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Contextuality from Single-State Representations: An Information-Theoretic Principle for Adaptive Intelligence

Song-Ju Kim

TL;DR

This work proves that any classical model reproducing contextual outcome statistics must incur an irreducible information-theoretic cost, and identifies contextuality as a general representational constraint on adaptive intelligence, independent of physical implementation.

Abstract

Adaptive systems often operate across multiple contexts while reusing a fixed internal state space due to constraints on memory, representation, or physical resources. Such single-state reuse is ubiquitous in natural and artificial intelligence, yet its fundamental representational consequences remain poorly understood. We show that contextuality is not a peculiarity of quantum mechanics, but an inevitable consequence of single-state reuse in classical probabilistic representations. Modeling contexts as interventions acting on a shared internal state, we prove that any classical model reproducing contextual outcome statistics must incur an irreducible information-theoretic cost: dependence on context cannot be mediated solely through the internal state. We provide a minimal constructive example that explicitly realizes this cost and clarifies its operational meaning. We further explain how nonclassical probabilistic frameworks avoid this obstruction by relaxing the assumption of a single global joint probability space, without invoking quantum dynamics or Hilbert space structure. Our results identify contextuality as a general representational constraint on adaptive intelligence, independent of physical implementation.

Contextuality from Single-State Representations: An Information-Theoretic Principle for Adaptive Intelligence

TL;DR

This work proves that any classical model reproducing contextual outcome statistics must incur an irreducible information-theoretic cost, and identifies contextuality as a general representational constraint on adaptive intelligence, independent of physical implementation.

Abstract

Adaptive systems often operate across multiple contexts while reusing a fixed internal state space due to constraints on memory, representation, or physical resources. Such single-state reuse is ubiquitous in natural and artificial intelligence, yet its fundamental representational consequences remain poorly understood. We show that contextuality is not a peculiarity of quantum mechanics, but an inevitable consequence of single-state reuse in classical probabilistic representations. Modeling contexts as interventions acting on a shared internal state, we prove that any classical model reproducing contextual outcome statistics must incur an irreducible information-theoretic cost: dependence on context cannot be mediated solely through the internal state. We provide a minimal constructive example that explicitly realizes this cost and clarifies its operational meaning. We further explain how nonclassical probabilistic frameworks avoid this obstruction by relaxing the assumption of a single global joint probability space, without invoking quantum dynamics or Hilbert space structure. Our results identify contextuality as a general representational constraint on adaptive intelligence, independent of physical implementation.
Paper Structure (16 sections, 1 theorem, 6 equations, 1 figure)

This paper contains 16 sections, 1 theorem, 6 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Representational Framework
  3. Single-State Representations and Interventions
  4. Classical Probabilistic Representations
  5. Information-Theoretic Obstruction
  6. Main Theorem
  7. Interpretation and Scope
  8. A Constructive Illustration
  9. Why Nonclassical Probability Appears
  10. Implications for Adaptive Intelligence
  11. Discussion and Outlook
  12. Proof of Theorem \ref{['thm:info_obstruction']}
  13. Setup
  14. Contextual Dependence and Incompatible Conditionals
  15. Information-Theoretic Bound
  16. ...and 1 more sections

Key Result

Theorem 1

Let $C$ denote the intervention, $S$ the internal state, and $O$ the outcome. Consider any classical single-state representation that reproduces the observed family of conditionals $\{p(o\mid c)\}_{c\in\mathcal{C}}$ via a single underlying probability space. If the observed statistics exhibit contex Equivalently, the required contextual information strictly exceeds what can be attributed to the in

Figures (1)

  • Figure 1: Schematic illustration of contextuality under the single-state constraint. Each pair of interventions admits a consistent marginal distribution, yet no global joint distribution exists that reproduces all pairwise marginals simultaneously. This incompatibility forces any classical single-state representation to introduce additional contextual information.

Theorems & Definitions (6)

  • Definition 1: Single-State Representation
  • Definition 2: Interventions
  • Definition 3: Representation Consistency
  • Definition 4: Classical Probabilistic Representation
  • Definition 5: Context Information
  • Theorem 1: Information-Theoretic Obstruction to Single-State Classical Representations