Lemma 31 ([A quantitative local inverse estimate]). Suppose a map Φ has, on a ball about q₀ , bounds \|DΦ(q₀)^-1 \|≤C r^-c , \|D^2Φ\|≤C r^-b , and the available source ball has radius at least c_0r^a . There are source and image balls …

Choose N>a and N>b+c . On a ball of radius ηr^N , for a sufficiently small fixed η , the norm of DΦ(q_0)^ -1 (DΦ(q)-DΦ(q_0)) is at most 1/2 . The map q q+DΦ(q_0)^ -1 (p-Φ(q)) is a contraction preserving that ball whenever |p-Φ(q_0)|≤c_1r^ N+c . Its fixed point gives the inverse and the displacement estimate. Differentiating the inverse equation expresses each fixed derivative in finite products of the inverse Jacobian and forward derivatives through the requested order. The additional hypotheses give fixed power bounds for these products. The inverse perturbation identity A^ -1 -B^ -1 =A^ -1 (B-A)B^ -1 transfers the estimates to a sufficiently close map. This argument also supplies explicit slack for smaller source and image balls.