# Verification report An explicit pair of odd zero-free exterior univalent functions f(z)=z-z^-1 and g(z)=z B(z^-2), B(w)=((1-i)sqrt(1-w^2)+(1+i)(1-iw))/2, has an analytic normalized geometric mean with exponent1/2 whose first four nonzero Laurent coefficients give weighted squared sum269463/262144>1. Radial dilation by1001/1000 makes both factors and the mean analytic across the unit circle, with full weighted sum finite and greater than101/100. # Originating-researcher mathematical audit 1. Both f and g have expansion z plus odd negative powers, so their constant terms vanish as required by the original chapter. An arbitrary translated Joukowski map would not satisfy this normalization and was not used. 2. k(w)=w/(1-w)^2 is a conformal bijection from the disk onto C minus(-infinity,-1/4], by the Cayley half-plane map and squaring. Half of this slit plane is C minus(-infinity,-1/8], a subset of the first slit plane. Thus phi=k^-1(k/2) is indeed an analytic univalent self-map of the disk, not just a formal power series. 3. p=-i phi(iw) is a univalent disk self-map with p'(0)=1/2. k_-(u)=u/(1+u)^2=-k(-u) is univalent. Therefore h=2 k_-(p) is normalized disk univalent and has only its simple zero at0. 4. For any such h, the analytic normalized square root b=sqrt(w/h(w)) exists and is nonzero on the disk. G(z)=z b(z^-2) is odd. Equality of two G values implies equality of h(z^-2); hence the inputs differ by at most a sign, which oddness/nonvanishing excludes. This is a full injectivity proof, not a sampled boundary plot or area-coefficient test. 5. The exact relation p/(1-ip)^2=w/[2(1-iw)^2] implies B=(1-iw)(1+p)/(1-ip), which is nonvanishing. It also implies (1-iw)^2((1+ip)/(1-ip))^2=1-w^2. Values at0 select the correct analytic root. The explicit algebraic formula for B is therefore the same factor as the root transform. SymPy independently checks both polynomial/rational identities. 6. f is the reciprocal odd root transform of k. Both factors omit0. Their product after factoring out z^2 is nonzero on the simply connected disk in w=z^-2. The normalized square root H defines a globally analytic mean F=zH(z^-2); individual multivalued square roots of f or g are never assumed globally defined. 7. Four coefficients obtained by exact Gaussian-rational recurrence match a separate SymPy expansion. Laurent indices are1,3,5,7, not1,2,3,4. Their weighted norm is269463/262144 with surplus7319/262144. Omitted terms are nonnegative, so truncating this sum is a rigorous lower bound. 8. Dilation f_R(z)=f(Rz)/R preserves normalization and univalence on |z|>R^-1. The mean coefficient scales by R^-(n+1), so its squared weight scales by R^-2(n+1). The smallest factor among the four terms is R^-16. For R=1001/1000, R^16<125/123 and the resulting lower bound33143949/32768000 exceeds101/100. All rational comparisons are checked exactly. 9. F_R is analytic on a neighborhood of the closed exterior disk. Cauchy on any circle R^-1