The Harmonic Range and Maximum Entropy: A Projective-Geometric Characterization of the Logistic Map at r = 4

The logistic map fᵣ(x) = rx(1-x) at parameter $r = 4$ is distinguished by a well-known constellation of dynamical properties: surjectivity on $[0,1]$, maximal topological entropy log 2, an absolutely continuous invariant measure, and postcritical finiteness. We observe that these properties are equivalent to a classical condition in projective geometry: the four structurally distinguished points of fᵣ on CP¹ — the critical point, critical value, non-trivial fixed point, and the point at infinity — form a harmonic range (cross-ratio $-1$) if and only if $r = 4$. This characterization emerges naturally from the realization of f₄ as a quotient of the doubling map on the circle by complex conjugation, whose symmetry group is the Klein four-group V₄ Z/2 × Z/2. The same group is the universal stabilizer of the cross-ratio in S₄ and underlies the enhanced D₄ symmetry specific to the harmonic value λ = -1. The harmonic condition is moreover equivalent to f₄ being the unique non-trivial Belyi map in the logistic family — its critical values lie in \0, 1, ∞\ — connecting the dynamical characterization to arithmetic geometry and the moduli space M0,4.