  We prove that the Heisenberg Riesz transform is $L_2$--unbounded on a family
of intrinsic Lipschitz graphs in the first Heisenberg group $\mathbb{H}$. We
construct this family by combining a method from \cite{NY2} with a stopping
time argument, and we establish the $L_2$--unboundedness of the Riesz transform
by introducing several new techniques to analyze singular integrals on
intrinsic Lipschitz graphs. These include a formula for the Riesz transform in
terms of a singular integral on a vertical plane and bounds on the flow of
singular integrals that arises from a perturbation of a graph. On the way, we
use our construction to show that the strong geometric lemma fails in
$\mathbb{H}$ for all exponents in $[2,4)$.
  Our results are in stark contrast to two fundamental results in Euclidean
harmonic analysis and geometric measure theory: Lipschitz graphs in
$\mathbb{R}^n$ satisfy the strong geometric lemma, and the $m$--Riesz transform
is $L_2$--bounded on $m$--dimensional Lipschitz graphs in $\mathbb{R}^n$ for
$m\in (0,n)$.
