(Prerequisites) While we have endeavoured to survey most of the previous works directly aimed at the Thomas-Yau conjecture, there is a very extensive literature on geometric measure theory and symplectic geometry in the background. We do not assume expertise on these matters, but some previous exposures such as F. Morgan’s introductory book [60], and the excellent surveys of Auroux [9] and Smith [73] would be useful. The most important background facts for our main purpose are also recalled in section 5.1 and the Appendix on the Fukaya category. While the brief summary therein is not completely sufficient for all arguments in this paper, we hope the casual reader could get the main gists, if not some sporadic remarks. The punctilious reader may wish to refer to the Floer degree and sign convention summarized in the Appendix, which is different from e.g. Seidel’s book [69]. The various allusions to Kähler geometry are mainly for motivational purposes, which can be skipped by readers less interested in these topics.