  Despite its unparalleled success, quantum mechanics (QM) is an incomplete
theory of nature. Longstanding concerns regarding its mathematical foundations
and physical interpretation persist, a full century beyond its conception.
Limited by these issues, efforts to move beyond QM have struggled to gain
traction within the broader physics community. One approach to progress in this
direction, which is deeply rooted in the tradition of physics, is the
development of new models for physical systems otherwise treated by QM. One
such model is presented here, which concerns the interaction of a spin system
with sequences of two Stern-Gerlach detectors that may be independently
rotated. Rather than employing the traditional formalism of QM, the proposed
model is supported by tools from discrete mathematics, such as finite groups,
set theory, and combinatorics. Equipped with this novel toolkit, an analog of
Wigner's d-matrix formula is derived and shown to deviate slightly from QM.
With these results, the proposed model is extended to an optical system
involving photon number states passing through a beam splitter. Leveraging
recent advancements in high precision experiments on these systems, we then
propose a means of testing the new model using a tabletop experiment. Hence,
the proposed model not only makes clear testable predictions, but also provides
valuable insight into the essential principles of quantum theory.
