  In previous work, we introduced the notion of binomial cup-one algebras,
which are differential graded algebras endowed with Steenrod $\cup_1$-products
and compatible binomial operations. Given such an $R$-dga, $(A,d_A)$, defined
over the ring $R=\mathbb{Z}$ or $\mathbb{Z}_p$ (for $p$ a prime), with
$H^0(A)=R$ and with $H^1(A)$ a finitely generated, free $R$-module, we show
that $A$ admits a functorially defined 1-minimal model, $\rho\colon
(\mathcal{M}(A),d)\to (A,d_A)$, which is unique up to isomorphism. Furthermore,
we associate to this model a pronilpotent group, $G(A)$, which only depends on
the 1-quasi-isomorphism type of $A$. These constructions, which refine
classical notions from rational homotopy theory, allow us to distinguish spaces
with isomorphic (torsion-free) cohomology rings that share the same rational
1-minimal model, yet whose integral 1-minimal models are not isomorphic.
