  How likely is it that there are particles in a vector-like pair of
representations in low-energy spectrum, when neither symmetry nor anomaly
consideration motivates their presence? We address this question in the context
of supersymmetric and geometric phase compactification of F-theory and
Heterotic dual. Quantisation of the number of generations (or net chiralities
in more general term) is also discussed along the way. Self-dual nature of the
fourth cohomology of Calabi--Yau fourfolds is essential for the latter issue,
while we employ Brill--Noether theory to set upper bounds on the number $\ell$
of vector-like pairs of chiral multiplets in the SU(5) 5+5bar representations.
For typical topological choices of geometry for F-theory compactification for
SU(5) unification, the range of $0 \leq \ell \lesssim 4$ for perturbative
unification is not in immediate conflict with what is already understood about
F-theory compactification at this moment.
