As a trilogy, this first paper along with the authors’ two companion articles introduces alternate approaches to standard -matrix parametrization of nuclear cross sections. The authors present first evidence of so-called shadow poles in -matrix theory for Xe. The number and location of shadow poles depends upon how one continues the -matrix operators to complex wave numbers. The authors also establish a new pole parametrization of -matrix cross sections, called the , which can be Doppler-broadened analytically. This averaging of cross sections over the thermal motion of target atoms is crucial as it ensures the stability of many nuclear reactors; yet, computing it is a major bottleneck. Doppler-broadened windowed multipole cross sections and their uncertainty can be stored and computed with breakthrough performance.