Multipartite entanglement underpins quantum technologies but its study is limited by the lack of universal measures, unified frameworks, and the intractability of convex-roof extensions. We establish an axiomatic framework for  $k$-entanglement measure which  specifies how to measure the $k$-entanglement hold by  part of system and ensures that the $k$-entanglement hold by part of system does not exceed that of the entire system. We also propose   firstly a \emph{true} $k$-entanglement measure, $E_w^{(k,n)}$, that satisfies all axioms, establishes $k$-entanglement as a multipartite quantum resource, avoids convex-roof constructions. A universal algorithm is designed to evaluate $E_w^{(k,n)}$ of arbitrary finite-dimensional states. An open-source software covering all partitions of four-qubit systems is provided, which  certifies $k$-entanglement within 200 seconds, consistent with necessary-and-sufficient criteria, tightening bounds and revealing new thresholds. This work offers a scalable, practical tool for rigorous multipartite entanglement quantification.

侯晋川，太原理工大学数学学院教授，博士生导师，研究方向为算子理论与算子代数、量子信息理论。曾担任山西师范大学校长，山西省科协主席兼太原理工大学副校长。两次获得山西省科技进步一等奖，两次获得山西省自然科学二等奖。享受国务院特殊津贴，曾获山西省优秀专家、山西省第二届科技功臣、全国做出突出贡献的回国留学人员、全国优秀教师、全国五一劳动奖章、全国先进工作者、山西省特级劳模等荣誉。