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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2604.21945 |
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| _version_ | 1866918464779714560 |
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| author | Lawrence, Neil D. |
| author_facet | Lawrence, Neil D. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_21945 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The No Barber Principle: Towards Formalised Selection in the Inaccessible Game Lawrence, Neil D. Category Theory Information Theory 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. |
| title | The No Barber Principle: Towards Formalised Selection in the Inaccessible Game |
| topic | Category Theory Information Theory |
| url | https://arxiv.org/abs/2604.21945 |