# Verification report

In A=C direct-product C, the coefficients e1=(1,1), e2=(i,2i) have a real-analytic right-regular degree-minus-one Cauchy kernel with unit flux for every admissible real-slice domain, but admit no Clifford-dual family. No real invertible coordinate change repairs the common scalar quadratic factorization. The homogeneous right-regular unit-flux kernel is unique.

Hayman-Lingham Problem 7.69(a), pp.181-182: the printed equivalence between kernel existence and fixed-coefficient Clifford relations is false even for real-analytic kernels, hence also under the proposed C1 weakening. Part (b), about nonassociative algebras, remains unaddressed. The whole frozen record is not claimed solved.

# Detailed originating-researcher self-audit

1. A=C direct-product C is a finite-dimensional unital associative complex algebra, hence meets the countable-basis condition. The determinant of e1=(1,1), e2=(i,2i) is i, so the basis is complex-linearly independent. This avoids the invalid shortcut of taking two dependent elements of C.
2. The real plane is identified by (x,y), not all of C2 as a four-real-dimensional domain. Both coordinate maps zeta1=x+iy and zeta2=2x+iy are bijective real-linear orientation-preserving maps to C, with positive determinants1 and2. Thus the two Cauchy boundary formulas use the same orientation.
3. The scalar differential equations become 2*dbar(h1)=0 and4*dbar(h2)=0. C1 suffices for holomorphy. The full oriented boundary Cauchy theorem works for multiply connected domains; no simply-connected assumption is hidden. A compact embedded2-manifold in the coordinate plane is handled by its interior components.
4. The source sign convention starts at j=1, yielding dsigma=-e1 dy+e2 dx. Each component equals i dzeta_j. The minus sign in G=-1/(2pi)*(1/zeta1,1/zeta2) therefore gives exactly1/(2pi i), not its negative. This checks normalization and flux.
5. Kernel denominators have no nonzero real zero. The kernel is jointly real analytic in source and base-point real coordinates away from the diagonal, right regular, homogeneous of degree-1 and decays as1/r. These are exact identities, not numerical evidence.
6. The diagonal relations uniquely force p1=e1 and p2=(-i,-i/2); the cross relation is(0,3i/2), nonzero. There is no untested alternative dual family. The algebra may have Clifford subalgebras; the failure concerns these prescribed coefficients.
7. A real scalar quadratic common to both symbol components must vanish at x=-iy and x=-2iy. Writing Q=r x^2+2s xy+t y^2 with real coefficients gives s=0,t=r=4r and thus Q=0. An invertible real coordinate change yielding Clifford dual relations would pull back the Euclidean quadratic to a nonzero positive definite Q, impossible. A complex basis change can change the real slice and is not this assertion.
8. Restricting an original holomorphic left-regular F on complex V to a translated real coordinate slice identifies the real partial derivatives with complex-coordinate partials. Hence the constructed formula includes that original function class; it does not require a hidden extension of arbitrary C1 data to C2.
9. Constants change no boundary integral because d(dsigma*f)=-Df dx wedge dy. This explains unnormalized nonuniqueness without using it as the counterexample. In the normalized class, each component is holomorphic on C*, homogeneous of degree-1; differentiating real scaling gives (z*k(z))'=0, so k=c/z. Unit flux forces c=-1/(2pi). Thus uniqueness in that class is proved, not assumed.
10. The q>0,q!=1 extension has determinant i(q-1), positive coordinate Jacobian q and cross relation(0,i(q-q^-1)); all are valid. The main counterexample uses q=2 only. Rational parameter checks are regression examples, not a proof of the entire family.
11. Exact SymPy checks verify algebra, right regularity, one-form identity, quadratic obstruction and circle flux by residues. The all-domain integral identity follows from the written analytic proof, not extrapolation from circle tests. No formal proof-assistant verification or independent review is claimed.

No gap was found in the written scoped counterexample. The source part(a) existence formulation is refuted; part(b) and the full record are not resolved.

# Bounded primary-source literature review,25-26October2026

Sources: Hayman-Lingham arXiv1809.07200v2, Problem7.69; John Ryan, Hypercomplex algebras, hypercomplex analysis and conformal invariance, Compositio Math61(1987),61-80, https://www.numdam.org/item/CM_1987__61_1_61_0/ ; Marin Genov, The generalized homological Cauchy integral formula for A-holomorphic functions, Serdica MathJ52(2)(2026),185-204, DOI10.55630/serdica.2026.52.185-204, publisher https://serdica.math.bas.bg/index.php/serdica/article/view/347 . Genov introduction and definition2.1 were read, not an exhaustive audit of every theorem.

Queries included Ryan Cauchy algebra real-analytic kernel; Ryan Cauchy integral associative algebra; the exact1987 title with counterexample/error/correction/erratum/incorrect; Cauchy integral formula commutative associative elliptic; Ryan Cauchy Clifford necessary algebra; Ryan Problem7.69; and Clifford elliptic systems. They did not identify this exact fixed-coefficient correction. That is a bounded search, not exhaustive novelty certification. Search snippets for octonionic and alternative-algebra Cauchy theories concern different function classes and were not used as mathematical premises.

The componentwise Cauchy formula is classical. No new general calculus is claimed. The manuscript supplies an explicit counterexample to the printed necessity, checks the real-slice source quantifiers, proves the common-real-quadratic obstruction, and specifies normalized uniqueness. The separate nonassociative question remains open in this work.

All four final PDF pages were directly visually inspected. No clipping, missing symbols, unresolved references or layout defects were found. The built-in compiler and Tectonic succeeded. Author name is prominent; affiliation/contact is a footnote; AI assistance appears in prose.