Existence of hypergraphs satisfying all three structural axioms

Construct, or otherwise establish the existence of, fixed-uniformity, large-degree hypergraph families that simultaneously satisfy one-overlap linearity, bounded local cycle rank at the required radii, and a global spectral expansion gap.

Background

The deterministic hSYK analysis assumes a q-uniform, d-regular interaction hypergraph satisfying three conditions: any two hyperedges overlap in at most one vertex; Levi-graph neighborhoods have only O(1) independent cycles up to radii relevant to the analysis; and the normalized line-graph random walk has a degree- and system-size-independent spectral gap. The paper finds suitable finite examples numerically, but does not prove that these local geometric properties can coexist with global expansion in the fixed-q, large-d regime. Establishing such families would provide the asymptotic constructions needed to underpin the model beyond finite-size evidence.

References

The main unresolved existence question is whether Axioms 1 and 2 can be realized simultaneously with the global spectral expansion required in Axiom 3, in the fixed-$q$, large-$d$ regime studied here. We conjecture that this is the case, although we do not pursue such an existence theorem here.

— Deterministic Hypergraph SYK: Melonic Dominance and Maximal Chaos Without Disorder Averaging  (2609.19255 - Prakash, 16 Sep 2026) in Section 2, paragraph beginning “The main unresolved existence question”