverses all these inequalities. Thus both bounds are strict whenever the actual curvature is nonconstant.