Summary: The classical Severi degree counts the number of algebraic curves of fixed genus and class passing through points in a surface. We express the Severi degrees of \(\mathbb{P}^1\times \mathbb{P}^1\) as matrix elements of the exponential of a single operator \(\mathrm{M}_S\) on Fock space. The formalism puts Severi degrees on a similar footing as the more developed study of Hurwitz numbers of coverings of curves. The pure genus 1 invariants of the product \(E \times \mathbb{P}^1\) (with \(E\) an elliptic curve) are solved via an exact formula for the eigenvalues of \(\mathrm{M}_S\) to initial order. The Severi degrees of \(\mathbb{P}^2\) are also determined by \(\mathrm{M}_S\) via the \((-1)^{d-1}/d^2\) disk multiple cover formula for Calabi-Yau threefold geometries.