In this paper, we begin by reviewing the calculus induced by the framework of [10]. In there, we extended Polylogarithm functions over a subalgebra of noncommutative rational power series, recognizable by finite state (multiplicity) automata over the alphabet \(X=\{x_0,x_1\}\). The stability of this calculus under shuffle products relies on the nuclearity of the target space [32]. We also concentrated on algebraic and analytic aspects of this extension allowing to index polylogarithms, at non positive multi-indices, by rational series and also allowing to regularize divergent polyzetas, at non positive multi-indices [10]. As a continuation of works in [10] and in order to understand the bridge between the extension of this “polylogarithmic calculus” and the world of harmonic sums, we propose a local theory, adapted to a full calculus on indices of Harmonic Sums based on the Taylor expansions, around zero, of polylogarithms with index \(x_1\) on the rightmost end. This theory is not only compatible with Stuffle products but also with the Analytic Model. In this respect, it provides a stable and fully algorithmic model for Harmonic calculus. Examples by computer are also provided.