From physics point of view, this paper focuses on evolver dynamics which is in fact somehow another interpretation of quantum dynamics' Heisenberg picture combined with the Probabilistic Evolution Theory (PREVTH) quite recently developed in Group for Science and Methods of Computing (Demiralp's group). Even though PREVTH has been based on conicality, during CMMSE 2017 a new point of view and a novel extension have been built for PREVTH, Single Monomial Probabilistic Evolution Theory has been made available by the second author. This work focuses on a one-degree-of-freedom quantum system whose potential may be come very steep (anharmonic symmetric exponential potential) for constructing, first, evolver dynamics equations, and then, single monomial PREVTH by using degree escalating procedure together with commutativity relation based addition to the single monomiality coefficient matrix. Both procedures introduce certain arbitrary parameters into the structure of that matrix by using Constancy Adding Space Extension. Parameters are determined to suppress the norm square of single monomiality coefficient matrix. A convergence analysis for this new PREVTH has also been presented.