围棋棋形数目的演化与终局邻域的结构分析 Structural Analysis of Go Shape Evolution and Terminal Neighborhood Dynamics

本文提出了“局部冻结模型”(Local Freezing Model),用于分析围棋终局附近的状态空间结构。模型将终局局面分解为若干弱交互的局部模块,通过构建邻接图刻画模块之间的依赖关系,并以奇环作为回溯扩展中产生分支的唯一结构性来源。上述结构性约束解释了为何平均父局面数始终接近 1,以及为何终局邻域仅呈现极缓慢的指数增长。模拟结果显示,log₁₀|Lₖ| 随回溯步数几乎线性增长,验证了模型的预测。研究表明,终局状态空间具有强局部性、强约束性与近似确定性。 This work presents the Local Freezing Model, a structural framework for analyzing Go positions near terminal endgames. The model decomposes a final position into weakly interacting local modules, builds an adjacency graph to capture their dependencies, and identifies odd cycles as the only source of branching during backward expansion. These structural constraints explain why the average number of parent positions remains close to one and why the endgame neighborhood grows only at a very slow exponential rate. Simulations confirm that the logarithm of the neighborhood size increases almost linearly with backward distance, revealing that the endgame state space is highly local, constrained, and nearly deterministic.