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We prove a recent conjecture of Borot et al. that a particular universalclosed algebraic formula recovers the correlation differentials of topologicalrecursion after the swap of $x$ and $y$ in the input data. We also show thatthis universal formula can be drastically simplified (as it was already done byHock).  As an application of this general $x-y$ swap result, we prove an explicitclosed formula for the topological recursion differentials for the case of anyspectral curve with unramified $y$ and arbitrary rational $x$.

A universal formula for the $x-y$ swap in topological recursion