The Origin of the Inaccessible Game

2026-01-18

AI Summary

This work addresses the limitation of classical Shannon entropy, which—due to the non-negativity of conditional entropy—cannot characterize “origin” states where joint entropy vanishes while marginal entropies remain positive. Building upon the Baez–Fritz–Leinster–Parzygnat categorical framework and von Neumann entropy, the authors construct a quantum information-geometric model incorporating an entropy-time parameterization to unify boundary dynamics. Under the constraint of marginal entropy conservation, they investigate maximum entropy production processes. Key contributions include the first explicit definition of an “inaccessible game” origin state (termed LME origin) in quantum information geometry, along with proofs of its existence and local unitary uniqueness; the revelation that constrained gradient flows possess a dual structure of symmetric dissipation and reversible unitary evolution; and the establishment of equivalence between marginal entropy conservation and modular energy conservation, which reduces to the classical energy constraint of non-equilibrium thermodynamics within Gibbs regions.

Technology Category

• Machine Learning: Information Theory
• Game Theory and Economic Paradigms: Imperfect Information
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Security and Privacy: Data transparency and provenance

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. Because the marginal-entropy sum is saturated termwise, the constraint gradient vanishes and first-order tangency is vacuous; admissible directions are selected by the kernel of the constraint Hessian, characterised by the marginal-preserving tangent space. 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 $c>0$. In this parametrisation, the infinite affine-time approach to the boundary maps to a finite entropy-time interval. The constrained dynamics split into a symmetric dissipative component realising SEA and a reversible component represented as unitary evolution. As in the classical game, marginal-entropy conservation is equivalent to conservation of a sum of local modular Hamiltonian expectations, a state-dependent"modular energy"; in Gibbs regimes where local modular generators become approximately parameter-invariant, this reduces to familiar fixed-energy constraints from nonequilibrium thermodynamics.

Problem

• inaccessible game
• origin
• von Neumann entropy
• marginal-entropy conservation
• zero joint entropy

Innovation

• von Neumann entropy
• locally maximally mixed state
• entropy time
• constrained gradient flow
• modular Hamiltonian