  Instability to high toroidal mode number ($n$) ballooning modes has been
proposed as the primary gradient-limiting mechanism for tokamak equilibria with
negative triangularity ($\delta$) shaping, preventing access to strong H-mode
regimes when $\delta\ll0$. To understand how this mechanism extrapolates to
reactor conditions, we model the infinite-$n$ ballooning stability as a
function of internal profiles and equilibrium shape using a combination of the
CHEASE and BALOO codes. While the critical $\delta$ required for avoiding
$2^\mathrm{nd}$ stability to high-$n$ modes is observed to depend in a
complicated way on various shaping parameters, including the equilibrium aspect
ratio, elongation and squareness, equilibria with negative triangularity are
robustly prohibited from accessing the $2^\mathrm{nd}$ stability region,
offering the prediction that that negative triangularity reactors should
maintain L-mode-like operation. In order to access high-$n$ $2^\mathrm{nd}$
stability, the local shear over the entire bad curvature region must be
sufficiently negative to overcome curvature destabilization on the low field
side. Scalings of the ballooning-limited pedestal height are provided as a
function of plasma parameters to aid future scenario design.
