Research-Stack/6-Documentation/papers/OTOM/05_Bind_Primitive.md

The Bind Primitive: Core Abstraction of OTOM

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

---

Abstract

The bind primitive is the fundamental operation underlying all computation in the OTOM framework. It provides a unified interface for state transitions across all domains, with explicit cost functions and invariant preservation guarantees.

---

1. The bind Signature

\text{bind} : (A \times B \times \text{Metric}) \rightarrow \text{Bind}\ A\ B

---

2. Five bind Classes

Class	Domain	Signature
informational_bind	Information theory	(\text{State} \times \text{Message} \times \text{Entropy}) \rightarrow \text{Bind}
geometric_bind	Geometry	(\text{Manifold} \times \text{Point} \times \text{Distance}) \rightarrow \text{Bind}
thermodynamic_bind	Thermodynamics	(\text{System} \times \text{Process} \times \text{Energy}) \rightarrow \text{Bind}
physical_bind	Physics	(\text{Field} \times \text{Particle} \times \text{Action}) \rightarrow \text{Bind}
control_bind	Control theory	(\text{Controller} \times \text{Plant} \times \text{Error}) \rightarrow \text{Bind}

---

3. Required Components

Every bind instance must provide:

3.1 Lawful Check

\text{lawful} : \text{Bind}\ A\ B \rightarrow \text{Bool}

Predicate verifying state transition validity.

3.2 Cost Function

\text{cost} : \text{Bind}\ A\ B \rightarrow \text{UInt32}

Q16.16 fixed-point cost metric.

3.3 Invariant Extractor

\text{invariant} : A \rightarrow \text{String}

Human-readable invariant description.

---

4. The Collapse Principle

All domain logic reduces to bind instances. There are no:

• Utility files
• Helper functions
• Ad-hoc state machines

Every function belongs to a specific bind instance.

---

5. Implementation

Lean 4 Module: Bind.lean

structure Bind (A B : Type) where
  source : A
  target : B
  metric : Metric
  lawful : Bool
  cost : UInt32
  invariant : String

---

6. Theorems

6.1 bind Preserves Invariant

\forall b : \text{Bind}\ A\ B, \text{lawful}(b) \implies \text{invariant}(\text{source}(b)) = \text{invariant}(\text{target}(b))

---

7. References

• Research Stack, AGENTS.md §4
• Research Stack, OTOM Ontology v2.2.