Kerr Soft Dressing and the $w_{1+\infty}$ Frame Algebra at Null Infinity

We construct the charge-generated intrinsic/canonical frame dictionary associated with the Kerr-selected soft dressing. Starting from the VV supertranslation, we formulate the higher-spin problem as an inverse problem at null infinity: the soft kernel ${\mathcal K}^{(s,0)}_{AB}[t]$, built from the parity-adapted maximally longitudinal scalar $\chi^{(s)}_t$, is matched to the Kerr-selected exponentiating projection of the universal soft contribution, thereby determining one parity component of the generator $t^{A_1\cdots A_s}$. Helicity conjugation fixes which one: the exponentiating source obeys $\overline{S^{(s)}_{+,{\rm exp}}}=(-1)^sS^{(s)}_{-,{\rm exp}}$, so the tower fixes the electric projection of the source at even levels and the magnetic projection at odd ones, matching the alternation of the Kerr mass and current moments; for aligned spin the projection is exhaustive and we solve the tower in closed form. The prescription reproduces the VV supertranslation at leading order and fixes the curl, not the divergence, of a smooth generalized-BMS vector at subleading order. The reason for the matching is physical: the same exponentiating soft factor is the classical limit of the Guevara--Ochirov--Vines spinning three-point operator and generates the Kerr multipole tower. We explain the corresponding hard flux charges and show how their external-state action gives the Ward representation of the soft theorem. The polynomial Poisson algebra on $T^\ast S^2$, with local $w_{1+\infty}$-type reductions, then acts on these frame-changing generators; it does not close on the Kerr-selected data alone. This gives the physical role of the $w_{1+\infty}$-like structure in Kerr black-hole scattering: it moves the intrinsic/canonical dictionary. Its observable imprint begins with displacement memory at $s=0$ and spin memory at $s=1$, followed by higher electric and magnetic memory moments.