Dynamical evidence of the sub-parsec counter-rotating disc for a close binary of supermassive black holes in the nucleus of NGC 1068

It arises a puzzle in \NGC\, how to secularly maintain the counter-rotating disc from $0.2$ to $7\,$pc unambiguously detected by recent ALMA observations of molecular gas. Upon further analysis of disc dynamics, we find that the Kelvin-Helmholtz (KH) instability (KHI) results in an unavoidable catastrophe of the disc developed at the interface between the reversely rotating parts, and demonstrate that a close binary of supermassive black holes provides tidal torques as the unique external sources to prevent the disc from the KH catastrophe. We are led to the inescapable conclusion that there must be a binary black hole at the center of NGC 1068, to prevent it from the KH catastrophe. The binary is composed of black holes with a separation of $0.1\,$pc from GRAVITY/VLTI observations, a total mass of $1.3\times 10^{7}\:M_{\odot}$ and a mass ratio of $\sim 0.3$ estimated from the angular momentum budge of the global system. The KHI gives rise to forming a gap without cold gas at the velocity interface which overlaps with the observed gap of hot and cold dust regions. Releases of kinematic energies from the KHI of the disc are in agreement with observed emissions in radio and $\gamma$-rays. Such a binary is shrinking with a timescale much longer than the local Hubble time via gravitational waves, however, the KHI leads to an efficient annihilation of the orbital angular momentum and speed up merge of the binary, providing a new paradigm of solving the long term issue of "final parsec problem". Future observations of GRAVITY+/VLTI are expected to be able to spatially resolve the CB-SMBHs suggested in this paper.