Diagrammatic proof of efficient contraction for general Hopf quadratic tensors

Develop a purely diagrammatic proof, using the (super) Hopf algebra axioms, that general Hopf quadratic tensors (including free-fermion cases with non-trivial embedding ε over the super Hopf algebra ℱ) can be contracted efficiently in terms of their representing coefficients.

Background

The authors show that free-fermion quadratic tensors over the super Hopf algebra ℱ can be efficiently contracted via Schur complements when the embedding is trivial, and they outline efficient procedures in specific cases. They also present a unified diagrammatic framework for quadratic tensors over (super) Hopf algebras.

However, providing a general, purely diagrammatic proof for efficient contraction remains unresolved, especially in the presence of non-trivial embeddings. Such a proof would unify the treatment of abelian-group and free-fermion cases and solidify the theoretical foundations of the proposed tensor formalism.

References

"Ideally, we would like to show that general Hopf quadratic tensors can be efficiently contracted in terms of the underlying data on a purely diagrammatic level. However, this turns out to be a bit tricky, especially with a non-trivial embedding $\epsilon$ involved. We thus leave this diagrammatic proof to future work, and comment on how to efficiently perform index contractions specifically for free-fermion quadratic tensors..."

— Quadratic tensors as a unification of Clifford, Gaussian, and free-fermion physics  (2601.15396 - Bauer et al., 21 Jan 2026) in Section 5.4 (Free-fermion index contraction)