摘要
By generalizing the construction of genuine multi-entropy\rm GM[𝚚]for genuine multi-partite entanglement proposed in the previous paper arXiv:2502.07995, we give a prescription on how to construct\rm GM[𝚚]systematically for any𝚚 . The crucial point is that our construction naturally fits to the partition numberp(𝚊)of integer𝚊 . For general𝚚 ,\rm GM[𝚚]containsN (𝚚) = p(𝚚)-p(𝚚-1)-1number of free parameters. Furthermore, these giveN (𝚚)+1number of new diagnostics for genuine𝚚 -partite entanglement. Especially for𝚚=4case, this reproduces not only the known diagnostics pointed out by arXiv:1406.2663, but also a new diagnostics for quadripartite entanglement. We also study these\rm GM[𝚚]for𝚚 = 4, 5in holography and show that these are of the order of𝓞\left{(}{1}{/}G_(N) \right{)}{}{b}oth analytically and numerically. Our results give evidence that genuine multipartite entanglement is ubiquitous in holography. We discuss the connection to quantum error correction and the role of genuine multipartite entanglement in bulk reconstruction.