Universal radius for ordered integer models

Determine whether the universal ordered-model radius satisfies H_m(q)=q^{m-3}(q+1) for every alphabet size m≥6 and every observation order q≥2.

Background

For an ordered real m-element alphabet, H_m(q) is the least diameter such that an increasing integer alphabet can preserve exactly all equalities between equal-length sums of at most q elements. The paper proves the proposed value for m=3,4,5 and for (m,q)=(6,2).

The corank-two construction establishes the lower bound q{m-3}(q+1) in every dimension, while the general upper bound leaves a gap for larger cases. In particular, q4+q3≤H_6(q)≤q4+q3+q2 and 48≤H_7(2)≤56. Resolving whether the lower bound is always sharp would determine the universal radius beyond the cases settled in the paper.

References

General equality at the lower endpoint of M-eq:two-sided for $m\ge6$ remains open; the case $(m,q)=(6,2)$ is established in the next section.

M-eq:two-sided:

qm−2+qm−3≤Hm(q)≤qm−2+Hm−1(q),and alsoHm(q)≤∑j=0m−2qj.q^{m-2}+q^{m-3}\le H_m(q)\le q^{m-2}+H_{m-1}(q), \qquad\text{and also}\qquad H_m(q)\le\sum_{j=0}^{m-2}q^j .

— Sharp order-preserving integer models for short additive equalities  (2609.08915 - Zhang, 8 Sep 2026) in Question M-q:six, Section 8 (Open problems); see also Section 6, immediately after Theorem M-thm:general-bounds