  We apply numerical conformal bootstrap techniques to the four-point function
of a Weyl spinor in 4d non-supersymmetric CFTs. We find universal bounds on
operator dimensions and OPE coefficients, including bounds on operators in
mixed symmetry representations of the Lorentz group, which were inaccessible in
previous bootstrap studies. We find discontinuities in some of the bounds on
operator dimensions, and we show that they arise due to a generic yet
previously unobserved fake primary effect, which is related to the existence of
poles in conformal blocks. We show that this effect is also responsible for
similar discontinuities found in four-fermion bootstrap in 3d, as well as in
the mixed-correlator analysis of the 3d Ising CFT. As an important byproduct of
our work, we develop a practical technology for numerical approximation of
general 4d conformal blocks.
