  Binary Chandrasekhar-mass white dwarfs accreting mass from non-degenerate
stellar companions through the single-degenerate channel have reigned for
decades as the leading explanation of Type Ia supernovae. Yet, a comprehensive
theoretical explanation has not yet emerged to explain the expected properties
of the canonical near-Chandrasekhar-mass white dwarf model. A simmering phase
within the convective core of the white dwarf leads to the ignition of one or
more flame bubbles scattered across the core. Consequently,
near-Chandrasekhar-mass single-degenerate SNe Ia are inherently stochastic, and
are expected to lead to a range of outcomes, from subluminous SN 2002cx-like
events, to overluminous SN 1991T-like events. However, all prior simulations of
the single-degenerate channel carried through the detonation phase have set the
ignition points as free parameters. In this work, for the first time, we place
ignition points as predicted by {\it ab initio} models of the convective phase
leading up to ignition, and follow through the detonation phase in fully
three-dimensional simulations. Single-degenerates in this framework are
characteristically overluminous. Using a statistical approach, we determine the
$^{56}$Ni mass distribution arising from stochastic ignition. While there is a
total spread of $\gtrsim 0.2 M_{\odot}$ for detonating models, the distribution
is strongly left-skewed, and with a narrow standard deviation of $\simeq 0.03
M_{\odot}$. Conversely, if single-degenerates are not overluminous but
primarily yield normal or failed events, then the models require fine-tuning of
the ignition parameters, or otherwise require revised physics or progenitor
models. We discuss implications of our findings for the modeling of
single-degenerate SNe Ia.
