  We study orbits of semigroups of $\text{SL}(2,\mathbb{Z})$, and demonstrate
reciprocity obstructions: we show that certain such orbits avoid squares, but
not as a consequence of obstructions inherited from an algebraic set, and not
as a consequence of congruence obstructions. This is in analogy to the
reciprocity obstructions recently used to disprove the Apollonian local-global
conjecture. We give an example of such an orbit which is known exactly, and
misses all squares together with an explicit finite list of sporadic values:
the corresponding semigroup is not thin, but is dense in an algebraic variety
that does not have such obstructions. We also demonstrate thin semigroups with
reciprocity obstructions, including semigroups associated to continued
fractions formed from finite alphabets. Zaremba's conjecture states that for
continued fractions with coefficients chosen from $\{1,\ldots,5\}$, every
positive integer appears as a denominator. Bourgain and Kontorovich proposed a
generalization of Zaremba's conjecture in the context of semigroups associated
to finite alphabets. We disprove their conjecture. In particular, we
demonstrate classes of finite continued fraction expansions which never
represent rationals with square denominator, but not as a consequence of
congruence obstructions, and for which the limit set has Hausdorff dimension
exceeding $1/2$. An example of such a class is continued fractions of the form
$[0; a_1, a_2, \ldots, a_n,1,1,2]$, where the $a_i$ are chosen from the set
$\{4,8,12,\ldots,128\}$. The object at the heart of these results is a
semigroup $\Psi\subseteq\Gamma_1(4)$ which preserves Kronecker symbols.
