The extension of the analytic fractal curve trees of (2601.17490} to analytic surface patch trees reveals a new geometric structure: branch points are replaced by interface curves that transmit the full analytical state of parent patches to their children. These interfaces prove to be central in determining the topology of the surface patch trees, including for the conditions for self-similarity of the interfaces, the patches and thus the trees. We establish the analytic conditions for the integrability and well-posedness of the surface patch trees and introduce further restrictions for conformality. We demonstrate that patch trees have a natural foliation that slices the trees into one dimensional curve trees, each of which has their own Hausdorff dimension, jointly creating a smooth dimension field. We extend the two dimensional surface model to arbitrary dimensions $n$ where $n-1$ interface manifolds transport the $n$ field state of the parent patches to their child branches. We note that the balance or discrepancy between patch field dimension and the dimensions in which the branches may evolve, determine the analytical regime from essentially geometrical to essentially operational.

翻译：将(2601.17490)中的解析分形曲线树扩展到解析曲面斑块树，揭示了一种新的几何结构：分支点被接口曲线取代，这些接口曲线将父斑块的全部解析状态传递给子斑块。这些接口被证明在确定曲面斑块树的拓扑结构中起着核心作用，包括接口、斑块乃至整棵树自相似性的条件。我们建立了曲面斑块树可积性与适定性的解析条件，并引入了共形性的进一步约束。我们证明斑块树具有一种自然的叶状结构，可将树分割为一维曲线树，每棵曲线树都有各自的豪斯多夫维数，共同形成一个光滑的维度场。我们将二维曲面模型推广到任意维度$n$，其中$n-1$维接口流形将父斑块的$n$维场状态传递给子分支。我们注意到，斑块场维度与分支可能演化维度之间的平衡或差异，决定了从本质上几何到本质上操作的解析机制。