Pro-unipotent harmonic actions and a computation of $p$-adic cyclotomic multiple zeta values

We obtain formulas relating $p$-adic cyclotomic multiple zeta values and cyclotomic multiple harmonic sums. In particular, we obtain a series formula for $p$-adic cyclotomic multiple zeta values, and conversely a formula for certain cyclotomic multiple harmonic sums in terms of $p$-adic cyclotomic multiple zeta values. Our formulas are related to the motivic framework via a new notion which we call pro-unipotent harmonic actions, which are ad hoc $p$-adic byproducts of the Ihara action. As an application, we prove a conjecture of Akagi, Hirose and Yasuda on the relation between $p$-adic multiple zeta values and multiple harmonic sums, and we generalize it to the cyclotomic case. We also deduce bounds on the dimension of the spaces of finite cyclotomic multiple zeta values.