Metadata

slug: jiang-wen-zhong-2026-alternating-polytope-gorenstein-pairs
bibkey: jiangwenzhong2026alternating
doi: null
url: https://arxiv.org/abs/2607.14887v1
triage: theorem
motivation_gids:
  - D5/S3/Combinatorics/Geometry/AlternatingAdjacentSumGorenstein.result

The Gorenstein pairs for alternating adjacent-sum polytopes

Problem

Xinru Jiang, Suzhen Wen and Yueming Zhong, Alternating adjacent-sum polytopes: transfer matrices and Ehrhart series, arXiv:2607.14887v1, §4, Question Q1 (q-gorenstein), ask:

For $s = 3$ , direct computation (Propositions 3.22 and 3.23) shows that only $d = 2$ is Gorenstein in even dimensions $d \leq 6$ ; we conjecture this extends to all $d \geq 4$ . Characterize all $(s, r)$ with $s \geq 2$ for which $P_{2 r}^{(s)}$ is Gorenstein.

Their set is $\mathcal P_d^{(s)}={x\in\mathbb R_{\ge0}^d: x_i+x_{i+1}\le s+\delta_i,\ 1\le i\le d-1}$, where $δ_{i} = 0$ for odd $i$ and $δ_{i} = 1$ for even $i$ . The Gorenstein characterization used here is the source’s interior-lattice translation condition: some integer $q \geq 1$ and $c \in Z^{d}$ satisfy $\operatorname{int}((n+q)P)\cap\mathbb Z^d =c+(nP\cap\mathbb Z^d) $f ore v ery in t e g er$ n\ge0$.

Motivation

D5/S3/Combinatorics/Geometry/AlternatingAdjacentSumGorenstein.result proves that for every $s \geq 2$ and $r \geq 1$ the condition holds exactly when $(s, r) = (3, 1)$ . This settles both the characterization and the stated $s = 3$ , $d \geq 4$ conjecture. Preregistration #14591 fixes the source, quantifiers and Lean conventions. The source note is Library/Geometry/jiangwenzhong2026alternating.md.

Gap

The literature readings attributed to the Claude Code orchestrator in #14591 found only arXiv v1, no matching MathDB page in the stated title, author and Gorenstein searches, and no resolution in the later same-author paper arXiv:2607.22008, which studies a different family. These readings establish absence within their searched scope; they do not establish an exhaustive priority claim.

Route

At $n = 0$ , the translation identity makes $c$ the unique interior lattice point of $qP$ . Every coordinate of $c$ is at least one, hence $q s \geq 3$ . The all-ones vector belongs to the interior, so uniqueness gives $c = 1$ . If $q s \geq 4$ , the vector $(2, 1, \dots, 1)$ is a different interior lattice point. Thus $q s = 3$ ; with $s \geq 2$ this forces $q = 1$ and $s = 3$ .

For $d \geq 3$ , put $w = (1, 4, 3, 1, \dots, 1)$ . Its adjacent sums are $5, 7, 4, 2, \dots$ , strictly below the corresponding capacities $6, 8, 6, 8, \dots$ of $2 P$ . But the second adjacent sum of $w - 1$ is $3 + 2 = 5 > 4$ . The $n = 1$ identity is impossible. For $d = 2$ , the triangle $P = {(u, v) \geq 0 : u + v \leq 3}$ has the required identity with $q = 1$ and $c = (1, 1)$ : integral strict inequalities at scale $n + 1$ become the non-strict inequalities at scale $n$ after translation. The actual zero dilation is included.

Falsifier

A different parity convention, relative rather than ambient interior, a different lattice, or omission of $n = 0$ changes the encoded problem. The public definitions use the literal adjacent inequalities, Mathlib ambient interior, real set dilation and pointwise integer casts. A single Gorenstein pair with $s \geq 2$ , $r \geq 1$ , different from $(3, 1)$ would contradict result under these definitions.

Evidence

The module contains result : claim, with $\forall s,r\in\mathbb N,\ s\ge2\Rightarrow r\ge1\Rightarrow (\operatorname{IsGorenstein}(s,2r)\iff s=3\land r=1)$. Its private lemmas index_center and not_gorenstein_large_dimension provide the uniqueness and parametric obstruction used by result; gorenstein_three_two supplies the positive direction for every dilation. The axiom closure of every public declaration is contained in ${propext, Classical.choice, Quot.sound}$ . The source-to-Lean interpretation is the convention stated in #14591; equivalence to other algebraic definitions of Gorenstein polytopes is not separately formalized.

Triage

What the settlement shows

Mechanism — proved in this module. index_center handles every $s \geq 2$ and $d \geq 2$ , and forces $q = 1$ , $s = 3$ , $c = 1$ from the translation identity. not_gorenstein_large_dimension uses the parametric $w$ for every $d \geq 3$ . The obstruction is one adjacent-sum excess after translation, caused by the alternating capacities; it does not require an Ehrhart-series computation.
Odd dimensions — paper argument from kernel-checked lemmas. For any $s \geq 2$ and $d \geq 3$ , index_center reduces a supposed Gorenstein translation to $s = 3$ ; not_gorenstein_large_dimension then contradicts it. This includes every odd $d \geq 3$ . The general-dimensional lemmas are in the module; a separate public odd-dimensional theorem is not claimed. Dimension one is outside the source’s domain.
The $s = 1$ family — paper argument. The source’s Theorem 1.9 (thm-gorenstein-s1) remains valid: for even $d = 2 r$ , the even-indexed capacity-two constraints follow from the adjacent disjoint capacity-one pairs. Thus $P$ is the product of $r$ unit triangles. For one triangle, positive integral coordinates in $(n + 3) P$ obey $u + v \leq n + 2$ ; subtracting $(1, 1)$ gives $u^{'}, v^{'} \geq 0$ and $u^{'} + v^{'} \leq n$ . Products give index $3$ and center $1$ for every $r \geq 1$ . This source proof is not a new Lean declaration here.
$h^{*}$ palindromicity — paper argument and exact finite computation. The eight explicit rows for $s = 2, 3, 4$ in Remark 3.18 agree with the characterization: only the $(s, d) = (3, 2)$ row is palindromic. The columns at $d = 8$ are ellipses, not printed vectors. The exact computation below supplies all twelve cases $s \in {2, 3, 4}$ , $d \in {2, 4, 6, 8}$ , and matches all eight printed rows. It also checks $s = 1$ , $d \in {2, 4, 6, 8}$ and the obstruction vector for every $3 \leq d \leq 32$ . These are finite computations; the unbounded classification is result. A general theorem about $h^{*}$ coefficients is not proved in this module.

The answer strengthens the source’s Corollary 3.24 from eventual failure to failure in every even dimension except the unique pair, and proves the $s = 3$ extension beyond dimensions four and six. The source’s transfer-matrix and Ehrhart identities, and its independent $s = 1$ theorem, keep their stated hypotheses and conclusions. Q2 on unimodality and real-rootedness remains open; this module does not settle it.

Exact finite computation

The computation is by the implementation seat. Command: python3 /tmp/op-jwz/check.py; exit code $0$ . The script can be saved from the complete source below and run using Python’s standard library. SHA-256: aafca8eecdaab339085feb4e6a0164f5fe208b7c73a35ecb1f54f2b77dd8f030.

For each $s, d$ , the recurrence enumerates every admissible integral coordinate prefix by its last coordinate. At dilation $n$ , the adjacent capacity is $n (s + δ_{i})$ , and prefix sums count every allowed predecessor exactly once. It computes $L (n)$ for $0 \leq n \leq d$ and $h_{k}^{*} = \sum_{j = 0}^{k} (- 1)^{j} (j d + 1) L (k - j)$ . Trailing zero coefficients are omitted from the displayed vectors.

$s$	$d$	$h^{*}$ vector	Palindromic
2	2	$(1, 3)$	no
2	4	$(1, 30, 55, 9)$	no
2	6	$(1, 197, 1818, 2786, 811, 27)$	no
2	8	$(1, 1180, 36610, 196846, 274966, 106075, 9271, 81)$	no
3	2	$(1, 7, 1)$	yes
3	4	$(1, 90, 284, 94, 1)$	no
3	6	$(1, 893, 13714, 31436, 14273, 973, 1)$	no
3	8	$(1, 8516, 456928, 3528084, 6770616, 3653018, 492328, 9692, 1)$	no
4	2	$(1, 12, 3)$	no
4	4	$(1, 205, 859, 381, 9)$	no
4	6	$(1, 2919, 59352, 166411, 92672, 8442, 27)$	no
4	8	$(1, 40746, 2986812, 27845877, 62524368, 39514968, 6452988, 174132, 81)$	no
1	2	$(1,)$	yes
1	4	$(1, 4, 1)$	yes
1	6	$(1, 20, 48, 20, 1)$	yes
1	8	$(1, 72, 603, 1168, 603, 72, 1)$	yes

from math import comb
import json


def lattice_count(s, d, n):
    """Exact enumeration by the last coordinate; all states are integers."""
    assert s >= 1 and d >= 2 and n >= 0
    counts = [1] * (n * s + 1)
    for i in range(1, d):
        capacity = n * (s + (i % 2 == 0))
        prefix = []
        total = 0
        for count in counts:
            total += count
            prefix.append(total)
        counts = [prefix[min(capacity - y, len(prefix) - 1)]
                  for y in range(capacity + 1)]
    return sum(counts)


def hstar(s, d):
    ehrhart = [lattice_count(s, d, n) for n in range(d + 1)]
    coefficients = [sum((-1) ** j * comb(d + 1, j) * ehrhart[k - j]
                        for j in range(k + 1)) for k in range(d + 1)]
    assert all(c >= 0 for c in coefficients)
    while coefficients[-1] == 0:
        coefficients.pop()
    return coefficients


source_rows = {
    (2, 2): [1, 3],
    (2, 4): [1, 30, 55, 9],
    (2, 6): [1, 197, 1818, 2786, 811, 27],
    (3, 2): [1, 7, 1],
    (3, 4): [1, 90, 284, 94, 1],
    (3, 6): [1, 893, 13714, 31436, 14273, 973, 1],
    (4, 2): [1, 12, 3],
    (4, 4): [1, 205, 859, 381, 9],
}
rows = []
for s in (2, 3, 4):
    for d in (2, 4, 6, 8):
        h = hstar(s, d)
        if (s, d) in source_rows:
            assert h == source_rows[s, d]
        palindromic = h == h[::-1]
        assert palindromic == (s == 3 and d == 2)
        rows.append(dict(s=s, d=d, hstar=h, palindromic=palindromic))
for d in (2, 4, 6, 8):
    h = hstar(1, d)
    assert h == h[::-1]
    rows.append(dict(s=1, d=d, hstar=h, palindromic=True))
for d in range(3, 33):
    w = [1, 4, 3] + [1] * (d - 3)
    assert all(a > 0 for a in w)
    assert all(w[i] + w[i + 1] < 2 * (3 + ((i + 1) % 2 == 0))
               for i in range(d - 1))
    shifted = [a - 1 for a in w]
    assert shifted[1] + shifted[2] > 4
print(json.dumps(dict(rows=rows, source_rows_checked=len(source_rows),
                     witness_dimensions=[3, 32]), indent=2))

ASSUMED-UNVERIFIED

Worldwide prior-art completeness beyond the literature searches stated in #14591 is unverified; no priority claim is made.
The separate algebraic equivalence of the stipulated interior-lattice translation condition is not formalized here.
Q2 and uniform $h^{*}$ coefficient properties beyond the supplied classification remain open.

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