  We fully resolve the Furstenberg set conjecture in $\mathbb{R}^2$, that a
$(s, t)$-Furstenberg set has Hausdorff dimension $\ge \min(s+t, \frac{3s+t}{2},
s+1)$. As a result, we obtain an analogue of Elekes' bound for the discretized
sum-product problem and resolve an orthogonal projection question of Oberlin.
