Research-Stack/6-Documentation/papers/OTOM/13_Metatyping_System.md

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.