  In this paper, we investigate three-term linear relations among theta series
of positive-definite integral binary quadratic forms. We extend Schiemann's
methods to characterize all possible three-term linear relations among theta
series of such forms, providing necessary and sufficient conditions for such
relations to exist. To accomplish this, we develop, implement, and execute a
novel extended refinement algorithm on polyhedral cones. We show that there is
exactly one non-trivial three-term linear relation: it involves quadratic forms
with discriminants $-3, -12, -48$, all in the same rational squareclass
$-3(\mathbb{Q}^\times)^2$.
