# Verification report There exists one bounded normalized univalent function f with log(f/z) in C^alpha on the closed disk for every alpha<1, but sup_n n|gamma_n(f)| is infinite. For this same f, every root transform has c_n^(p)=O(n^(-1+eta)) for every eta>0 and therefore satisfies the source bound O(n^(2/p-1)) for all positive integers p. The converse in Hayman-Lingham Problem6.84, UnsolvedMath AMR-022-6084 record2306084: do the root coefficient bounds for every p force gamma_n=O(1/n) for an individual f in S? Constants may depend on p. # Self-audit of the complete counterexample Date: 26 October 2026. Accepted scope: a negative answer to the exact individual-function converse in Hayman–Lingham Problem 6.84. The limit f is one bounded normalized univalent function, not a family depending on p. Its logarithm is C^alpha for every alpha<1; nevertheless n|gamma_n| is unbounded. No independent review or formal verification is claimed. ## Classical seed input The proof needs arbitrarily large n times the nth coefficient of log(g/z) over smooth g in S. Girela (2000), printed p.338, explicitly states the stronger existence of a bounded g in S with gamma_n not O(n^-0.83), citing Duren Theorem 8.4. That page was inspected visually. The exact question's source independently attributes a failure of O(1/n) to Pommerenke. Selecting one finite coefficient and then dilating g(rz)/r preserves a strict lower bound and supplies analyticity and injectivity past the closed disk. No effective formula for the seed coefficients is claimed. ## Injection and normalization The Nth power of u_N=z exp(H(z^N)/N) is g(z^N), not g(z)^N. Equality of two u_N values implies equality of z^N values and then the root-of-unity covariance forces the two points to coincide. The zero case is separate and harmless. Multiplication by exp(-A/N) preserves injectivity, while Re H<=A gives |phi_N(z)|<=|z|. Thus the old function is always evaluated inside its univalence disk. Multiplication of the composition by exp(A/N) restores derivative one. Its normalized log is H(z^N)/N+F(phi_N), with no missing additive A or branch constant. ## Boundary regularity and neighborhoods Every finite old function is analytic past the closed disk. Its normalized log is also analytic there: the growth lower bound for S excludes zeros of f/z on the closed disk, and compactness supplies a larger zero-free disk. The composition extends to a neighborhood of the closed disk even if phi_N touches its boundary, since the old function is defined beyond it. The sup norm of the log increment is O(1/N), and its derivative norm is O(1); the z^N derivative cancels the divisor N. Interpolation gives O(N^(alpha-1)), with old functions and alpha fixed before choosing N. No estimate at alpha=1 is used or asserted. ## Coefficient transfer Since phi_N=a z times a function of z^N, its jth power has support only in degrees j modulo N, with minimum degree j. At Nm only j=N,2N,...,mN contribute; the j=0 term is absent because F(0)=0. The Cauchy coefficient bound is at most one for each phi_N^j because its disk supremum is at most one. The coefficients of the fixed old F decrease geometrically beyond the unit circle. Multiplying their tail by Nm still tends to zero. Thus the transferred weighted coefficient converges to m[z^m]H, with no factor of N left and no assumption on its phase. ## Diagonal order and persistence At each step the seed, target degree m, old function, Hölder exponent, and tolerance are fixed before N is increased. The two limiting estimates therefore satisfy all finitely many conditions at once. Each later sup-norm tolerance is at most 2^-k/n_i for every earlier selected degree n_i. The later total change in that weighted coefficient is at most 2^-i, while its initial magnitude exceeds 4i+1. This proves unboundedness for the limit, rather than merely for its finite approximants. For any alpha<1, the late alpha_k exceed alpha and the norm embedding constant is at most two on a disk of diameter two. All limits coincide by uniform convergence. The limit f=z exp(F) has derivative one, so the standard univalent-limit theorem cannot degenerate to a constant. ## The root bounds and quantifiers The same bounded F is used for every p. Exponentiation preserves each Hölder class on its bounded range. The Fourier shift h=pi/n gives |[z^n]U| <= [U]_alpha (pi/n)^alpha/2. Hence the root coefficients are O(n^-1+eta) for every eta>0. For p>=3, alpha=1-1/p is stronger than the required exponent; bounded coefficients suffice for p=1,2. Constants may depend on p and eta. The uniform pA(p) converse is Milin's Theorem 2 and is credited, not presented as new. ## Checks and limits The standard-library checker passed 15 independent direct-composition versus logarithmic-expansion cases, comparing 351 exact coefficients and 276 predicted zero entries. It separately checks the generalized binomial expression for transferred terms and the transfer bound. The initial checker attempt used Python's integer zero for an empty sum, which introduced floating-point values; setting the starting sum to Fraction(0) repaired that code issue. The proof did not depend on it. The checker also brackets a critical point of the discarded lacunary proposal. That failed proposal is retained in failed_routes.md and is not used as a source counterexample. Finite checks do not prove the infinite construction, the univalence theorem, or the seed input. All four final PDF pages were visually inspected. No clipping, missing symbols, unresolved references or formula errors were found. The built-in editor compiler and Tectonic both succeeded without diagnostic warnings. The author is the only prominent author-line text; company and contact are in a footnote; AI assistance is disclosed in prose. Conclusion: the written analytic construction is internally complete. One exact source question is answered negatively and is counted once. Classical input is indispensable and credited. A bounded literature search found no matching solution; absolute priority is not certified. # Bounded literature and scope review — 26 October 2026 The exact source is Hayman–Lingham Problem6.84. The search distinguished the converse for one f from the universal Szegő problem for all f, from the Bieberbach–Milin averaged inequality, from Fekete–Szegő functionals, and from individual sharp bounds for small logarithmic coefficients. Queries included the exact problem number and author; Milin, converse, Szegő and logarithmic coefficients; logarithmic coefficients with root transforms, composition, Hölder regularity and gliding constructions; and the exact titles of the Milin and Pommerenke references. The bounded search located no matching root-composition resolution of the converse. This is not a comprehensive or absolute-priority certification. The following inputs are credited: 1. The existence of univalent functions with unbounded weighted logarithmic coefficients is classical, attributed by the source to Pommerenke and explicitly recorded by Girela on p.338, citing Duren. Our argument uses this theorem and does not reproduce or improve its original coefficient-growth exponent. 2. Milin (1985), Theorem1, proves the forward implication; Theorem2 supplies a converse under an extra uniformity condition on pA(p). The latter confirms that a naive p-to-infinity limit is not new. 3. Root transforms preserve univalence, univalent normalized limits stay univalent, and Hölder boundary regularity gives Fourier coefficient decay. These standard facts are proved or explained directly where needed. The proposed contribution is the combination of normalized root insertion, the coefficient congruence filter, and a summable diagonal budget. It produces a single bounded univalent function with normalized logarithm in every C^alpha, alpha<1, while n|gamma_n| is unbounded. This satisfies every root bound in the exact source and negates its conclusion. It does not settle universal p=5,...,11 bounds, supply an effective numerical formula for the seed sequence, or establish any new sharp logarithmic coefficient bound. See source_audit.md for primary URLs and retrieval limits. The initial public HF snapshot had 671 events; exact deduplication will be refreshed before any public notice. A manuscript is one result here, not a count of separate results for every p or every Hölder exponent.