  We present new applications of parity inversion and time-reversal to the
emergence of complex behavior from simple dynamical rules in stochastic
discrete models. Our parity-based encoding of causal relationships and
time-reversal construction efficiently reveal discrete analogs of stable and
unstable manifolds. We demonstrate their predictive power by studying
decision-making in systems biology and statistical physics models. These
applications underpin a novel attractor identification algorithm implemented
for Boolean networks under stochastic dynamics. Its speed enables resolving a
longstanding open question of how attractor count in critical random Boolean
networks scales with network size, and whether the scaling matches biological
observations. Via 80-fold improvement in probed network size ($N=16,384$), we
find the surprisingly low scaling exponent of $0.12\pm 0.05$ -- approximately
one tenth the analytical upper bound. We demonstrate a general principle: a
system's relationship to its time-reversal and state-space inversion constrains
its repertoire of emergent behaviors.
