  Let $f$ be a monic quadratic polynomial over a finite field of odd
characteristic. In 2012, Boston and Jones constructed a Markov process based on
the post-critical orbit of $f$, and conjectured that its limiting distribution
explains the factorization of large iterates of $f$. Later on, Xia, Boston, and
the author did extensive Magma computations and found some exceptional families
of quadratics that do not seem to follow the original Markov model conjectured
by Boston and Jones. They did this by empirically observing that certain
factorization patterns predicted by the Boston-Jones model never seem to occur
for these polynomials, and suggested a multi-step Markov model which takes
these missing factorization patterns into account. In this note, we provide
proofs for all these missing factorization patterns. These are the first
provable results that explain why the original conjecture of Boston and Jones
does not hold for all monic quadratic polynomials.
