  We prove a complexity dichotomy theorem for Holant Problems on 3-regular
graphs with an arbitrary complex-valued edge function. Three new techniques are
introduced: (1) higher dimensional iterations in interpolation; (2) Eigenvalue
Shifted Pairs, which allow us to prove that a pair of combinatorial gadgets in
combination succeed in proving #P-hardness; and (3) algebraic symmetrization,
which significantly lowers the symbolic complexity of the proof for
computational complexity. With holographic reductions the classification
theorem also applies to problems beyond the basic model.
