Lemma 6. The reference functions are smooth in (κ,s,z) for s>0 and are defined at every finite positive width. They satisfy ∂_s j_κ=-s^2Θ_κ, ∂_sΘκ =j_κ-(2Θ_κ)s, ∂_s L_κ =(2Θ_κ)s, with limiting values j_κ(0,z)=z and Θ_κ(0,z)= L_κ(0,z)=0 . …

The entire-profile construction in Lemma 1 ([Global arc coordinates]) gives an arc at every lower base height. Its width tends to zero at the peak and tends to infinity as the base tends to minus infinity, since s^2≥2(t-h) . Its strictly negative base derivative gives the smooth inversion from base height to any s>0 , including smooth dependence on the quadratic coefficients. Thus no finite-width existence claim is being inferred from the singular equations at s=0 . Equation gives Equation and, by integration of the slope equation, the first bound in Equation. For κ≠0 , consider instead the single profile κu^2/2 . Let h(t,s) be its base height when the peak height is t and the width is s . Its peak slope is z=κt , and its base slope is κ h(t,s) . Therefore ∂_z j_κ(s,z)=∂_t h(t,s)>0 by Lemma 1 ([Global arc coordinates]). This calculation is valid for either sign of κ . For κ=0 , one has j_0(s,z)=z . The smooth scaling in Lemma 3 ([Smooth scaling at zero width]) gives, locally uniformly in the finite parameters, Θ_κ(s,z)=(z)/(3)s+O(s^2), ∂_zΘ_κ(s,z)=(s)/(3)+O(s^2). Differentiating the angle equation at positive width gives ∂_s(∂_zΘ_κ) =∂_z j_κ -2(2Θ_κ)s∂_zΘ_κ. The initial sign supplied by Equation and the positive forcing prove ∂_zΘ_κ>0 by the scalar integrating-factor formula. To verify the range, fix r>0 . For 0<s≤r , j_κ(s,z)≥z-(|κ|r^2)/(2). For sufficiently large positive z , the angle is positive near zero and cannot cross zero downwards, since its derivative there would be positive. Given N>0 , on [r/2,r] and whenever 0≤Θ_κ≤N , ∂_sΘ_κ ≥z-(|κ|r^2)/(2)-2(2N)r. For large enough z , this lower bound forces a crossing of N within that interval if the angle has not already exceeded N ; it also prohibits a downward crossing of N . Thus Θ_κ(r,z)→+∞ as z→+∞ . Applying the same argument to (-j,-Θ,-κ) proves the negative limit as z→-∞ . Strict monotonicity and smoothness give the asserted bijection. Integrating the last equation in Equation gives L_κ(s,z) =∫_0^s((2Θ_κ(q,z)))/(q)\,dq. Equation, including its locally uniform parameter derivative, bounds the differentiated integrand near zero. Differentiation under this integral is therefore valid and yields Equation. Its integrand is strictly positive at every positive width.