Sharp bounds for monogenizations of quartic trinomial orders

Let ξ be a root of an irreducible quartic trinomial X⁴+aX²+b∈ Z[X]. We determine sharp bounds for the number of monogenizations of the order Z[ξ], modulo integral translation and sign. In the totally real range, writing $a=-A$ and $b=B$ with $A,B>0$ and A²>4B, there are at most ten classes, with equality exactly for $(A,B)=(5,5)$; every other such pair gives at most nine. If $b<0$, there are at most four classes, with equality exactly when $b=-1$ and a=± c² for some positive integer c; if $b<-1$, the bound improves to three. As a byproduct of the algebraic reduction, we also prove two conjectures of Ga\'al concerning power integral basis generators in specified trinomial families. The proof factors the monogenizer condition into principal and exceptional contributions and reduces the counting problem to quartic Pell equations and square conditions in Lucas sequences. In the totally real case, the sharp bound requires combining the two contributions: when the exceptional count reaches six, the principal count is at most three. No maximality assumption on Z[ξ] is needed for the two sharp bounds.