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.

翻译：不可达博弈是一种信息几何框架，其中信息损失的动力学源于边际熵守恒下的最大熵产生。我们研究该博弈的起始状态——起源。经典香农熵禁止了具有零联合熵与正边际熵的表示：条件熵的非负性排除了这种可能性。在Baez Fritz Leinster Parzygnat范畴框架内，将香农熵替换为冯·诺依曼熵则消除了这一障碍，并允许一个明确定义的起源：一个具有最大混合边际的全局纯态，该态在局域幺正等价意义下被选定。在此LME起源处，边际熵守恒转化为二阶几何条件。由于边际熵和逐项饱和，约束梯度为零，一阶相切条件失效；允许方向由约束Hessian矩阵的核选定，其特征表现为边际保持切空间。我们推导了矩阵指数族中的约束梯度流，并证明当趋近起源时，仿射时间参数会发生退化。这激发了一种公理上可区分的重新参数化——熵时间$t$，其定义为$dH/dt = c$（$c>0$为固定常数）。在此参数化下，趋近边界的无限仿射时间映射到一个有限的熵时间区间。约束动力学分解为一个实现SEA的对称耗散分量，以及一个表示为幺正演化的可逆分量。与经典博弈类似，边际熵守恒等价于局域模哈密顿量期望值之和的守恒，这是一种依赖于状态的“模能量”；在局域模生成元近似保持参数不变的吉布斯区域中，该约束简化为非平衡热力学中常见的固定能量约束。