Chapter 50: The Hinge Theorem

The Complete Dependency Chain (the relevant definition, Ch. 60) established that every structure in Book III descends from five generators and seven axioms. This chapter draws the consequence: every result in Books IV–VII is a sector instantiation of Book III’s enrichment structure. Section 1 states the Hinge Theorem , which asserts that each downstream book is the enrichment functor Enr^k restricted to a specific sector of the 4+1 decomposition. Section 2 proves the Sector Instantiation Lemma (Proposition [prop:sector-instantiation-lemma]): restricting Enr^k to a single sector produces a well-defined sub-theory. Section 3 treats each of the four downstream books as a specific instantiation. Section 4 proves that the seven-book architecture is not a design choice but a mathematical necessity: the enrichment levels E₀–E₃ combined with the tripartite structure of E₁ yield exactly 1 + 3 + 1 + 1 + 1 = 7 books.