Recall $\hat{\theta}_{i}=\arg\int_{L_{i}}\Omega$. The caveat is that $\max\{\hat{\theta}_{i}^{\pm}\}$ and $\min\{\hat{\theta}_{i}^{\pm}\}$ do not quite control $\hat{\theta}_{i}$, so the above Theorem 3.28 does not imply the phase angle inequality (31). In this sense the conclusion of Theorem 3.28 is weaker than Theorem 3.21, illustrating the power of the positivity condition.

On the other hand, we will heuristically argue in section 3.6 that in the Thomas-Yau-Joyce picture, once we assume the existence of Joyce’s Bridgeland stability condition, the phase angle inequality (31) can be deduced without the positivity condition in Theorem 3.21.

We think it is very interesting to either prove the positivity condition as a consequence of the other assumptions, or to find another Floer theoretic argument for (31) that requires neither the positivity condition, nor the a priori knowledge of special Lagrangian representatives.