Higher-spin algebras from soft theorems I: the wedge condition

In this article we use the sub${}^n$-soft graviton theorems to construct the map $\text{Top}$ from the spin-graded set of holomorphic functions on local celestial sphere patches to differential operators acting on the asymptotic data for massless particles at $\text{scrip}$, in analogy with previous results in the literature for the sub${}^n$-soft photon theorems. The result is an explicit closed-form formula. We show that the wedge subalgebras for both Yang-Mills and gravity are the natural domain on which $\text{Top}$ becomes a representation.