A Counterexample to the Area Inequality for Geometric Means of Exterior Univalent Functions

A question of D. K. Thomas, recorded as Problem 6.50 in Hayman and Lingham's Research Problems in Function Theory, asks whether geometric means of normalized exterior univalent functions satisfy the area-theorem coefficient inequality. We give an explicit counterexample at exponent 1/2. The two factors are odd, zero-free and univalent on |z|>1, and their normalized geometric mean is analytic throughout that domain. Its first four nonzero negative Laurent coefficients have weighted squared sum 269463/262144>1. A fixed radial dilation by 1001/1000 gives factors and a mean analytic across the unit circle, with full coefficient sum finite and greater than 101/100. The proof uses elementary conformal maps and exact rational arithmetic. This gives a negative answer to the entire single-question frozen UnsolvedMath record AMR-022-6050. The package includes the complete four-page proof, exact and symbolic checks, and a source/scope audit. AI-assisted self-audited preprint; no independent peer review, formal verification or absolute-priority certification is claimed.