GL fixed point for a provability implication ($F=(\square q\to q),\mathrm{GL}⊢F\leftrightarrow (\square F\to q)$)

The Löb axiom gives $\square F\to \square q$. The tautology $q\to F$, necessitated and distributed, gives $\square q\to \square F$. Thus $\square F\leftrightarrow \square q$, and replacing the antecedent in $\square F\to q$ gives exactly $F$. Setting $q=\perp$ produces a negative-provability fixed point, provably equivalent to $\neg \square \perp$.