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We introduce and consider the notion of stable degeneracies of translationinvariant energy functions for finite Ising models. By this term we mean thelack of injectivity that cannot be lifted by changing the interaction.  We show that besides the symmetry-induced degeneracies, related to spin flip,translation and reflection, there exist additional stable degeneracies, due tomore subtle symmetries. One such symmetry is the one of the Singer group of afinite projective plane. Others are described by combinatorial relations akinto trace identities.  Our results resemble traits of the length spectrum for closed geodesics on aRiemannian surface of constant negative curvature. There stable degeneracy isdefined w.r.t. Teichm\"uller space as parameter space.

Stable Degeneracies for Ising Models