We give a construction of an odd spectral triple on the Cuntz algebra $O_{ N}$, whose $K$-homology class generates the odd $K$-homology group $K^1(O_{ N})$. Using a metric measure space structure on the Cuntz-Renault groupoid, we introduce a singular integral operator which is the formal analogue of the logarithm of the Laplacian on a Riemannian manifold. Assembling this operator with the infinitesimal generator of the gauge action on $O_{ N}$ yields a $\theta$-summable spectral triple whose phase is finitely summable. The relation to previous constructions of Fredholm modules and spectral triples on $O_{ N}$ is discussed.
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