TY - RPRT TI - Mathematical Foundations of Beta Diversity: Why Common Metrics Fail in Microbiome Analysis AU - Zhu, Z. AU - Zhang, Y. AU - Li, W. AU - Greenacre, M. AU - Satabdi, S. AU - Shi, Y. AU - Zhang, L. PY - 2025 DO - 10.1101/2025.08.14.670326 UR - https://www.biorxiv.org/content/10.1101/2025.08.14.670326v1 ID - 10.1101/2025.08.14.670326 ER -