The Origin of the Inaccessible Game

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.