# Metatyping System: Type-Level Metaprogramming

**Authors:** Research Stack Team  
**Date:** April 2026  
**Domain:** TTM Layer M (Lean Semantics - Core)  
**OTOM Version:** 2.2

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## Abstract

The Metatyping System (Sigma) provides type-level metaprogramming capabilities for OTOM. It enables compile-time computation, type-directed code generation, and formal verification of type safety properties.

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## 1. Metatype Hierarchy

### 1.1 Type Universe

$$\mathcal{U}_0 : \mathcal{U}_1 : \mathcal{U}_2 : \cdots$$

### 1.2 Metatype Constructor

$$\text{Meta}(A) = \{\text{operations}(A), \text{properties}(A), \text{instances}(A)\}$$

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## 2. Type-Level Computation

### 2.1 Type Functions

$$F : \text{Type} \rightarrow \text{Type}$$

### 2.2 Type Classes

$$\text{Class}(C) = \{(T, \text{impl}_T) \mid T : C\}$$

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## 3. Sigma Notation

### 3.1 Trajectory Quality

$$\Sigma = \int_0^T \sqrt{g_{ij} \dot{x}^i \dot{x}^j}\, dt$$

### 3.2 Metatyping Judgment

$$\Gamma \vdash e : \tau \leadsto \Sigma$$

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## 4. Generic Programming

### 4.1 Type Specialization

$$\text{specialize}(f, T) = f_T$$

### 4.2 Template Instantiation

$$\text{instantiate}(\text{template}, \text{args}) = \text{code}$$

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## 5. Implementation

**Lean 4 Modules:**
- `Metatype.lean` — Core metatype system
- `Canon.lean` — Canonical form normalization
- `Transition.lean` — State transitions

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## 6. Theorems

### 6.1 Type Safety

$$\text{wellTyped}(e) \implies \text{noRuntimeTypeError}(e)$$

### 6.2 Parametricity

$$\forall f : \forall a. a \rightarrow a, \forall x, f(x) = x$$

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## 7. References

- Wadler, P. (1989). Theorems for free!
- Pierce, B.C. (2002). Types and Programming Languages.
- Research Stack, OTOM Ontology v2.2.