Uniqueness of Beltrami solutions fails without the three-point normalization

Assume the Axiom of Choice. It implies Countable Choice (The Axiom of Choice, AC implies DC implies countable choice, The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)).

Statement refuted. For a fixed Beltrami coefficient $\mu$ on the sphere, the equation $f_{\bar{z}}=\mu f_z$ has at most one quasiconformal homeomorphic solution $f:\widehat{\mathbb{C}}\to \widehat{\mathbb{C}}$ (Measurable Beltrami coefficients and measurable conformal structures, Weak solutions of the Beltrami equation, The ACL and Sobolev analytic definition of quasiconformality).

Counterexample. Set $\mu \equiv 0$. The identity $f_1(z)=z$ and inversion $f_2(z)=1\mathrm{/}z$ on the sphere are distinct Möbius transformations (Möbius transformations of the Riemann sphere), hence biholomorphisms (Every Möbius transformation is a biholomorphism of the Riemann sphere). They are $1$-quasiconformal with Beltrami coefficient zero (Every 1-quasiconformal homeomorphism is conformal, The ACL and Sobolev analytic definition of quasiconformality, The Beltrami coefficient and the maximal dilatation) and solve $f_{\bar{z}}=0$ weakly (Weak solutions of the Beltrami equation). Thus the same coefficient has at least two quasiconformal solutions without normalization.

More generally, the measurable Riemann mapping theorem says that for a fixed coefficient all solutions are the Möbius postcompositions of one solution, and exactly one solution remains after fixing three distinct image points (The measurable Riemann mapping theorem on the sphere).

Lyubich, Ch. 2 §14, printed p. 195, was read in full; it states uniqueness only up to Möbius postcomposition and exact uniqueness after fixing three points. Bishop, Ch. 3 §2, printed p. 88, Theorem 2.11, was also read in full but is context only: its printed $K=(k+1)\mathrm{/}(k-1)$ is negative for $0\le k<1$, and the proof invokes an unresolved “Theorem ??” for coefficient convergence. The counterexample itself is verified directly from the sphere charts and quasiconformal definitions.

The explicit maps and calculations above remain unchanged. Their exact normalized uniqueness and solution-family uses now consume the complete stable MRMT proof, and their analytic conventions consume the earlier12 definitions/equivalence. Root decisions and full-run certification are separate.