Consequences of the Bootstrap Assumptions

This chapter discusses the proof for Theorem M0, together with other first consequences of the bootstrap assumptions. The only bootstrap assumption used in the proof of Theorem M0 is the bootstrap assumption BA-D on decay for k = 0, 1 derivatives. The chapter then relies on (4.1.5) and the assumptions (4.1.1) on the initial data layer. This observation allows one to use the conclusions of Theorem M0, not only for the bootstrap spacetime M in Theorem M1–M5, but also for the extended spacetime in the proof of Theorem M8, where the only assumption is the one on decay (which is established for the extended spacetime in Theorem M7). The chapter not only improves the bootstrap assumption (4.1.7), but also gains derivatives iteratively.