Intermediate left-right gauge symmetry, unification of couplings and fermion masses in SUSY $SO(10)times S_4$

If left-right gauge theory occurs as an intermediate symmetry in a GUT then, apart from other advantages, it is possible to obtain the see-saw scale necessary to understand small neutrino masses with Majorana coupling of order unity. Barring threshold or non-renormalizable gravitational effects, or assumed presence of additional light scalar particles of unprescribed origin, all other attempts to achieve manifest one-loop gauge coupling unification in SUSY SO(10) with left-right intermediate symmetry have not been successful so far. Attributing this failure to lack of flavor symmetry in the GUT, we show how the spontaneous symmetry breaking of $SO(10)times S_4$ leads to such intermediate scale extending over a wide range, $M_R simeq 5times 10^{9}$ GeV to $10^{15}$ GeV. All the charged fermion masses are fitted at the see-saw scale, $M_Nsimeq M_R simeq 4 times 10^{13}$ GeV which is obtained with Majorana coupling $f_0 simeq 1$. Using a constrained parametrization in which CP-violation originates only from quark sector, besides other predictions made in the neutrino sector, the reactor mixing angle is found to be $theta_{13} simeq 3^{circ} - 5^{circ}$ which is in the range accessible to ongoing and planned experiments. The leptonic Dirac phase turns out to be $delta sim 2.9- 3.1$ radians with Jarlskog invariant $J sim 2.95 times 10^{-5} - 10^{-3}$.