arXiv subjects

Eliot Hodges

Publications and source records attributed to Eliot Hodges.

5 recordsLinked to original sources

arXiv2026

Measuring Progress in Reasoning Toward Mathematical Discovery with Automatic Verification

Can AI make progress on important, unsolved mathematical problems? Large language models are now capable of sophisticated mathematical and scientific reasoning, but whether they can perform novel research is still widely debated and underexplored. We introduce HorizonMath, a benchmark of 113 predominantly unsolved problems spanning eight domains in mathematics and the mathematical sciences, paired with an open-source evaluation framework for automated verification. Our benchmark targets the generator-verifier gap: problems where discovery is hard and requires meaningful mathematical insight, but verification is computationally straightforward. This contrasts with most existing research-level benchmarks, which instead rely on formal proof verification or manual review, both of which are expensive to scale. Because these solutions are unknown, HorizonMath is resistant to data contamination, and most state-of-the-art models score under 10%. Using this framework, we identify six novel solutions to research problems that either resolve previously open questions or improve on the best-known published results, with GPT-5.4 Pro and GPT-5.6 Sol each discovering three of these solutions. Across seven frontier model families, reasoning efficiency and behavior also vary substantially. We release HorizonMath as an open challenge and a growing community resource, where each verified solution is a candidate contribution to the mathematical literature.

cs.LG↗

arXiv2025

A parametrization of $3$-class groups of quadratic rings over Dedekind domains

Let $R$ be a Dedekind domain with field of fractions $K$ and $\operatorname{char}(R)\neq3$. In this paper, we generalize Bhargava's parametrization of $3$-torsion ideal classes by binary cubic forms to work over $R$. Specifically, we construct arithmetic subgroups of $\operatorname{GL}_2(K)$ whose actions on certain lattices of binary cubic forms over $K$ parametrize $3$-torsion ideal classes in class groups of quadratic rings over $R$.

math.NT↗

arXiv2024

Bender--Knuth Billiards in Coxeter Groups

Let $(W,S)$ be a Coxeter system, and write $S=\{s_i:i\in I\}$, where $I$ is a finite index set. Fix a nonempty convex subset $\mathscr{L}$ of $W$. If $W$ is of type $A$, then $\mathscr{L}$ is the set of linear extensions of a poset, and there are important Bender--Knuth involutions $\mathrm{BK}_i\colon\mathscr{L}\to\mathscr{L}$ indexed by elements of $I$. For arbitrary $W$ and for each $i\in I$, we introduce an operator $τ_i\colon W\to W$ (depending on $\mathscr{L}$) that we call a noninvertible Bender--Knuth toggle; this operator restricts to an involution on $\mathscr{L}$ that coincides with $\mathrm{BK}_i$ in type $A$. Given a Coxeter element $c=s_{i_n}\cdots s_{i_1}$, we consider the operator $\mathrm{Pro}_c=τ_{i_n}\cdotsτ_{i_1}$. We say $W$ is futuristic if for every nonempty finite convex set $\mathscr{L}$, every Coxeter element $c$, and every $u\in W$, there exists an integer $K\geq 0$ such that $\mathrm{Pro}_c^K(u)\in\mathscr{L}$. We prove that finite Coxeter groups, right-angled Coxeter groups, rank-3 Coxeter groups, affine Coxeter groups of types $\widetilde A$ and $\widetilde C$, and Coxeter groups whose Coxeter graphs are complete are all futuristic. When $W$ is finite, we actually prove that if $s_{i_N}\cdots s_{i_1}$ is a reduced expression for the long element of $W$, then $τ_{i_N}\cdotsτ_{i_1}(W)=\mathscr{L}$; this allows us to determine the smallest integer $\mathrm{M}(c)$ such that $\mathrm{Pro}_c^{\mathrm{M}(c)}(W)=\mathscr{L}$ for all $\mathscr{L}$. We also exhibit infinitely many non-futuristic Coxeter groups, including all irreducible affine Coxeter groups that are not of type $\widetilde A$, $\widetilde C$, or $\widetilde G_2$.

math.CO↗

arXiv2023

The Distribution of Sandpile Groups of Random Graphs with their Pairings

We determine the distribution of the sandpile group (also known as the Jacobian) of the Erdős-Rényi random graph $G(n,q)$ along with its canonical duality pairing as $n$ tends to infinity, fully resolving a conjecture from 2015 due to Clancy, Leake, and Payne and generalizing the result by Wood on the groups. In particular, we show that a finite abelian $p$-group $G$ equipped with a perfect symmetric pairing $δ$ appears as the Sylow $p$-part of the sandpile group and its pairing with frequency inversely proportional to $|G||\mathrm{Aut}(G,δ)|$, where $\mathrm{Aut}(G,δ)$ is the set of automorphisms of $G$ preserving the pairing $δ$. While this distribution is related to the Cohen-Lenstra distribution, the two distributions are not the same on account of the additional algebraic data of the pairing. The proof utilizes the moment method: we first compute a complete set of moments for our random variable (the average number of epimorphisms from our random object to a fixed object in the category of interest) and then show the moments determine the distribution. To obtain the moments, we prove a universality result for the moments of cokernels of random symmetric integral matrices whose dual groups are equipped with symmetric pairings that is strong enough to handle both the dependence in the diagonal entries and the additional data of the pairing. We then apply results due to Sawin and Wood to show that these moments determine a unique distribution.

math.CO↗

arXiv2022

On Promotion and Quasi-tangled Labelings of Posets

In 2022, Defant and Kravitz introduced extended promotion (denoted $\partial$), a map that acts on the set of labelings of a poset. Extended promotion is a generalization of Schützenberger's promotion operator, a well-studied map that permutes the set of linear extensions of a poset. It is known that if $L$ is a labeling of an $n$-element poset $P$, then $\partial^{n-1}(L)$ is a linear extension. This allows us to regard $\partial$ as a sorting operator on the set of all labelings of $P$, where we think of the linear extensions of $P$ as the labelings which have been sorted. The labelings requiring $n-1$ applications of $\partial$ to be sorted are called tangled; the labelings requiring $n-2$ applications are called quasi-tangled. In addition to computing the sizes of the fibers of promotion for rooted tree posets, we count the quasi-tangled labelings of a relatively large class of posets called inflated rooted trees with deflated leaves. Given an $n$-element poset with a unique minimal element with the property that the minimal element has exactly one parent, it follows from the aforementioned enumeration that this poset has $2(n-1)!-(n-2)!$ quasi-tangled labelings. Using similar methods, we outline an algorithmic approach to enumerating the labelings requiring $n-k-1$ applications to be sorted for any fixed $k\in\{1,\ldots,n-2\}$. We also make partial progress towards proving a conjecture of Defant and Kravitz on the maximum possible number of tangled labelings of an $n$-element poset.

math.CO↗