A Cauchy Kernel without the Fixed-Coefficient Clifford Relations

We give an explicit counterexample to the fixed-coefficient necessity in the existence formulation of Hayman-Lingham Problem 7.69(a). In the commutative complex algebra C direct-product C, the coefficients e1=(1,1) and e2=(i,2i) define two differently scaled Cauchy-Riemann operators. The ordinary Cauchy formula in two orientation-preserving real-linear coordinates yields a real-analytic reproducing kernel for every admissible real-slice domain. The kernel is right regular, homogeneous of degree -1 and normalized by unit flux, and is unique within that normalized class. Nevertheless no elements p1,p2 satisfy p_j e_l+p_l e_j=2 delta_jl. A common-real-quadratic obstruction excludes repairing the relations by an invertible real coordinate change. Thus the printed existence criterion fails even for real-analytic kernels, and remains false under the proposed C1 weakening. The original Ryan theorem includes a uniqueness assertion omitted in the problem-book formulation; this distinction is explicit. The classical Cauchy formula and commutative-algebra literature are credited. The nonassociative part (b) is not addressed, so the whole UnsolvedMath record AMR-022-7069 is not claimed solved. The package contains the four-page proof, exact symbolic checks and a source/scope audit. AI-assisted author-self-audited preprint; no independent peer review, formal verification or exhaustive priority certification is claimed.