We specify the \emph{LoTT Rigidity Prover}, a complete proof system grounded in the Rigidity Constraint on coherent ML-algebra semantics. Within the rigid universe $\Rigid$, the prover: (1) \emph{proves theorems} --- because $J$ is a function with a unique canonical value, proof search terminates and every derivation has a canonical normal form; (2) \emph{replaces Lean and Coq} --- existing systems carry the Mathlib dimensional bound silently, working propositionally where $\Rigid$ works definitionally; (3) \emph{handles non-rigid types} --- $\mathbf{2}$, $S^1$, $\mathrm{Fin}(n)$ are not ignored but \emph{classified}, with monodromy certificates specifying exactly what additional structure is required for admission; and (4) \emph{supersedes Mathlib} --- every result Mathlib proves in its valid domain is reproved in $\Rigid$ with an explicit rigidity certificate, and the undisclosed $\pione = 0$ assumption is made a verified precondition rather than a silent collapse. The prover is grounded in three established results: the Rigidity Constraint (Eden 2025, Theorem 3.1); the Monodromy Gap in Awodey--Hua (arXiv:2601.06567), where Proposition 2.7 implicitly assumes lift canonicity that Axioms A1 and A2 do not supply; and the Mathlib Dimensional Bound. The prover consists of four audit primitives, a rigid universe $\Rigid$ defined as their joint fixpoint, and a complete proof engine whose decidability is a consequence of the rigidity constraint itself.