The Riesz–Titchmarsh transform at p = 3

This paper gives a direct proof that the Riesz--Titchmarsh transform$R$ has sharp $\ell^3$ norm $\sqrt{3}$. An exact positive cubicremainder identity also yields the sharp real inequality$\|R(aDa)\|_{3/2}\le \|a\|_3^2$, where $D$ is the discrete Hilberttransform. We characterize equality in the mixed cubic estimateand prove that the operator norm has no nonzero extremizer.The integer propagation principle of Ba\~nuelos and Kwa\'snickigives the sharp norms at exponents $3k$ and $3k/(3k-1)$,where $k\ge 1$ is an integer. The direct $p=3$ proof was obtained during the author's researchleading to \href{https://arxiv.org/abs/2610.10158}{arXiv:2610.10158}.The argument is independent of the operator-theoretic proofsin that preprint.