发表机构
Department of Orthopaedic Surgery, Nozaki Tokushukai Clinic; Daito, Osaka, Japan（野崎特修会诊所骨科; 日本大阪府大东市）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对最近邻图在扰动下的稳定性，提出有限配置的显式保证，并应用于冻结粒子数据，证明其有限结构不变性，但动态时间界需额外证据。
AI 中文摘要
最近邻图是离散对象，其成员关系在底层坐标的微小扰动下可能发生变化。我们在任意度量空间中为有限标记配置建立了显式的稳定性保证。如果第k个和第(k+1)个距离之间的间隙为正，则每个标签的同步扰动小于该间隙的四分之一时，能保持有向k最近邻成员关系。在所述的均匀位移假设下，因子四是最优的，且任何仅基于标记邻居族的固定确定性构造因此保持不变。在标记信息空间轨迹的区间有效Lipschitz界下，同一结果给出了首次可能重新布线时间的认证下界，并针对标签特定运动界进行了细化。我们将有限定理应用于冻结的标准化Taylor-Green未来信息坐标[d_B, log A_B]，涉及585个粒子在三个观测时间层。向外舍入的区间算术认证了约1.023 x 10^-6的充分扰动半径，同一研究工作流程中单独实现的检查器验证了秩和距离边际计算。随后，一项结果盲审检查了该构造是否支持数值连续时间界。由于完整信息映射涉及离散聚类和边界重建，且缺乏解析导数界、验证的稠密时间界或认证的连续性模量，该应用被正确分类为DISCRETE_ONLY，并带有STOP_NO_INTERVAL_VALID_BOUND。因此，冻结配置的有限局部结构不变性被证明并数值认证，而时间特化和预测泛化仍是需要额外证据的独立问题。
英文摘要
Nearest-neighbor graphs are discrete objects whose membership may change under small perturbations of the underlying coordinates. We establish an explicit stability guarantee for finite labeled configurations in an arbitrary metric space. If the gap between the kth and (k+1)st distances is positive, simultaneous per-label perturbations smaller than one quarter of that gap preserve the directed k-nearest-neighbor membership. The factor four is sharp under the stated uniform displacement assumptions, and every fixed deterministic construction based solely on the labeled neighbor family is consequently invariant. Under an interval-valid Lipschitz bound for labeled information-space trajectories, the same result yields a certified lower bound on the first possible rewiring time, with a refinement for label-specific motion bounds. We apply the finite theorem to frozen standardized Taylor-Green future-information coordinates [d_B, log A_B] for 585 particles at three observed time strata. Outward-rounded interval arithmetic certified a sufficient perturbation radius of approximately 1.023 x 10^-6, and a separately implemented checker within the same research workflow verified the rank and distance-margin calculations. An outcome-blind audit then examined whether the construction supported a numerical continuous-time horizon. Because the complete information map involved discrete clustering and boundary reconstruction and lacked an analytic derivative bound, validated dense-time bound, or certified modulus of continuity, the application was correctly classified as DISCRETE_ONLY with STOP_NO_INTERVAL_VALID_BOUND. Thus finite local structural invariance is proved and numerically certified for the frozen configuration, while temporal specialization and predictive generalization remain separate questions requiring additional evidence.
Comments16 pages. Includes a certified Taylor-Green application and an outcome-blind audit of continuous-time applicability
