  Motivated by Arveson's conjecture, we introduce a notion of hyperrigidity for
a partial order on the state space of a $C^*$-algebra $B$. We show how this
property is equivalent to the existence of a boundary: a subset of the pure
states which completely encodes maximality in the given order. In the classical
case where $B$ is commutative, such boundaries are known to exist when the
partial order is induced by some well-behaved cone. However, the relevant order
for the purposes of Arveson's conjecture is the dilation order, which is not
known to fit into this framework. Our main result addresses this difficulty by
showing that the dilation maximal states are stable under absolute continuity.
Consequently, we obtain the existence of a boundary projection in the bidual
$B^{**}$, on which all dilation maximal states must be concentrated. The
topological regularity of this boundary projection is shown to lie at the heart
of Arveson's conjecture. Our techniques do not require $B$ to be commutative.
