Nonmonotone Argument and Radial Angle of a Support Slit

Every global maximizer of the real part of a2+i*10^-24(a4-3a3) in the normalized schlicht class omits a slit with nonmonotone continuous argument and nonmonotone signed and ordinary unsigned radial angles. This gives negative answers to the two questions recorded as Hayman-Lingham Problem 6.72. Compactness provides genuine extremizers, the area theorem bounds the actual slit tip, and a controlled Schiffer natural coordinate identifies a subarc of the actual omitted slit. No explicit maximizing function or uniqueness is required. The research note credits classical support-slit geometry, Schiffer methods and the Friedland-Schiffer coefficient/control framework. The accompanying package contains manuscript source, immutable authored input, independent analytic and quantitative derivations, exact rational and directed interval checks, and a bounded primary-source priority audit. Complete comparisons include Zemyan (2003, 2005, 2007), Duren-Leung-Schiffer (1983), Zhang-Ma (1991), Brickman (1984), and Goh (1992, 1994). The later Duren commentary (2013 online/2014 print) and Hibschweiler (1995) could not be inspected. Priority therefore cannot be fully cleared. These limitations and other uninspected leads are disclosed; no historical-first priority or continued open status in 2026 is asserted. Finite controls supplement the uniform analytic proofs and are not formal certification of the classical inputs. AI tools were used extensively in development, writing, reproduction and adversarial verification. This is an unrefereed preprint with internal AI adversarial reviews; it has not received human peer review.