  We prove the inverse conjecture for the Gowers U^{s+1}[N]-norm for all s >=
3; this is new for s > 3, and the cases s<3 have also been previously
established. More precisely, we establish that if f : [N] -> [-1,1] is a
function with || f ||_{U^{s+1}[N]} > \delta then there is a bounded-complexity
s-step nilsequence F(g(n)\Gamma) which correlates with f, where the bounds on
the complexity and correlation depend only on s and \delta. From previous
results, this conjecture implies the Hardy-Littlewood prime tuples conjecture
for any linear system of finite complexity.
  Erratum (added April 2024): a 6-page erratum is available as a separate PDF.
