Three Mechanisms for Rigidity: Why Lean and Coq are Rigid Without Knowing It

We identify three distinct mechanisms by which a type theory achieves $\pi_1(\mathcal{U}, A) = 0$---the condition required for strict initiality, canonicity, and coherent semantics. The Rigidity Constraint (Eden 2025) established that $\isContr(\Aut(A))$ is necessary for coherent interpretation. We show this is one of three routes to the same endpoint. \emph{Mechanism 1} (Contractible Aut): types genuinely have no non-trivial symmetries. \emph{Mechanism 2} (Identity Collapse): identity types are collapsed to propositions (UIP/Axiom K), making symmetries invisible to the identity structure. \emph{Mechanism 3} (Non-Univalent Universe): types may have non-trivial symmetries, but the universe does not see them because there is no univalence axiom. We prove that the kernels of Lean~4 and Coq achieve rigidity via Mechanism~3: they are rigid not because their types lack symmetries, but because their universes are blind to them. We further identify a new decoupling: Lean with \texttt{Classical.choice} has strict initiality without canonicity---the exact inverse of cubical HoTT. The partial order (initiality $\Rightarrow$ canonicity) is preserved because the failure is in the extended language, not the interpreted one. The complete classification extends the $3 \times 3$ table of the preceding papers to a $3 \times 7$ table covering all major formulations of type theory.