  Despite the rich and fruitful history of the integrability approach to string
theory on the $AdS_3\times S^3\times T^4$ background, it has not been possible
to extract many concrete predictions from integrability, except in a strict
asymptotic regime of large quantum numbers, due to the severity of wrapping
effects. The situation changed radically with two independent and identical
proposals for the Quantum Spectral Curve (QSC) for this system in a background
of pure Ramond-Ramond flux. This formulation is expected to capture all
wrapping effects exactly and describe the full planar spectrum. Massless modes
conjecturally manifest themselves in a new property of this QSC: the
non-quadratic nature of the branch-cut singularities of the QSC Q-functions.
This feature implies new technical challenges in solving the QSC equations as
compared to the well-studied case of N=4 SYM. In this paper we resolve these
difficulties and obtain the first ever predictions for generic unprotected
string excitations. We explain how to extract a systematic expansion around the
analogue of the weak 't Hooft coupling limit in N=4 SYM and also obtain
high-precision numerical results. This concrete data and others obtainable from
the QSC could help to identify the so-far mysterious dual CFT.
