The Bind Primitive: Core Abstraction# Bind Primitive

Authors: Research Stack Team Date: April 2026 Domain: TTM Layer A (Compression/Routing) + Bind Theory OTOM Version: 2.2

References: See 00_Master_References.md for complete source mapping

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. The bind primitive is now extended with geometric structure folding (Torus-Menger-Horn) and Mass Number gates for manifold merging admissibility.

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.

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