A proof of a conjecture of Gukov-Pei-Putrov-Vafa

In this paper, we prove Gukov-Pei-Putrov-Vafa’s conjecture that the Witten-Reshetikhin-Turaev invariants are radial limits of homological blocks, which are $q$-series introduced by them for plumbed $3$-manifolds with negative definite linking matrices. In our previous work, the author attributed this conjecture to the holomorphy of certain rational functions by developing an asymptotic formula based on the Euler-Maclaurin summation formula. However, it is challenging to prove holomorphy for general plumbed manifolds. In this paper, we address this challenge using induction on a sequence of trees obtained by repeating “pruning trees.”