= Solution
The Riemann sphere has three complex <Conformal Killing vector fields>, forming the Möbius group $PSL(2,\mathbb C)$. Gauge fixing this residual conformal symmetry permits three insertion points to be fixed arbitrarily. Each unintegrated <string vertex operator> contains $c\widetilde c$, and the three insertions exactly saturate the three holomorphic and three antiholomorphic ghost zero modes. Since every full vertex has weight $(0,0)$, the correlator is Möbius invariant and independent of the chosen three positions.
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