The goal of this project is to initiate a unified program for Tauberian theory centered around a convolution average procedure. The ultimate objective is to undertake a systematic study of this procedure. Ideally this program transparently recovers all known significant Tauberian theorems while also removing certain hiatuses in the field. This project shall focus on the following topics, essential for this program. 1) Tauberian lemmas for a variety of flexible Tauberian conditions 2) The development of a complete complex general remainder theory 3) New sharp finite forms 4) A multiplier calculus for exact behavior for integral transforms, related to certain (quantified) Tauberian theorems under flexible Tauberian conditions 5) Optimality of the acquired quantified Tauberian theorems