Proof of the d=4 hypergeometric series identity for log 2

Prove the hypergeometric series identity for log 2 given by log 2 = sum_{n=1}^{∞} [P(n) / (3 n (2 n − 1) (3 n − 1) (3 n − 2))] · (1 / (2^{13} · 3^{3}))^{n} · ((1)_n (1/2)_n (1/3)_n (2/3)_n / (1/12)_n (5/12)_n (7/12)_n (11/12)_n), where P(n) = 686430 n^{3} − 742257 n^{2} + 223397 n − 13858 and (ν)_n denotes the rising factorial (Pochhammer symbol). Establish a rigorous proof of this identity, which was discovered via LLL search and numerically validated but presently remains unproven.

Background

In the catalogue of fast Ramanujan-type series for logarithms, most identities are proven via Beta integral techniques and Wilf–Zeilberger certificates. However, one d=4 identity (Eq.(13)) for log 2 was found by the LLL lattice reduction search and has resisted proof.

This identity has low binary splitting cost and was used as a secondary digits validation algorithm for log 2 in y-cruncher v0.8.5, successfully verifying 3 trillion decimal digits. Despite numerical evidence and computational use, no formal proof has been established.