A complete characterization of Bridgeland stability conditions on a triangulated category, known since the inception of the subject [13, Prop 5.3], is that $\mathcal{P}(\phi_{0}<\phi\leq\phi_{0}+1)$ defines an abelian subcategory (‘the heart of a bounded $t$-structure’), and the central charge function on this abelian subcategory satisfies the Harder-Narasimhan condition.

In the Thomas-Yau-Joyce picture, $\mathcal{P}(-\frac{1}{2}<\phi\leq\frac{1}{2})$ essentially is the same as the subcategory of almost calibrated Lagrangians, if we assume there is no special Lagrangian of phase $\frac{\pi}{2}$, which holds as long as the discrete set of values of $\Omega$-periods on $H_{n}(X,\mathbb{Z})$ miss the phase angle $\frac{\pi}{2}$. Then this picture would predict almost calibrated Lagrangians to form an abelian category, which morever generate the entire derived Fukaya category using the shift operator. Morally, this is asserting that there are sufficiently many almost calibrated Lagrangians, which is evidently very deep since constructing geometric Lagrangian objects is known to be a difficult problem in symplectic topology. Another deep prediction of the existence of Bridgeland stability condition [41, conjecture 3.6], is that the derived Fukaya category $D^{b}Fuk(X)$ (after incorporating immersed and singular Lagrangians with local systems) is automatically idempotent complete, so agrees with $D^{\pi}Fuk(X)$. These predictions, if correct, are very interesting structural results on the Fukaya category, but at the moment they are controversial.

In Chapter 5, we will set up a variational framework to find special Lagrangian representatives of $D^{b}Fuk(X)$ classes under the assumption of Thomas-Yau semistability. By restricting only to the subcategory of almost calibrated Lagrangians, our program evades these difficult structural claims on the entire $D^{b}Fuk(X)$. It would thus not have the same strength as the Joyce program, nor is it subject to the same falsification criteria.