  It was conjectured by Bergelson, Tao, and Ziegler that every Host--Kra
${\mathbf F}_p^\omega$-system of order $k$ is an Abramov system of order $k$.
This conjecture has been verified for $k \leq p+1$. In this paper we show that
the conjecture fails when $k=5, p=2$. We in fact establish a stronger
(combinatorial) statement, in that we produce a bounded function $f: {\mathbf
F}_2^n \to {\mathbf C}$ of large Gowers norm $\|f\|_{U^6({\mathbf F}_2^n)}$
which (as per the inverse theorem for that norm) correlates with a
non-classical quintic phase polynomial $e(P)$, but with the property that all
such phase polynomials $e(P)$ are ``non-measurable'' in the sense that they
cannot be well approximated by functions of a bounded number of random
translates of $f$.
