  We propose that quantum physics is the continuous approximation of a more
fundamental, discrete graph theory (theory X). Accordingly, the Euclidean
transition amplitude Z provides a partition function for geometries over the
graph, which is characterized topologically by the difference matrix and source
vector of the discrete graphical action. The difference matrix and source
vector of theory X are related via a graphical self-consistency criterion (SCC)
based on the boundary of a boundary principle on a graph. In this approach, the
SCC ensures the source vector is divergence-free and resides in the row space
of the difference matrix. Accordingly, the difference matrix will necessarily
have a nontrivial eigenvector with eigenvalue zero, so the graphical SCC is the
origin of gauge invariance. Factors of infinity associated with gauge groups of
infinite volume are excluded in our approach, since Z is restricted to the row
space of the difference matrix and source vector. Using this formalism, we
obtain the two-source Euclidean transition amplitude over a (1+1)- dimensional
graph with N vertices fundamental to the scalar Gaussian theory.
