The unverifiable problem in polynomial time within NP-hard problems is a fundamental scientific challenge behind the fault recovery optimization of new distribution systems. Traditional exact and approximation algorithms cannot reconcile the contradiction between speed and accuracy. In response to the shortcomings of traditional algorithms

this paper proposes a novel lossless conversion algorithm that can reconcile the inherent contradiction by converting the unverifiable problem into a verifiable problem without any loss. Firstly

it's demonstrated that the fault recovery optimization of the new distribution system belongs to the unverifiable problem in polynomial time within non-deterministic polynomial-time hard (NP-hard) problems fault recovery

and pointing out that the lossless conversion from unverifiable to verifiable is the key to solving the contradiction. Secondly

a lossless conversion algorithm is proposed based on Hidden Markov Models and the forward algorithm

which constructs a new algorithmic logic by searching the reverse process of the operating state-driven scene

which achieves a reduction in time complexity from exponential level to polynomial level. Finally

in case studies

the proposed algorithm only takes 1.58% of the computing time to obtain a zero-error solution

demonstrating the excellent algorithmic performance of balancing speed and accuracy.