Non-unimodal h^*-vectors of very ample polytopes: the Lasoń–Michałek family in closed form
Abstract
Lasoń and Michałek constructed, for every k ≥ 2 and a ≥ 0, a very ample lattice polytope P_(k,a) of dimension 2k-1 as a lattice segmental fibration over the edge polytope of the even cycle C₂ₖ. Hofscheier, Kurylenko and Nill recently observed that P_(15,332) × P_(15,332) has a non-unimodal h^*-vector, answering a question of Ferroni and Higashitani and one of Balletti; their example was found with a language model and their proof is the display of three 22-digit integers computed with software the paper does not name. We compute the Ehrhart polynomial and the h^*-polynomial of P_(k,a) in closed form for all k and a, show that putting an arbitrary integer height vector c over the cycle changes nothing — the Ehrhart polynomial depends only on the alternating sum of the heights — and use the resulting arithmetic to settle four points the source leaves open. Throughout, a violation at index i means h^*ᵢ₋₁>h^*ᵢ<h^*ᵢ₊₁, which is the failure of unimodality the source exhibits; every negative statement below asserts the absence of one, which is formally weaker than unimodality. For a ≤ 4000 the h^*-vector of P_(15,a) × P_(15,a) has a violation exactly for a ∈ {332,…,338}, always at index 26; the minimality of k=15, which the source asserts without proof, holds over the ranges searched; there are non-unimodal products of two very ample polytopes that are not isomorphic and not even equidimensional, namely P_(14,559) × P_(16,205), with factors of dimensions 27 and 31; and, for the source's closing question, no violation in the ranges searched sits above d/2 — every one sits at least 3 below it, the same distance as in all three published examples. All statements are machine-checked in Lean 4.
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Archived files
- Version 1 · current (opens in a new tab)
Source snapshot 2026-08-30 15:34 UTC
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Claim ledger
Stated results
V1known2026-08-28
The source's main theorem: h*₂5 > h*₂6 < h*₂7 for the 58-dimensional very ample polytope P_(15,332) x P_(15,332), with the three 22-digit values, and 26 as the only violated index
V2known2026-08-28
The source's Proposition 1.2(3), h*_(P_(15,332))(t) = 1 + 30(t +... + t¹4) + 346 t¹5, recomputed from the derived Ehrhart polynomial, with the lattice-point and normalised-volume cross-checks
V3candidate2026-08-28
Closed form for the Lasoń-Michałek family: E(m) = (m+1)C(m+2k-1,2k-1) - (m+1-a)C(m+k-1,2k-1) and h*(t) = 1 + 2k(t +... + tᵏ⁻¹) + (a+k-1)tᵏ, verified for 2 <= k <= 20, a <= 200
V4candidate2026-08-28
The non-unimodality window of P_(15,a) x P_(15,a) is exactly a in 332,...,338 for a <= 4000, and the violated index is 26 for every one of them
V5candidate2026-08-28
Minimality of k = 15: for every k <= 14 and every a <= 2200 the h*-vector of P_(k,a) x P_(k,a) is unimodal; and the k = 16 square window is exactly [370,394]
V6candidate2026-08-28
New non-unimodal very ample products with two different factors: P_(15,a) x P_(15,332) exactly for a in [332,344], and P_(14,a) x P_(16,205) exactly for a in [554,561] - factors of different dimensions 27 and 31
V7measurement2026-08-28
The source's closing question, searched: no violation above d/2 anywhere in the Lasoń-Michałek product family - every violation is at least 3 below d/2, for 2 <= k <= 26 and a <= 2200 and on two asymmetric slices
V8routine2026-08-28
Negative controls: binom pinned by Pascal's rule on [0,350]², the h* transform pinned against four textbook h*-vectors, the dip predicate pinned both ways, and too-large / too-small refutations of the window
V9candidate2026-08-28
The whole even-cycle construction is one-parameter: for every integer height vector c, the segmental fibration P(k,c) over the edge polytope of C₂k has E_(P(k,c)) = E_(P_(k,|gamma|)) with gamma = sumᵢ (-1)ⁱ⁻¹ cᵢ; certified on 8192 height vectors at k = 2, 3
Provenance
- Generated by
- Machina Mathematica
- Released by
- Korea Superintelligence Labs
- Source context
- A lattice polytope P ⊂ Rⁿ of dimension d has Ehrhart polynomial E_P(k) = |kP ∩ Zⁿ| and h*-polynomial h*_P(t) = (1−t)ᵈ⁺¹ Σ_(k≥0) E_P(k) tᵏ = Σᵢ₌₀ᵈ h*ᵢ tⁱ. The h*-vector is unimodal when h*₀ ≤ ⋯ ≤ h*ⱼ ≥ ⋯ ≥ h*_d for some j.
- Snapshot
- 2026-09-07 03:53 UTC
- Ledger commit
801848d7