Relate sign-rank to Z2-index for total matrices

Determine whether there exists a function f: N → N such that for every total sign matrix A, the sign-rank satisfies srank(A) ≤ f(ind(S(A))), where ind(S(A)) is the Z2-index of the sign complex S(A) associated to A.

Background

The paper develops a topological framework linking sign-rank of a (partial) sign matrix A to the Z2-index of an associated free Z2–simplicial complex S(A), proving ind(S(A)) ≤ srank(A)−1. They then exhibit separations showing the two parameters can differ substantially for both random and structured matrices, and provide techniques to upper bound index.

For partial matrices, they construct examples where ind(S(A)) is constant while sign-rank grows polynomially, ruling out a general equivalence. For total matrices, however, they leave open whether a functional dependence always exists, which would imply a qualitative equivalence in the constant regime and allow many sign-rank questions to be reframed topologically.

References

We leave the following remaining important question as open. Does there exist a function $f\colon \mathbb{N}\to\mathbb{N}$ such that for every total sign matrix~$A$,

\srank(A)\leq f(ind(S(A))) ?

— A $\mathbb{Z}_2$-Topological Framework for Sign-rank Lower Bounds  (2604.01510 - Frick et al., 2 Apr 2026) in Section 6, Separations – Index vs. sign-rank for total matrices (Question \ref{question:totalseparation})