We verify that persistent observers in causally invariant hypergraph substrates satisfy the conditions of the Conant-Ashby Good Regulator Theorem. Building on Wolfram's hypergraph physics and Vanchurin's neural network cosmology, we formalize persistent observers as entities that minimize prediction error at their boundary with the environment. Applying a modern reformulation of the Conant-Ashby theorem, we demonstrate that hypergraph observers satisfy Good Regulator conditions, requiring them to maintain internal models. Once an internal model with loss function exists, the emergence of a Fisher information metric follows from standard information geometry. Invoking Amari's uniqueness theorem for reparameterization-invariant gradients, we show that natural gradient descent is the unique admissible learning rule. Under the ansatz M=F^2 for exponential family observers and one specific convergence time functional, we derive a closed-form formula for the regime parameter alpha in Vanchurin's Type II framework, with a quantum-classical threshold at kappa(F)=2. However, three alternative convergence models do not reproduce this result, so this prediction is strongly model-dependent. We further introduce the directional regime parameter alpha_{v_k} and the trace-free deviation tensor, showing that a single observer can simultaneously occupy different Vanchurin regimes along different eigendirections of the Fisher metric. This connects Wolfram and Vanchurin frameworks through established theorems, providing approximately 25-30% novel contribution.

翻译：我们验证了因果不变超图基底中的持久观测者满足Conant-Ashby良好调节器定理的条件。基于Wolfram的超图物理学和Vanchurin的神经网络宇宙学，我们将持久观测者形式化为在其与环境边界处最小化预测误差的实体。应用Conant-Ashby定理的现代重构，我们证明超图观测者满足良好调节器条件，这要求它们维持内部模型。一旦存在具有损失函数的内部模型，Fisher信息度量的出现便遵循标准信息几何学。通过引用Amari关于重参数化不变梯度的唯一性定理，我们证明自然梯度下降是唯一可采纳的学习规则。在指数族观测者的假设M=F^2及特定收敛时间泛函下，我们推导出Vanchurin类型II框架中体系参数alpha的闭式公式，其量子-经典阈值位于kappa(F)=2。然而，三种替代收敛模型未能复现此结果，因此该预测具有强烈的模型依赖性。我们进一步引入方向性体系参数alpha_{v_k}和无迹偏差张量，表明单个观测者可沿Fisher度量的不同特征方向同时占据不同的Vanchurin体系。这通过既定定理将Wolfram与Vanchurin框架相连接，提供了约25-30%的新贡献。