Integrality of Constrained Polytope Families

Definitions and mathematical statement

Let $r\in{\mathbb{Z}}_{>0}$, and put $N_r=(r+1)(r+2)/2$. The source refers to the integral matrices $B=B_r\in{\mathbb{Z}}^{p_r\times N_r}$ and $R=R_r\in{\mathbb{Z}}^{q_r\times N_r}$ defining type-$A_r$ hive polytopes. Their entries, row conventions, boundary normalization and numbers of rows are not printed in the supplied record; these matrices must be fixed from the intended hive construction, rather than allowed to range over arbitrary integral matrices.

For integral right-hand sides $b\in{\mathbb{Z}}^{p_r}$ and $c\in{\mathbb{Z}}^{q_r}$, write \[ P_r(b,c)=\{h\in{\mathbb{R}}^{N_r}:B_rh=b,\ R_rh\leq c\}, \] where the inequality is componentwise. The conjectural assertion is \[ \forall r\in{\mathbb{Z}}_{>0}\ \forall b\in{\mathbb{Z}}^{p_r}\ \forall c\in{\mathbb{Z}}^{q_r},\qquad P_r(b,c)\ne\varnothing\ \Longrightarrow\ P_r(b,c)\cap{\mathbb{Z}}^{N_r}\ne\varnothing. \] Thus real feasibility of every indicated system with integral data is to imply integral feasibility. The integral vector must satisfy the same equalities and inequalities, with no dilation or change of right-hand sides. The displayed assertion asks for at least one integral point; it does not require every vertex to be integral or assert a unimodular triangulation.

The original wording describes an extension of an earlier integrality result for an "indicated polytope," but that earlier polytope and result are not supplied. The literature assessment mentions both a unimodular-triangulation conjecture and an integrality conjecture, without stating the triangulation assertion or its relation to this question. Those stronger or differently scoped assertions cannot be reconstructed here. The record calls $P_r(b,c)$ a polytope; whether boundedness follows from the intended normalization of $B_r$, or restricts the quantified data, also needs the missing matrix conventions. The clean statement otherwise quantifies over all integral $b,c$.

The supplied assessment, checked on 2026-08-21, reports computational verification in finite ranks and no located general proof or counterexample. It does not specify the ranks, data ranges or computational certificates. Its open-status label and source-verification claim are retained as dated metadata, not independently verified conclusions.

Short English statement

For the particular matrices from the intended hive construction, the question asks whether every real feasible system with integral right-hand sides has an integral feasible solution. The matrices and the earlier special-case result are absent, so the exact family requires source confirmation. Existence of an integral point is the stated target; vertex integrality and unimodular triangulations require separate statements.

Sources

• Local UnsolvedMath registry, permanent ID 30000107, source code catalogue entry, "Integrality of Constrained Polytope Families." Original and cleaned statements, background, verification metadata and dated literature assessment are the supplied local source record.
• Ehrhart Quasipolynomials: Algebra, Combinatorics, and Geometry (2005), Oberwolfach Reports, pp. 2071--2102, DOI 10.4171/owr/2004/39: https://doi.org/10.4171/owr/2004/39. The citation labels the report 2005, while the DOI, proposed-year field and year tag use 2004; both are retained.
• Supplied reference labelled "De Loera--McAllister": https://escholarship.org/uc/item/0tf8b069. No fuller bibliographic details are supplied.
• Literature triage checked 2026-08-21; local record updated 2026-09-16. The supplied verification field reports checking the official report and normalizing the statement. No external source has been visited for this draft.