Chow groups of symmetric products of projective space

We prove that the integral cycle class map for ${\text{SP}}^d({\text{PP}}^n)$ is split injective, with finite cokernel annihilated by $d!$,and we determine the ranks of Chow groups for ${\mathrm{S}\mathrm{P}}^d({\mathbb{P}}^n)$. Compatible integral transfer retractions show that ${\text{CH}}_q({\text{SP}}^d({\text{PP}}^n))$ stabilizes for $d\ge q$. An explicit nonalgebraic order two class persists under addition of points, producing a split cokernel summand even in the range $q\le d$ and in the infinite symmetric product. For the smooth configuration open set, we identify the codimension two cokernel with third unramified cohomology and detect its ramification on a rational resolution. In contrast to addition, even multiplicities annihilate a nonzero Chow class.