Mixed partition functions are exactly the graph parameters of exponentially bounded edge-connection rank

Abstract

We prove a conjecture of Regts and Sevenster: a complex-valued graph parameter $f$ with $f(\varnothing)=1$ has exponentially bounded edge-connection rank if and only if it is a mixed partition function; moreover, the model may be chosen with its numbers of even and odd colours explicitly bounded in terms of the rank bound. From $f$ we construct a connection category, a rigid symmetric $\mathbb{C}$-linear monoidal category whose morphism spaces have the connection ranks as dimensions and whose trace pairings are nondegenerate. The rank hypothesis forces moderate tensor growth, and a recent theorem of Etingof and Penneys then shows that every nilpotent endomorphism has trace zero; together with the nondegeneracy of the trace pairing, this makes the category semisimple, and a theorem of Deligne provides a faithful symmetric tensor functor to finite-dimensional super vector spaces. We then identify the resulting super tensor network with the Regts-Sevenster model exactly, viz. with its Eulerian-subgraph expansion and its sign of $-1$ for every fermionic circuit. An appendix gives an independent and direct proof of the nilpotent-trace step, showing that in a rigid symmetric $\mathbb{C}$-linear category with $\mathrm{End}(\mathbf{1})=\mathbb{C}$, exponentially bounded endomorphism growth makes the trace zeta function of every endomorphism rational, with explicit degree bounds.

Disclosure

“ndomorphisms universally of trace zero — the converse direction to Corollary A.2 — under a sign-property hypothesis. Finally, Guletskiı̆ [17] shows that rationality of the zeta function does not imply Kimura- finiteness. Acknowledgements Claude Fable 5 and GPT-5.6 Sol Pro were used extensively in the development and prepa- ration of this work. References [1] Yves André. Motifs de dimension finie (d’après S.-I. Kimura, P. O’Sullivan . . . ). Astérisque, 299:115–145, 2005.”

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Proof ideas or individual proof-step assistance

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2026-08-23T19:12:17.582751Z

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