The small Davenport constant of the Heisenberg group of order 125

Technical Summary: The Small Davenport Constant of the Heisenberg Group of Order 125

Problem Statement
The paper addresses the determination of the small Davenport constant, $d(G)$, for the Heisenberg group $H_p^3$ of order $p^3$ over the finite field $\mathbb{F}_p$. The constant $d(G)$ is defined as the maximal length of a product-one-free sequence over $G$ (a sequence where no nonempty subsequence admits an ordering whose product is the identity). While Godara and Sarkar established $d(H_{27}) = 6$ for $p=3$ and conjectured $d(H_p^3) = 3p - 3$ for all odd primes $p$, the case for $p \ge 5$ remained open. This work specifically settles the first open case: $p=5$, determining $d(H_{125})$.

Methodology
The proof combines structural group theory with exhaustive computational verification, reducing a non-commutative problem to additive combinatorics over $\mathbb{F}_p$.

1. Product-One Criterion: The authors establish a necessary and sufficient condition for a sequence to have a product-one ordering. For a sequence $I$ with elements $g_i = M(a_i, b_i, c_i)$, a product-one ordering exists if and only if:

   • The sum of the quotient vectors vanishes: $\sum (a_i, b_i) = (0, 0)$.
   • The negative sum of the central coordinates, $-\sum c_i$, lies within the "cross-sum set" $\Omega(I)$. This set is defined as the set of all possible values of $\sum_{s<t} a_{\sigma(s)}b_{\sigma(t)}$ over all permutations $\sigma$ of the sequence.
   • This criterion reduces the non-abelian problem to analyzing the geometry of quotient vectors in $\mathbb{F}_p^2$ and the arithmetic of central coordinates.

2. Reduction to Additive Combinatorics:

   • Line Caps: Using Olson's result on the Davenport constant of $C_p^2$, the authors derive a "line cap" bound: for any line $L$ through the origin in $\mathbb{F}_p^2$, the number of non-central terms with quotients on $L$ plus the number of central terms is bounded by $2p-2$.
   • Central Forcing: A lemma is proven stating that if a sequence contains a zero-sum-free set of central terms $Z$ and a collection of disjoint zero-quotient-sum blocks $B_j$, a product-one subsequence is forced if $|\Sigma_0(Z)| + \sum (|\Omega(B_j)| - 1) \ge p$. Here, $\Sigma_0(Z)$ is the set of subset sums of the central coordinates.

3. The Spread Bound and Exhaustive Search:

   • The core of the upper bound proof relies on a "spread" parameter, $\sigma(Q)$, defined as the maximum value of $\sum (|\Omega(B_j)| - 1)$ over disjoint zero-sum blocks of a quotient multiset $Q$.
   • For $p=5$, the authors reduce the problem of proving $d(H_{125}) \le 12$ to verifying a single finite statement: Proposition 2. This proposition asserts that for any multiset $Q$ of length $13-z$ (where $z$ is the number of central terms) satisfying the line cap constraints, $\sigma(Q) \ge 4-z$.
   • This proposition is verified via an exhaustive, memory-flat depth-first search in C. The search enumerates all valid multisets over the 24 nonzero vectors of $\mathbb{F}_5^2$, pruning branches that fail to meet the spread requirement. The result (zero "bad leaves") was independently reproduced by a second implementation with a different pruning strategy.

4. Verification Rigor: Every numerical claim, including the group multiplication logic, the product-one criterion, and the auxiliary lemmas, was machine-checked. The authors explicitly note that while the result is not formally verified in a proof assistant, the independent reproduction of the load-bearing search and machine-checked lemmas serve as the reproducibility guarantee.

Key Results

• Theorem 1: $d(H_{125}) = 12$.
  • Lower Bound: The sequence $x^4 y^4 v^4$ (length 12) is explicitly shown to be product-one-free.
  • Upper Bound: Every sequence of length 13 over $H_{125}$ contains a product-one subsequence. This is proven by showing that any such sequence must satisfy the conditions of the Central Forcing lemma, which is guaranteed by the verified Spread Bound (Proposition 2).
• Limitations for $p \ge 7$: The authors demonstrate that the method does not generalize directly to $p=7$. A specific counterexample multiset $Q_0$ is identified where the Chevalley–Warning theorem forces a zero-sum block, but that block is collinear (yielding $|\Omega|=1$) rather than "wide" (yielding $|\Omega|=p$). Consequently, the spread cannot be forced by polynomial methods alone, and the exhaustive search space becomes computationally intractable ($\sim 10^{16}$ multisets). This leaves the range $18 \le d(H_{343}) \le 24$ open.

Significance and Claims
The paper claims to be the first resolution of the open case for the Heisenberg group Davenport constant conjecture ($d(H_p^3) = 3p-3$) beyond $p=3$.

• Novelty: The contribution lies not in the introduction of new theorems (Cauchy–Davenport, Chevalley–Warning, and Olson's results are standard) but in their specific assembly to address a non-abelian target and the rigorous finite verification that closes the gap for $p=5$.
• Specificity: The argument is highlighted as "genuinely $p$-specific." The authors identify the precise step where the logic fails for $p \ge 7$ (the inability of Chevalley–Warning to force a wide, non-collinear block), providing an honest assessment of the method's limits.
• Reproducibility: The paper emphasizes the role of computational verification, noting that the "verdict" of the upper bound is an empirical fact derived from exhaustive search, independently reproduced to ensure correctness.
• Attribution: The paper explicitly discloses that the results were obtained via an AI research collaboration, with specific tasks (problem framing, verification apparatus, and the final write-up) attributed to an AI system under human direction, while the logical reduction and correction of initial hypotheses were performed by a "cold, tool-free" reasoner.