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Thunderstorms are difficult to predict because of their small length scale and fast predictability destruction. A cell's predictability is constrained by properties of the flow in which it is embedded (e.g., vertical wind shear), and associated instabilities (e.g., convective available potential energy). To assess how predictability of thunderstorms changes with environment, two groups of 780 idealized simulations (each using a different microphysics scheme) were performed over a range of buoyancy and shear profiles. Results were not sensitive to the scheme chosen. The gradient in diagnostics (updraft speed, storm speed, etc.) across shear-buoyancy phase space represents sensitivity to small changes in initial conditions: a proxy for inherent predictability. Storm evolution is split into two groups, separated by a U-shaped bifurcation in phase space, comprising 1) cells that continue strengthening after 1 h versus 2) those that weaken. Ensemble forecasts in regimes near this bifurcation are hence expected to have larger uncertainty, and adequate dispersion and reliability is essential. Predictability loss takes two forms: (i) chaotic error growth from the largest and most powerful storms, and (ii) tipping points at the U-shaped perimeter of the stronger storms. The former is associated with traditional forecast error between corresponding grid points, and is here counterintuitive; the latter is associated with object-based error, and matches the mental filtering performed by human forecasters for the convective scale.