  Bak, Tang, and Wiesenfeld developed their theory of self-organized
criticality in the late 1980s to explain why many real-life processes exhibit
signs of critical behavior despite the absence of a tuning parameter. A decade
later, Dickman, Mu\~noz, Vespignani, and Zapperi explained self-organized
criticality as an external force pushing a hidden parameter toward the critical
value of a traditional absorbing-state phase transition. As evidence, they
observed empirically that for various sandpile models, the particle density in
a finite box under driven-dissipative dynamics converges to the critical
density of an infinite-volume version of the model. We give the first proof of
this well-known density conjecture in any setting by establishing it for
activated random walk in one dimension. We prove that two other natural
versions of the model have the same critical value, further establishing
activated random walk as a universal model of self-organized criticality.
