The No Barber Principle: Towards Formalised Selection in the Inaccessible Game

2026-04-21

AI Summary

This work addresses the impredicative circularity arising in unattainable games due to distinctions that cannot be internally represented within the system. To resolve this, the paper introduces the “No Barber Principle,” which prohibits dynamical rules relying on external arbiters or structures. Inspired by Russell’s paradox and Lawvere’s diagonalization argument, it incorporates this paradoxical structure into information-theoretic dynamical systems for the first time, thereby constructing an endogenous selection mechanism. Using tools from category theory (FinProb and NCFinProb), information theory (Shannon and von Neumann entropy), and marginal entropy conservation constraints, the study demonstrates that the classical probability category FinProb violates the principle due to the existence of copying morphisms, whereas the noncommutative probability category NCFinProb—lacking canonical copying maps—is better suited as an internal language, thus establishing a consistent foundation for selection rules in unattainable games.

Technology Category

• Game Theory and Economic Paradigms: Imperfect Information
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Machine Learning: Probabilistic Circuits and Graphical Models

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Semantics and Knowledge: Data modeling to support human-machine intelligence, including LLMs agents, intelligent system behavior, explanations, and user-friendly interactions

Abstract

The inaccessible game (Lawrence, 2025, 2026) is an information-theoretic dynamical system governed by three information loss axioms, a marginal entropy conservation constraint and maximum entropy dynamics. In this paper we look at selection in the game. Our aim is to develop a selection policy for the game rules based on a minimal set of assumptions. We seek necessary consistency constraints for self-determining dynamical systems. Specifically, we suggest that rules that quantify over distinctions they cannot internally represent risk impredicative-style circularity. Our criterion is motivated by an analogy with Russell's paradox. We formulate a no-barber principle which prohibits dynamics that appeal to external adjudicators or structure lying outside the system. To motivate our principle we examine Russell's paradox through its structural formalisation as a Lawvere diagonalisation. The marginal-entropy conservation in the game is a nontrivial entropy constraint which prohibits external structure. Through the no-barber principle we argue (i) the classical category FinProb, in which Shannon entropy is characterised, is cartesian and provides canonical diagonal (copying) maps that make Lawvere-style constructions expressible and is structurally incompatible with the no-copying instantiation of the no-barber principle studied here. (ii) the noncommutative category NCFinProb, in which von Neumann entropy is characterised, is symmetric monoidal and lacks canonical copying maps, making it a more natural candidate for the game's internal language.

Problem

• inaccessible game
• selection policy
• impredicativity
• no-barber principle
• information-theoretic dynamical system

Innovation

• No Barber Principle
• Inaccessible Game
• Lawvere Diagonalisation
• NCFinProb
• Impredicativity