The Origin of the Inaccessible Game

Neil D. Lawrence

doi:10.48550/arXiv.2601.12576

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The Origin of the Inaccessible Game

Neil D. Lawrence

, 2026.

Abstract

The inaccessible game is an information-geometric framework where dynamics of information loss emerge from maximum entropy production under marginal-entropy conservation. We study the game’s starting state, the origin. Classical Shannon entropy forbids a representation with zero joint entropy and positive marginal entropies: non-negativity of conditional entropy rules this out. Replacing Shannon with von Neumann entropy within the Baez–Fritz–Leinster–Parzygnat categorical framework removes this obstruction and admits a well-defined origin: a globally pure state with maximally mixed marginals, selected up to local-unitary equivalence. At this LME origin, marginal-entropy conservation becomes a second-order geometric condition. We derive the constrained gradient flow in the matrix exponential family and show that, as the origin is approached, the affine time parameter degenerates. This motivates an axiomatically distinguished reparametrisation, entropy time t, defined by dH/dt = c for fixed constant c > 0. The constrained dynamics split into a symmetric dissipative component realising steepest entropy ascent and a reversible component represented as unitary evolution.

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BibTeX


@Misc{publications/the-origin-of-the-inaccessible-game,
  title = 	 {The Origin of the Inaccessible Game},
  author = 	 {Lawrence, Neil D.},
  year = 	 {2026},
  doi = 	 {10.48550/arXiv.2601.12576},
  url = 	 {/publications/the-origin-of-the-inaccessible-game.html},
  abstract = 	 {The inaccessible game is an information-geometric framework where dynamics of
information loss emerge from maximum entropy production under marginal-entropy
conservation. We study the game's starting state, the origin. Classical Shannon
entropy forbids a representation with zero joint entropy and positive marginal
entropies: non-negativity of conditional entropy rules this out. Replacing
Shannon with von Neumann entropy within the Baez–Fritz–Leinster–Parzygnat
categorical framework removes this obstruction and admits a well-defined
origin: a globally pure state with maximally mixed marginals, selected up to
local-unitary equivalence. At this LME origin, marginal-entropy conservation
becomes a second-order geometric condition. We derive the constrained gradient
flow in the matrix exponential family and show that, as the origin is
approached, the affine time parameter degenerates. This motivates an
axiomatically distinguished reparametrisation, entropy time t, defined by
dH/dt = c for fixed constant c > 0. The constrained dynamics split into a
symmetric dissipative component realising steepest entropy ascent and a
reversible component represented as unitary evolution.
}
}

Endnote

%0 Generic
%T The Origin of the Inaccessible Game
%A Neil D. Lawrence
%D 2026	
%F publications/the-origin-of-the-inaccessible-game
%R 10.48550/arXiv.2601.12576
%U /publications/the-origin-of-the-inaccessible-game.html
%X The inaccessible game is an information-geometric framework where dynamics of
information loss emerge from maximum entropy production under marginal-entropy
conservation. We study the game's starting state, the origin. Classical Shannon
entropy forbids a representation with zero joint entropy and positive marginal
entropies: non-negativity of conditional entropy rules this out. Replacing
Shannon with von Neumann entropy within the Baez–Fritz–Leinster–Parzygnat
categorical framework removes this obstruction and admits a well-defined
origin: a globally pure state with maximally mixed marginals, selected up to
local-unitary equivalence. At this LME origin, marginal-entropy conservation
becomes a second-order geometric condition. We derive the constrained gradient
flow in the matrix exponential family and show that, as the origin is
approached, the affine time parameter degenerates. This motivates an
axiomatically distinguished reparametrisation, entropy time t, defined by
dH/dt = c for fixed constant c > 0. The constrained dynamics split into a
symmetric dissipative component realising steepest entropy ascent and a
reversible component represented as unitary evolution.

RIS


TY  - GEN
TI  - The Origin of the Inaccessible Game
AU  - Neil D. Lawrence
DA  - 2026/01/18	
ID  - publications/the-origin-of-the-inaccessible-game
DO  - 10.48550/arXiv.2601.12576
UR  - /publications/the-origin-of-the-inaccessible-game.html
AB  - The inaccessible game is an information-geometric framework where dynamics of
information loss emerge from maximum entropy production under marginal-entropy
conservation. We study the game's starting state, the origin. Classical Shannon
entropy forbids a representation with zero joint entropy and positive marginal
entropies: non-negativity of conditional entropy rules this out. Replacing
Shannon with von Neumann entropy within the Baez–Fritz–Leinster–Parzygnat
categorical framework removes this obstruction and admits a well-defined
origin: a globally pure state with maximally mixed marginals, selected up to
local-unitary equivalence. At this LME origin, marginal-entropy conservation
becomes a second-order geometric condition. We derive the constrained gradient
flow in the matrix exponential family and show that, as the origin is
approached, the affine time parameter degenerates. This motivates an
axiomatically distinguished reparametrisation, entropy time t, defined by
dH/dt = c for fixed constant c > 0. The constrained dynamics split into a
symmetric dissipative component realising steepest entropy ascent and a
reversible component represented as unitary evolution.

ER  -

APA


Lawrence, N.D.. (2026). The Origin of the Inaccessible Game.  doi:10.48550/arXiv.2601.12576 Available from /publications/the-origin-of-the-inaccessible-game.html.

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Neil D. Lawrence

The DeepMind Professor of Machine Learning, Cambridge University

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Information Topography

Information topography is the geometry of how information flows through a complex system. The programme develops the mathematical foundations of that geometry — from information-theoretic axioms to predictive accounts of how communication structure emerges and reorganises.