Machine-verified formalization in Lean 4 / Mathlib v4.28.0. In any monoidal closed, locally finitely presentable category where the tensor product preserves finite presentability, the internal hom endofunctor has a fixed point L that is unique up to isomorphism. This fixed point supports universal computation. The proof is a single causal chain: ∅ → M(∅) → M²(∅) → ⋯ → L Forced development (Adamek chain from initial object) ↓ L ≅ [A, L] Identity (Lambek iso: the fixed point IS its function space) ↓ Closed container Containerization (boundary persists under the generator) ↓ Identity loop Identity modulation (fold/unfold IS the computational core) ↓ Lambda model Universal computation (app + abs + β + η, no ℕ needed) Every arrow is a Lean theorem. The reflexive fixed point L ≅ [L, L] is already a model of the untyped lambda calculus, which is Turing-complete. Computation is not added to the fixed point; it IS the fixed point. Parallel to the categorical construction, the project proves the computability theory side independently: the Church-Turing characterization theorem, the Effective Myhill Isomorphism Theorem, and the strong Rogers isomorphism. These connect to the categorical side via the three-layer Kleene bridge. The uniqueness statement is tower initiality: the Adamek chain from ∅ is initial among all M-generated chains. Any process that generates structural levels by iterating M receives a unique chain morphism from the canonical chain. 42 files. 8051 lines. 0 sorry. 0 custom axioms.