Rank-one five-letter profiles by primitive relation mass

Determine, for each 2≤μ≤q, the maximum of M_q(T) over five-element rank-one profiles whose primitive balanced relation has mass μ, and determine whether the maximum over all such rank-one profiles equals qL_4(q).

Background

A rank-one five-letter profile is one whose complete space of vanishing q-relations is generated by a single primitive balanced relation of mass μ. The paper proves the upper bound μq(q+1) for the minimum model diameter M_q(T).

The cluster family has mass q and minimum diameter qL_4(q), suggesting a possible extremal value for all rank-one profiles. However, the bound is known to imply qL_4(q) only for relatively small masses, while the proper-power family demonstrates that some higher-mass profiles can still have model diameter below qL_4(q). The remaining high-mass profiles are unresolved.

References

For $2\le\mu\le q$, determine the maximum of $M_q(T)$ over five-letter rank-one profiles of primitive mass $\mu$, and in particular whether the maximum over all rank-one five-letter profiles is $qL_4(q)$, the value of the cluster family of Corollary~\ref{M-thm:cluster-model}.

— Sharp order-preserving integer models for short additive equalities  (2609.08915 - Zhang, 8 Sep 2026) in Question M-q:rank-one-mass, Section 8 (Open problems)