鄰接法（Neighbor-Joining, NJ）是一種廣泛應用於系統發育樹 重建的距離式演算法，但其決策機制與誤差傳遞行為長期缺乏 統一的幾何解釋。本文提出一個基於幾何結構的NJ分析框架， 將距離矩陣空間中的拓樸判定問題重構為由Q-criterion所誘導 的超平面分割結構。在此框架下，每一種orderedtopology可被 表示為一個由線性不等式定義的凸錐體（topology cone），而不 同拓樸之間的邊界則對應於Q值相等所形成的超平面。
基於此幾何表示，NJ演算法可被視為在錐體分割空間中的序列式決策過程，其中每一步聚合操作對應於部分座標的固定與決策自由度的逐步消除。由此引入「點飄移（pointdrifting）」的概念，用以描述在迭代過程中距離矩陣在不同等價表示之間的轉換，而非傳統意義下的數值誤差累積。
進一步地，本文從幾何角度定義並刻畫NJ的穩定性結構，將安全半徑（saferadius）表示為以距離矩陣為中心、完全包含於同一拓樸錐體內的最大𝐿∞球半徑，從而提供NJ局部拓樸不變性的幾何量化方式。該結果揭示NJ的穩定性本質上來自於距離空間中與超平面分界的幾何距離，而非單純的演算法誤差控制。
總體而言，本文建立了一個統一的幾何框架，用以描述NJ的決策結構、拓樸穩定性與誤差行為，為距離式系統發育樹重建方法提供新的幾何視角。

Neighbor-Joining (NJ) is a widely used distance-based algorithm for phylogenetic tree reconstruction. However, its decision mechanism and error propagation behavior have long lacked a unified geometric interpretation. In this work, we propose a geometric framework for analyzing NJ by reformulating the topology inference problem in the space of distance matrices as a hyper plane partition induced by the Q-criterion. Within this framework, each ordered topology can be represented as a convex cone defined by a system of linear in equalities (a topology cone), while the boundaries between different topologies correspond to hyperplanes induced by equality of Q-values.
Based on this geometric representation, the NJ algorithm can be viewed as a sequential decision process in a partitioned cone space, where each agglomeration step corresponds to fixing a subset of coordinates and progressively eliminating degrees of freedom. This leads to the notion of point drifting, which describes the evolution of distance matrix representations across equivalent forms during iterations, rather than interpreting it as conventional numerical error accumulation.
Furthermore, we characterize the stability structure of NJ from a geometric perspective by defining the safe radius as the maximal 𝐿∞-ball centered at a given distance matrix that is fully contained within the corresponding topology cone. This provides a geometric quantification of local topology invariance. The result reveals that the stability of NJ is fundamentally determined by geometric distances to hyperplane boundaries in the space of distance matrices, rather than purely by algorithmic error control.
Overall, we establish a unified geometric framework that describes the decision structure, topological stability, and error behavior of NJ, providing a new geometric perspective on distance-based phylogenetic reconstruction methods.

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