Chow and augmented Chow polynomials as evaluations of Poincar\'e-extended ab-indices

We show that Chow polynomials and augmented Chow polynomials of matroids, and more generally of finite graded posets admitting R-labelings, are obtained as evaluations of their Poincar\'e-extended ab-indices. This implies in particular explicit combinatorial $\gamma$-positive expansions for both, providing the first proof of the $\gamma$-positivity not relying on the K\"ahler package for the Chow ring. We then evaluate this expansion to obtain an explicit closed formula for the braid arrangement.