Jeffs’ conjecture on convexity for codes with up to four maximal codewords

Establish whether every neural code on any number of neurons that has at most four maximal codewords is convex if and only if it has neither local obstructions nor wheels. Equivalently, prove or refute that for all neural codes with at most four maximal codewords, avoiding both Giusti–Itskov local obstructions and Ruys de Perez et al. wheels characterizes convexity.

Background

The paper studies convex realizability of neural codes via combinatorial criteria, particularly local obstructions and wheels. Prior results show sufficiency of avoiding local obstructions for codes with up to three maximal codewords, but counterexamples exist for four maximal codewords that require consideration of wheels. Jeffs’ conjecture proposes that for all codes with at most four maximal codewords, avoiding both types of obstructions exactly characterizes convexity.

The authors make substantial progress by proving the conjecture for all nerve cases L1–L23 and for minimal L24 codes, but unresolved cases remain.