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

Technical Deep Dive: Contextuality from Single-State Representations

Overview

This paper presents a fundamental information-theoretic result showing that classical probabilistic models with a fixed internal state space cannot faithfully represent contextual dependence without incurring an irreducible cost in additional contextual information. This result reframes contextuality as a general representational constraint, rather than a peculiarity of specific physical theories.

Problem & Context

Adaptive systems often operate across multiple contexts while reusing a fixed internal state space due to constraints on memory, representation, or physical resources. This "single-state reuse" is ubiquitous in natural and artificial intelligence, but its fundamental representational consequences remain poorly understood.

The authors aim to isolate the constraints on representation itself, independent of dynamics, learning rules, or physical implementation. They formalize a minimal framework for single-state representations and classical probabilistic models, and then establish an information-theoretic obstruction to reconciling contextual dependence within this framework.

Methodology

The authors introduce a formal representational framework with the following key elements:

• Single-State Representation: A fixed internal state space S that cannot be indexed, duplicated or partitioned according to context. Contextual variation must be mediated solely through "interventions" that act on S.
• Classical Probabilistic Representation: Outcome statistics are generated by conditional probability distributions over observables given the internal state s ∈ S, with no additional contextual degrees of freedom beyond S.
• Context Information: The additional information required to account for contextual dependence in observable statistics beyond what can be encoded in S alone.

Results

The main result is an information-theoretic theorem:

Theorem 1: Any classical single-state representation that reproduces observed contextual dependence must introduce an additional contextual variable M such that:

`` H(M) > I(C; O | S) ``

where C is the intervention, S the internal state, and O the observable outcome. This inequality establishes an irreducible information-theoretic cost associated with representing contextual dependence within a fixed internal state space.

The authors also provide a minimal constructive illustration showing how this information-theoretic obstruction arises in a concrete model.

Interpretation

The key insights are:

• Contextuality is not a peculiarity of quantum mechanics, but a general representational constraint on adaptive systems operating under single-state reuse.
• The obstruction does not reflect insufficient state capacity, but rather a fundamental incompatibility between classical probabilistic representations and the requirement of a fixed internal state space across interventions.
• Nonclassical probabilistic frameworks, such as quantum probability, naturally accommodate contextual dependence by relaxing the assumption of a single global probability space.

Limitations & Uncertainties

• The results are representation-theoretic and do not depend on specific dynamics, learning rules, or physical implementation details.
• However, the analysis assumes "memoryless" interventions that leave no persistent trace in the internal state. Allowing interventions to modify the internal state in a lasting way would introduce additional memory resources, violating the single-state constraint.
• The authors do not explore the relationship between the information-theoretic cost and other measures of representational complexity or efficiency.

What Comes Next

The authors suggest several avenues for future work, including:

• Sharpening the connection between information-theoretic cost and representational complexity
• Exploring extensions to dynamical and learning settings
• Examining how similar obstructions manifest in multi-agent or hierarchical systems
• Identifying the minimal mathematical ingredients required for nonclassical representations that accommodate contextual dependence under single-state constraints

Overall, this work establishes contextuality as a fundamental representational constraint on adaptive intelligence, with broad implications for the design and analysis of intelligent systems operating under resource limitations.