Three Harmonic Log-Distances Characterize Holomorphic and Antiholomorphic Maps

Let Ω be a connected open subset of the complex plane and let F: Ω → ℂ be continuous. We prove that if a₀, a₁, a₂ are three noncollinear points and each function log|F − aⱼ| is harmonic wherever it is finite, then F is holomorphic or antiholomorphic on all of Ω. The map may attain the anchors, and no differentiability, local injectivity, or orientation assumption is imposed. The proof combines reconstruction from squared distances with the standard factorization theorem for units in a tensor product. Harmonic log-distances give squared moduli of holomorphic functions; three anchors then place both F − aⱼ and its reciprocal in an algebra of finite sums of holomorphic times antiholomorphic functions. Unit factorization yields a differential identity whose comparison at two anchors forces a single complex orientation. Two anchors do not suffice, even for a real-analytic, orientation-preserving local diffeomorphism. Noncollinearity is essential: a folding map passes every log-distance test with an anchor on one line. The result is positioned relative to classical finite tests for harmonic morphisms, without a claim of established historical priority.