Root-Transform Bounds Do Not Force Logarithmic Coefficient Bounds

We construct a bounded normalized univalent function whose logarithmic coefficients are not O(1/n), although every pth root transform satisfies the coefficient estimate O(n^(2/p-1)). This gives a negative answer to the converse question recorded as Hayman-Lingham Problem 6.84 (UnsolvedMath AMR-022-6084). The normalized logarithm of the same function is Holder continuous of every order below one on the closed disk; consequently every root transform has coefficients O(n^(-1+eta)) for every eta>0. Starting from the classical existence of arbitrarily large weighted logarithmic coefficients, the proof inserts dilated seeds by high-order root transforms and normalized composition. A congruence filter transfers selected coefficients while a summable diagonal budget preserves earlier coefficients and gives boundary regularity. This is an analytic existence construction using a credited classical seed theorem. The four-page English manuscript includes the complete proof and supporting exact rational checks. AI-assisted, self-audited preprint; no independent peer review, formal verification or exhaustive priority certification is claimed.