The Initiality Conjecture is Resolved by Partition: True In The Rigid Universe, False Outside It

Abstract. We resolve the Initiality Conjecture for Martin-Löf type theory, open since Streicher (1991) and the subject of Voevodsky's unfinished program (2016). The conjecture asserts that the term model Ctx(MLTT) is initial among all categories with families (CwFs) modeling MLTT. We prove the conjecture decomposes into two parts along a boundary identified by the Rigidity Constraint (Eden 2025): true for CwFs in which every type has contractible automorphism group (the rigid universe U_Rigid), and false for CwFs containing types with non-trivial automorphisms (U_HoTT \ U_Rigid). The obstruction to initiality is monodromy: when π₁(U, A) ≠ 0, the interpretation functor depends on a choice of cleavage, destroying uniqueness. We identify this as the same obstruction discovered in Awodey–Hua (2026), where Axioms A1 and A2 are insufficient to ensure that the J-eliminator is a function rather than a relation. The third axiom required to close their gap (monodromy triviality) is equivalent to the rigidity constraint. The conjecture was open for thirty years because the boundary between the two cases was undisclosed.