LarsenClose/fixed-point-formalization: Substrate-Independent Computation from Categorical Fixed Points

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.