# Verification report — OWR-741-001 (De Loera and McAllister: hive polytopes with arbitrary integral right-hand sides)

Verification date: 2026-10-12.

**Verdict.** The note answers the question of the record for every rank r (the r of i + j + k = r; a pattern of rank r
has (r+1)(r+2)/2 entries, 3r of them on the boundary).
- **No for every r ≥ 6 (Theorem 1.1).** There is an integral c such that {h : Bh = 0, Rh ≤ c} is a single point with
  four coordinates 1/2 and all others 0. For r = 6 the point φ has the value 1/2 at (1,4,1), (2,3,1), (2,1,3),
  (3,1,2), and c = −⌊excess(φ)⌋ on each of the 45 rhombi (c = −1 for two rhombi, +1 for ten, 0 for 33). The proof
  is written out: fifteen of the 45 inequalities force h = φ (Proposition 3.1); Lemma 4.1 passes from rank r to
  rank r + 1. No computation enters this proof.
- **Yes for r ≤ 5 (Theorem 1.2).** Every nonempty polytope {h : Bh = b, Rh ≤ c} with integral b and c contains an
  integral point, also with the condition h ≥ 0. Proof: a reduction (Lemmas 2.1 and 2.2) and an exact computation
  (`note/small_ranks.py`: 394 and 418 classes for r = 5, an integral solution for each). A second proof, which does
  not use Lemma 2.2, was given by the second verification run (a partition of the unit cube into 4, 64 and 21,484
  boxes; `verification_run_B/`).
- **The conjecture (Corollary 1.3).** For r ≥ 6 the columns of the homogenized hive matrix [B 0; R I] have no
  unimodular cover, neither in the sense of De Loera–McAllister, Section 4.1 (sets of m columns of determinant ±1
  whose cones cover the cone of the columns), nor for their convex hull (the wording of Conjecture 1 of the
  Oberwolfach abstract). For the second form the note uses examples in which the sum of all coordinates, slack
  variables included, is an integer (Propositions 3.2 and 4.2: 13 for rank 6, and 18, 25, 33, … for ranks 7, 8,
  9, …).
- **Not new.** Theorem 1.2 is contained in Theorem 4.6 of De Loera–McAllister (with an integral vertex), in each
  reading of the range of that theorem; the proof of the note does not use it.
- **New, to our knowledge and relative to the literature found.** The failure of the conjecture for every r ≥ 6
  (Theorem 1.1, Corollary 1.3), and with it the exact range of the integer point property.
- **Limits.** For r ≤ 5 the note proves an integral point, not an integral vertex. The journal version of the paper
  of De Loera and McAllister (Experiment. Math. 15 (2006), 7–19) was not accessible to us and was not read; the
  arXiv version (v1) was read. The published version of the survey of Haase, Paffenholz, Piechnik and Santos was
  not compared with its arXiv version. No counterexample to the conjecture and no statement that it fails was
  found in the literature accessible to us; a search that finds nothing is not a proof of novelty, and no priority
  is claimed.

The note is unrefereed.

## Statement checked
- **Primary source.** T. B. McAllister (joint work with J. De Loera), "Two Conjectured Generalizations of the
  Saturation Theorem", Oberwolfach Reports 1 (2004), Report No. 39/2004, doi:10.4171/owr/2004/39, pp. 2072–2074.
  Read in the publisher's file. The hive polytope is {h ∈ R^((r+1)(r+2)/2) : Bh = b(λ,μ,ν), Rh ≤ 0} with integral
  matrices B, R and a vector b(λ,μ,ν) with 3r entries; the abstract conjectures that the integrality result of
  Knutson and Tao extends to all nonempty polytopes {h : Bh = b, Rh ≤ c} with integral b and c, and states
  Conjecture 1 (the convex hull of the columns of [B 0; R I] has a unimodular triangulation with vertices among
  these points), "verified up to r = 6".
- **The paper.** J. A. De Loera, T. B. McAllister, "On the computation of Clebsch–Gordan coefficients and the
  dilation effect", arXiv:math/0501446v1 (the only arXiv version). Definition 2.2: the index set i + j + k = r,
  non-negative values, and three families of rhombus inequalities; `note/hive_system.py` generates them as printed
  and checks that they are exactly the unit rhombi. Section 4.1: the homogenized hive matrix M, the g-hive
  polytopes {h~ ≥ 0 : M h~ = b}, Lemma 4.3 and Corollary 4.4, Conjecture 4.5, Theorem 4.6 ("true for r ≤ 6").
- **Corpus record.** ulamai/UnsolvedMath, OWR-741-001 (version 1.7.0 of the dataset; upstream status `open`). Its original statement is the
  conjecture of the abstract; its current statement asks whether every nonempty polytope {h : Bh = b, Rh ≤ c} with
  integral b and c contains an integral point.
- **Conventions fixed in the note (Section 1.3).** The rank r as above; both versions with and without h ≥ 0;
  integral point (record) versus integral vertex (sources); B = restriction to the boundary, R = one row per
  rhombus with entries 0, ±1. The examples have b = 0, which is integral in every encoding of the boundary
  equations.

## Readings
- Read: the Oberwolfach abstract (publisher's file); arXiv:math/0501446v1: abstract, Sections 1 and 2, the passages
  of Section 3.1 on the ranks, Section 4.1; McAllister's thesis (University of California, Davis, 2006):
  Definition 1.2 with the example after it, the passages of Section 2.2.1 on the ranks, Section 3.1;
  arXiv:1405.1687v3 (Haase, Paffenholz, Piechnik, Santos): Section 1.2.1 and the list of open questions; the
  abstract of the talk of C. Haase on the web page of the seminar meeting of 26 June 2015.
- Not read: the journal version of the paper of De Loera and McAllister (not accessible to us); the published
  version of the survey; Knutson–Tao and the TOPCOM paper (cited as in the sources).

## Results in the paper
- Theorem 1.1, Theorem 1.2, Corollary 1.3 as above.
- Lemma 2.1 (reduction to boundary 0 and right-hand sides ⌊excess(x)⌋; both versions). Lemma 2.2 (if a bad point
  exists, a bad vertex of the arrangement {form = integer} exists; the vertices form finitely many classes modulo
  Z^n).
- Proposition 3.1 (P_6(0, w) = {φ}); Proposition 3.2 (P_6(0, w^ψ) = {ψ}, ψ = φ + (1/2)e_(4,1,1), total 13).
- Lemma 4.1 (extension from rank r to rank r + 1, with the formula for the total); Proposition 4.2 (for every r ≥ 6
  a single non-integral point with integral total); Lemma 4.3 (the columns generate Z^m, 0 is an integral
  combination with coefficient sum 1; a cover of the convex hull by unimodular simplices gives integral solutions);
  the paragraph after the proof of Corollary 1.3 (every integral point of the convex hull of the columns is a
  column, so that the restriction to simplices with vertices among the columns is no restriction).
- Remarks 6.1 to 6.4: the range of the earlier verification (facts from the texts only); integral vertices (a
  test); rank 6 (a computation not used in the proofs: 173,839 classes, 162 without integral solution); open
  questions.

## Computations (programs and outputs in reproducibility/)
| Program | Statement | Kind | Result |
|---|---|---|---|
| `note/hive_system.py` | set-up | exact | ranks 1–10: 3r(r−1)/2 printed inequalities = the unit rhombi, each once |
| `note/example_rank6.py` | Proposition 3.1 | exact check | φ satisfies the 45 inequalities (w = −1, 0, 1 for 10, 33, 2 rhombi); the fifteen inequalities are among them; twenty certificates for P = {φ}; no integral point with interior values in [−4, 5] |
| `note/example_psi_and_totals.py` | Propositions 3.2, 4.2 | exact check | the eighteen inequalities; totals 25/2 (φ) and 13 (ψ); totals 18, 25, 33, 42, 52, 63 for ranks 7–12 |
| `note/extension.py` | Lemma 4.1 | exact test | ranks 7–12: extended point feasible, row k = 1 forced to 0, old system recovered; rank 7: no integral point in [−4, 5]^15 |
| `note/small_ranks.py` | Theorem 1.2 | exact, part of the proof | rank 4: 9 forms, 84 sets, 4 classes; rank 5: 18 forms, 18,564 sets, 394 classes; with h ≥ 0: 21 forms, 54,264 sets, 418 classes; every class solved |
| `note/integral_vertices_rank5.py` | Remark 6.2 | exact test | all 4 and 394 polytopes have an integral vertex |
| `note/rank6_vertices.py` | Remark 6.3 | floating-point determinants, exact certificates | 30,045,015 sets of ten forms; 173,839 classes; 173,677 solved; 162 not, each with exact bounds on one coordinate that exclude every integer |
| `verification_run_B/t2_cover_check.py` | Theorem 1.2, second proof | exact, integers | partitions into 4, 64 and 21,484 boxes verified |

## Independent verification runs
Three independent, AI-assisted verification runs. The first two did not see each other and examined the first
written version.
- **Run A (statement and proofs).** Compared the formalisation with the two sources (own download and extraction).
  Checked each of the fifteen inequalities and the derivation of Proposition 3.1; derived Lemmas 2.1, 2.2, 4.1 and
  Corollary 1.3 (a) again; read and re-ran `small_ranks.py`; reproduced Table 2 with its own implementation and a
  second enumeration of the classes from their definition; tested all 184,820 classes with denominators up to 7
  and random points in rank 5. No mathematical error. Its findings, all taken over: the fifteen inequalities give
  the single point (three further lines); the wording of Conjecture 1 of the Oberwolfach abstract needs examples
  with integral total, for which it supplied ψ and the extension of the reflected φ; the two wordings of the
  record; wording of the remark on the range.
- **Run B (own computations, literature).** Built B, R, M with its own code; exact simplex method with certificates
  checked against M: the g-hive polytope of rank 6 is one point with 25 non-integral coordinates; the same for the
  extended examples of ranks 7 and 8; the second proof of Theorem 1.2 by box partitions; all classes with
  denominators up to 20 (rank 5), 80 (rank 4), 200 (rank 3); random right-hand sides; re-run of the programs of the
  note (outputs reproduced). No mathematical error. Its findings, all taken over: the passages of the thesis on the
  ranks; the abstract of the talk of 2015 names the same matrix; Theorem 1.2 is contained in Theorem 4.6 of the
  sources; the name "bases" for sets of n forms was corrected.
- **Run 2 (the third run, on the final text).** Read the final text line by line; compared every statement about the
  sources with the Oberwolfach abstract, arXiv:math/0501446v1, the thesis, arXiv:1405.1687v3 and the page of the
  talk of 2015, and the record with the dataset. Own programs, exact arithmetic (`independent_run_2/`): Table 1; the
  inequalities (1)–(20); own rational combinations which give h = φ from the fifteen inequalities and h = ψ from
  the eighteen; the totals 25/2 and 13; φ* and the totals 18, 25, 33, 42, 52, 63, with single points certified for
  ranks 7, 8, 9; Lemma 4.3 (a) for ranks 6 and 7; Table 2 reproduced; the box partitions checked with its own forms;
  the recorded bounds of the 162 classes of Remark 6.3. Re-ran `run_note.sh` and `t2_cover_check.py` (outputs
  reproduced). No mathematical error. Its findings, all taken over: the version of the dataset in the
  bibliography; the floor in the public notice; working names and old statement numbers in the package. It also
  proved that every integral point of the convex hull of the columns is a column; the paragraph after the proof of
  Corollary 1.3 was added from this. Not checked by this run: the literature searches, the completeness of the
  enumeration of the 173,839 classes of rank 6, the programs that need numpy or scipy.

## Relation to the literature, novelty and scope
- The conjecture: De Loera and McAllister (Oberwolfach abstract 2004; arXiv 2005, Experiment. Math. 2006,
  Conjecture 4.5; thesis 2006, Conjecture 3.5). The three texts state that it was verified for r ≤ 6. In the
  indexing of the definitions (dimension (r+1)(r+2)/2) the example of the note has r = 6, so that the statement of
  Theorem 4.6 for r = 6 and Corollary 1.3 cannot both hold. The texts do not use the letter r in one way (the
  example labelled r = 5 in the arXiv paper has five rows and is labelled r = 4 in the thesis; the ranks of the
  experiments differ by one between the two texts), and none gives the size of a triangulated matrix. The note
  records these facts in Remark 6.1 and draws no conclusion from them.
- The survey of Haase, Paffenholz, Piechnik and Santos (arXiv:1405.1687v3; Mem. Amer. Math. Soc. 2021) lists the
  question as open. The abstract of a talk of C. Haase (Berlin, 26 June 2015) announces a regular unimodular
  triangulation; no written account was found.
- Searches (12 October 2026): arXiv, zbMATH Open, Crossref, OpenAlex, Semantic Scholar (lists of works citing the
  paper, examined by title), nine web searches. MathSciNet and Google Scholar were not searched; the authors of
  the conjecture were not contacted.

## Public release and license
Unrefereed preprint, CC BY 4.0. Files: paper (PDF), source archive (LaTeX source, this report, reproducibility
package), verification report. AI-assisted tools supported research, computation, proof development and manuscript
preparation; the author is responsible for all claims.