  We present the first complete NLO prediction with full jet algorithm
implementation for the single inclusive jet production in $pA$ collisions
within the CGC effective theory. Our prediction is fully differential over the
final state physical kinematics, which allows the implementation of any IR safe
observable including the jet clustering procedure. The NLO calculation is
organized with the aid of the power counting proposed in [1] which gives rise
to the novel soft contributions in the CGC factorization. We achieve the
fully-differential calculation by constructing suitable subtraction terms to
handle the singularities in the real corrections. The subtraction contributions
can be exactly integrated analytically. We present the NLO cross section with
the jets constructed using the anti-$k_T$ algorithm. The NLO calculation
demonstrates explicitly the validity of the CGC factorization in jet
production. Furthermore, as a byproduct of the subtraction method, we also
derive the fully analytic cross section for the forward jet production in the
small-$R$ limit. We show that in the small-$R$ approximation, the forward jet
cross section can be factorized into a semi-hard cross section that produces a
parton and the semi-inclusive jet functions. We argue that this feature holds
for generic jet production and jet substructure observables in the CGC
framework. Last, we show numerical analyses of the derived formula to validate
our calculations. We justify when the small-$R$ approximation is appropriate.
Like forward hadron production, the obtained NLO result also exhibits the
negativity of the cross section in the large jet transverse regime, which
signals the need for the threshold resummation. A sketch of the threshold
resummation in the CGC framework is presented based on the multiple emission
picture.
