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The dataset generation failed because of a cast error
Error code: DatasetGenerationCastError
Exception: DatasetGenerationCastError
Message: An error occurred while generating the dataset
All the data files must have the same columns, but at some point there are 3 new columns ({'index', 'correctness_reason', 'correctness_score'})
This happened while the json dataset builder was generating data using
hf://datasets/ai4he/conjecture-context-ablation/evaluation/expert_validation/gemma-4-31b Alignment.json (at revision 5eedb5902220e28bf8be29c5b82dd6c27da22087), ['hf://datasets/ai4he/conjecture-context-ablation@5eedb5902220e28bf8be29c5b82dd6c27da22087/evaluation/expert_validation/Validation Dataset.json', 'hf://datasets/ai4he/conjecture-context-ablation@5eedb5902220e28bf8be29c5b82dd6c27da22087/evaluation/expert_validation/gemma-4-31b Alignment.json', 'hf://datasets/ai4he/conjecture-context-ablation@5eedb5902220e28bf8be29c5b82dd6c27da22087/evaluation/expert_validation/gemma-4-31b Quality.json'], ['hf://datasets/ai4he/conjecture-context-ablation@5eedb5902220e28bf8be29c5b82dd6c27da22087/evaluation/expert_validation/Validation Dataset.json', 'hf://datasets/ai4he/conjecture-context-ablation@5eedb5902220e28bf8be29c5b82dd6c27da22087/evaluation/expert_validation/gemma-4-31b Alignment.json', 'hf://datasets/ai4he/conjecture-context-ablation@5eedb5902220e28bf8be29c5b82dd6c27da22087/evaluation/expert_validation/gemma-4-31b Quality.json']
Please either edit the data files to have matching columns, or separate them into different configurations (see docs at https://hf.co/docs/hub/datasets-manual-configuration#multiple-configurations)
Traceback: Traceback (most recent call last):
File "/usr/local/lib/python3.14/site-packages/datasets/builder.py", line 1856, in _prepare_split_single
writer.write_table(table)
~~~~~~~~~~~~~~~~~~^^^^^^^
File "/usr/local/lib/python3.14/site-packages/datasets/arrow_writer.py", line 811, in write_table
self._write_table(pa_table, writer_batch_size=writer_batch_size)
~~~~~~~~~~~~~~~~~^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
File "/usr/local/lib/python3.14/site-packages/datasets/arrow_writer.py", line 822, in _write_table
pa_table = table_cast(pa_table, self._schema)
File "/usr/local/lib/python3.14/site-packages/datasets/table.py", line 2381, in table_cast
return cast_table_to_schema(table, schema)
File "/usr/local/lib/python3.14/site-packages/datasets/table.py", line 2309, in cast_table_to_schema
raise CastError(
...<3 lines>...
)
datasets.table.CastError: Couldn't cast
index: int64
author: string
input: string
actual_output: string
expected_output: string
correctness_score: double
correctness_reason: string
to
{'author': Value('string'), 'expected_output': Value('string'), 'actual_output': Value('string'), 'input': Value('string')}
because column names don't match
During handling of the above exception, another exception occurred:
Traceback (most recent call last):
File "/src/services/worker/src/worker/job_runners/config/parquet_and_info.py", line 1379, in compute_config_parquet_and_info_response
parquet_operations, partial, estimated_dataset_info = stream_convert_to_parquet(
~~~~~~~~~~~~~~~~~~~~~~~~~^
builder, max_dataset_size_bytes=max_dataset_size_bytes
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
)
^
File "/src/services/worker/src/worker/job_runners/config/parquet_and_info.py", line 956, in stream_convert_to_parquet
builder._prepare_split(split_generator=splits_generators[split], file_format="parquet")
~~~~~~~~~~~~~~~~~~~~~~^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
File "/usr/local/lib/python3.14/site-packages/datasets/builder.py", line 1702, in _prepare_split
for job_id, done, content in self._prepare_split_single(
~~~~~~~~~~~~~~~~~~~~~~~~~~^
gen_kwargs=gen_kwargs, job_id=job_id, **_prepare_split_args
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
):
^
File "/usr/local/lib/python3.14/site-packages/datasets/builder.py", line 1858, in _prepare_split_single
raise DatasetGenerationCastError.from_cast_error(
...<4 lines>...
)
datasets.exceptions.DatasetGenerationCastError: An error occurred while generating the dataset
All the data files must have the same columns, but at some point there are 3 new columns ({'index', 'correctness_reason', 'correctness_score'})
This happened while the json dataset builder was generating data using
hf://datasets/ai4he/conjecture-context-ablation/evaluation/expert_validation/gemma-4-31b Alignment.json (at revision 5eedb5902220e28bf8be29c5b82dd6c27da22087), ['hf://datasets/ai4he/conjecture-context-ablation@5eedb5902220e28bf8be29c5b82dd6c27da22087/evaluation/expert_validation/Validation Dataset.json', 'hf://datasets/ai4he/conjecture-context-ablation@5eedb5902220e28bf8be29c5b82dd6c27da22087/evaluation/expert_validation/gemma-4-31b Alignment.json', 'hf://datasets/ai4he/conjecture-context-ablation@5eedb5902220e28bf8be29c5b82dd6c27da22087/evaluation/expert_validation/gemma-4-31b Quality.json'], ['hf://datasets/ai4he/conjecture-context-ablation@5eedb5902220e28bf8be29c5b82dd6c27da22087/evaluation/expert_validation/Validation Dataset.json', 'hf://datasets/ai4he/conjecture-context-ablation@5eedb5902220e28bf8be29c5b82dd6c27da22087/evaluation/expert_validation/gemma-4-31b Alignment.json', 'hf://datasets/ai4he/conjecture-context-ablation@5eedb5902220e28bf8be29c5b82dd6c27da22087/evaluation/expert_validation/gemma-4-31b Quality.json']
Please either edit the data files to have matching columns, or separate them into different configurations (see docs at https://hf.co/docs/hub/datasets-manual-configuration#multiple-configurations)Need help to make the dataset viewer work? Make sure to review how to configure the dataset viewer, and open a discussion for direct support.
author string | expected_output string | actual_output string | input string |
|---|---|---|---|
Alexandra Seceleanu | \label{thm: reg} Let $G$ be a bipartite graph with adjacency matrix $A$. Then \[ \reg(R/I_G)\leq \brank(A). \] | \begin{theorem}
Let $G=(X\sqcup Y,E)$ be a bipartite graph with edge ideal $I_G\subset R$ and biadjacency matrix $A(G)$. Suppose that $\{C_1,\dots,C_r\}$ is a biclique cover of $G$ so that $bc(G)=r$, where by definition $bc(G)=\operatorname{brank}(A(G))$. Since each biclique $C_i$ has edge ideal $I_{C_i}$ satisfying $\... | Given the following results, definitions, and notation, write a theorem that would be useful in proving the paper. You do not need to use all of these examples, but select the ones that you feel are the most relevant. Definitions: ["\\begin{definition}\nThe \\textit{Booleans}, $\\Bc$, are defined to be the semiring on ... |
Alexandra Seceleanu | \label{thm: identity complement} Let $A$ be an $n\times n$ Boolean identity complement matrix and let $G$ be the corresponding bipartite graph. Then \begin{enumerate} \item The regularity of $R/I_G$ is 2. \item The isolation number of $A$ is 3. \end{enumerate} | \begin{theorem}
Let $G$ be a bipartite graph with no isolated vertices and let $A$ be its Boolean (i.e., $0$--$1$) adjacency matrix. Suppose that
\begin{enumerate}
\item the edge ideal $I_G$ of $G$ may be decomposed as a sum of biclique edge ideals---that is, there exists a cover $\{C_1,\dots,C_m\}$ of $G$ by bicliques... | Given the following previous results, write a theorem that will be useful in proving the paper. Previous results: ["Proposition{\\cite{Herzog}}\\label{thm: reg sum} Let $I$ and $J$ be monomial ideals. Then \\[ \\reg(I+J)\\leq reg(I)+reg(J) \\]","Corollary\\label{2.9} For a finite collection $\\{I_\\... |
Alexandra Seceleanu | \label{thm: isolation number = reg} Let $A$ be a Boolean matrix and let $S=k[A]/J_A$ be the quotient ring of the isolation ideal. Then \[ \reg(S) = \iota(A) = \dim(\Delta_A) +1 \leq \brank(A). \] | \begin{theorem}[Regularity Equals the Isolation Number]
Let $A$ be an $m\times n$ Boolean matrix and let $J_A\subset k[A]$ be its isolation ideal as defined in (\ref{eq: J}). Suppose that $S= k[A]/J_A$ is the corresponding Artinian quotient. Then the Castelnuovo--Mumford regularity of $S$ is given by
\[
\operatorname{r... | Given the following previous results, write a theorem that will be useful in proving the paper. Previous results: ["{\\cite{Herzog}}\\label{thm: reg sum} Let $I$ and $J$ be monomial ideals. Then \\[ \\reg(I+J)\\leq reg(I)+reg(J) \\]","\\begin{corollary}\\label{2.9} For a finite collection $\\{I_\\la... |
Jordan Bounds | \label{central} Let $f:N_r\rightarrow N_r$ be a linear map such that $[f(x),x]\in Z(N_r)$ for all $x\in N_r$. Then there exist $\lambda\in F$ and additive $\mu:N_r\rightarrow\Omega$ such that $f(x)=\lambda x+\mu(x)$ for all $x\in N_r$. | \begin{theorem}
Let $F$ be a field of characteristic zero and let $N_n=N_n(F)$ denote the ring of $n\times n$ strictly upper triangular matrices (with $n\ge 4$) over $F$, whose center is $Z(N_n)$. Suppose that $f\colon N_n\to N_n$ is a linear map satisfying the commuting property
\[
[f(A),A]=0\quad\text{for all }A\in N... | Given the following results, definitions, and notation, write a theorem that would be useful in proving the paper. You do not need to use all of these examples, but select the ones that you feel are the most relevant. Definitions: ["Let F be a field of characteristic zero and N_r=N_r(F) the ring of strictly upper trian... |
Elizabeth Gross | [The Maximum Likelihood Degree of $\beta$-SBMs] \label{thm:RecursionTheorem} Given positive integers $n,k,n_1,n_2,\dots, n_k$ such that $n,k>1$, \begin{enumerate} \item $\MLdeg(n)= \begin{cases} 1 & n=2\\ 2^{n-1}-n & n>2 \end{cases}$, \item $\MLdeg(n_1... | \begin{theorem}[ML Degree Invariance and Quadratic Structure]
Let $k$ be a positive integer and let $n_1,n_2,\dots,n_k$ be positive integers. Consider the $\beta$-Stochastic Block Model $M=\mathcal{M}(n_1,\dots,n_k)$ with design matrix $A$, and let $V(A)$ be the Zariski closure of its associated log-linear model with d... | Given the following previous results, write a theorem that will be useful in proving the paper. Previous results: ["For a log-linear model given by a design matrix $A$, we denote by $V(A)$ the Zariski closure of the model $\\mathcal{M}_A$, and we denote by $I_A:= I(V(A))$ the defining ideal of $V(A)$. For log-linear m... |
Christopher O’Neill | \label{thm: computing aw in fixed directions} Let $P\subset\RR^d$ be a rational polytope and let $L$ be the input bit-size of $P$. In fixed dimension, given a nonzero integer vector $c\in\ZZ^d\setminus\{0\}$, there is an algorithm to compute $\aw_c(P)$ that is polynomial time in~$L$ and the input bit-size of $c$. | \begin{theorem}[Structure and Efficient Enumeration of the Arithmetic Range]
Let $K\subset\mathbb{R}^d$ be a convex body with $K\cap\mathbb{Z}^d\neq\emptyset$ and let $c\in\mathbb{Z}^d\setminus\{0\}$. Define the arithmetic range of $K$ in direction $c$ by
\[
\mathrm{AR}_c(K)=\{ n\in\mathbb{Z}:\ \exists x\in K\cap\mathb... | Given the following definitions and notation, write a theorem that would be useful in proving the paper. You do not need to use all of these examples, but select the ones that you feel are the most relevant. Definitions: ["Fix a nonzero $c \\in \\ZZ^d$. The \\emph{hyperplane cover of $K\\cap\\ZZ^d$ in direction $c$} is... |
Christopher O’Neill | \label{thm: eql in fixed directions} Let $P$ be a rational polytope in $\mathbb{R}^d$. Then, for a fixed direction $c \in \mathbb{Z}^d$, the arithmetic width $\aw_c(nP)$ is an eventually quasilinear function of $n \in \mathbb{Z}_{\geq 0}$. | \begin{theorem}
Let \(K\subset\mathbb{R}^d\) be a convex body whose affine span contains a lattice point, and let \(c\in\mathbb{Z}^d\setminus\{0\}\) be an integer vector so that the linear functional \(\ell_c:\mathbb{Z}^d\to\mathbb{Z}\) is defined by \(\ell_c(x)=c^T x\). Assume further that \(K\) is a rational polytope... | Given the following previous results, write a theorem that will be useful in proving the paper. Previous results: ["\\begin{definition}\n Let $K\\subseteq \\mathbb{R}^d$ be a convex body and let $\\ell_c: \\mathbb{Z}^d\\to \\mathbb{Z}$ be a linear functional given by $\\ell_c(x)=c^Tx$. The \\emph{arithmetic range} o... |
Dwight Anderson Williams II | \label{thm: tower} Let $x=(x_0,x_1,\ldots,x_n)\in\{0,1,2,\ldots,n\}^{n+1}$. Then $x\in \ToH_{n+1}$ if and only if $x$ satisfies: \begin{enumerate} \setcounter{enumi}{-1} \item $x_n=0$, \item there exists $j\in[n-1]$ and distinct $k,k' \in [n-1]$ with $x_k = x_{k'} = j$, and \item if $i \neq k, k'$, then... | \begin{theorem}
Let $n\in\mathbb{N}$ and let $\alpha=(a_1,a_2,\dots,a_n)\in\mathrm{PF}(n)$ be a parking function with total displacement
\[
d(\alpha)=\sum_{i=1}^n k_i,
\]
where $k_i$ is the number of spots by which car $i$ is bumped from its preferred spot. For $d\ge0$ denote by $\mathrm{dPF}(n,d)$ the subset of $\ma... | Given the following definitions, write a theorem that will be useful in proving the paper. Definitions: ["Fix $n\\in\\mathbb{N}=\\{1,2,3,\\ldots\\}$ and consider $n$ cars entering sequentially to park on a one-way street consisting of $n$ spots numbered consecutively from 1 to $n$. \tThe cars enter the street one at a ... |
Dwight Anderson Williams II | {\label{thm: pf}} Let $\alpha=(a_1,\ldots,a_n)\in[n]^n$. Then $\alpha\in PF_n(1)$ if and only if $\alpha$ satisfies: \begin{enumerate} \item there exists $j\in[ n-1]$ and distinct $k,k'\in [n]$ such that $a_k=a_{k'}=j$ and $a_i\neq j$ for any other index $i\neq k, k'$ and \item $ \{a_i:i\in[n] \mbox{... | \begin{theorem}
Let $n\in\mathbb{N}$ and consider any parking function $\alpha\in\mathrm{PF}(n)$ with displacement
\[
d(\alpha):=\sum_{i=1}^n k_i,
\]
where $k_i$ is the number of spots car $i$ is bumped from its preferred spot. Then the following statements hold:
\begin{enumerate}
\item The displacement is bounded by $... | Given the following results, definitions, and notation, write a theorem that would be useful in proving the paper. You do not need to use all of these examples, but select the ones that you feel are the most relevant. Definitions: ["Fix $n\\in\\mathbb{N}=\\{1,2,3,\\ldots\\}$ and consider $n$ cars entering sequentially ... |
Sara Jamshidi | [The Spark Randomizer] \label{thm:main} Let $I=\ideal{f_1,\dots,f_s}\subset K[x_1,\dots,x_n]$ be an ideal and fix a monomial ordering $\prec$. Let $\mH\subset I$ be any finite set guaranteed to contain a minimal Gr\"obner basis of $I$. % \marginpar{\sonja{SEE margin note at thm\ref{thm:grobner violator basis is a ... | \begin{theorem}
Let $K$ be a field and let $R=K[x_1,\dots,x_n]$ be a polynomial ring equipped with a fixed monomial order $\prec$. Let $f_1,\dots,f_s\in R$ and denote by $I=\langle f_1,\dots,f_s\rangle$ the ideal they generate. Suppose that $H\subseteq I$ is a finite set of polynomials (for example, $H$ may be taken to... | Given the following definitions and notation, write a theorem that would be useful in proving the paper. You do not need to use all of these examples, but select the ones that you feel are the most relevant. Definitions: ["Denote by $K$ a field, for example the reader may keep $K=\\mathbb C$ in mind. Let $f_1=0,\... |
Sara Jamshidi | \label{thm:grobner violator basis is a GB} Fix a finite set of polynomials $\mH\subset R$ and a monomial order $\prec$. Then a largest basis of the violator space $(\mH,\vi_\prec)$ contains all the elements in the minimal Gr\"obner basis of $\ideal{\mH}$ which live in $\mH$. %%$\mH$-truncated (minimal) Gr\"obner ... | \begin{theorem}[Gr\"obner Bases via Violator Spaces]
Let $\mathcal{H}$ be a finite set of polynomials in $R=K[x_1,\dots,x_n]$ and fix a monomial order $\prec$. Define the mapping $\mathsf{V}_{\prec}$ on subsets $S\subseteq\mathcal{H}$ by
\[
\mathsf{V}_{\prec}(S):=\{h\in\mathcal{H}:\ \init(S)\subsetneq\init(S\cup\{h\})\... | Given the following results and notation, write a theorem that would be useful in proving the paper. You do not need to use all of these examples, but select the ones that you feel are the most relevant. Results: ["Fix a monomial order $\\prec$ on $R$ and an ideal $I$. Denote by $\\init(h)$ the initial term of $h\\... |
Craig Larson | \label{bracelets-Eger} Every bracelet with a perfect matching is Egerv\'{a}ry. | \begin{theorem}[Bracelet Egerv\'ary]
Let $G$ be a bracelet, that is, a graph consisting of three distinct nodes $u$, $v$, and $w$ together with three weak bananas joining $u$ to $v$, $v$ to $w$, and $w$ to $u$. If $G$ has a perfect matching, then $G$ is Egerv\'ary.
\end{theorem} | Given the following previous results, write a theorem that will be useful in proving the paper. Previous results: ["The graph $K_4$ is the smallest instance of what we call a \\textit{weak wheel}, i.e., a cycle $C$ with three or more nodes, a node $w$ not in $C$, and a path (\\textit{spoke}) from each node of $C$ to $w... |
Craig Larson | \label{weak-bananas-Eger} Every weak banana with a perfect matching is Egerv\'{a}ry. | \begin{theorem}[Unified Egerv\'ary Criterion]
Let $G$ be a matchable graph that admits a perfect matching. Then $G$ is Egerv\'ary provided that at least one of the following conditions holds:
\begin{enumerate}
\item $G$ is a K\H{o}nig--Egerv\'ary (KE) graph, i.e. $\alpha(G)+\mu(G)=|V(G)|$;
\item $G$ contains a subg... | Given the following results and notation, write a theorem that would be useful in proving the paper. You do not need to use all of these examples, but select the ones that you feel are the most relevant. Results: ["Every KE graph is Egerv\\'{a}ry (Corollary~\\ref{cor:kayll-KE-implies-eger}). But there are also non-KE g... |
Martin Malandro | \label{ThmLegitPartition} Let $e_1, e_2, \in E(S)$ with $e_1 \D e_2$. Suppose $f\in E(S)$ with $f\leq e_1$. Then \[ |h\in E(S):h\leq e_1 \up{ and }h\D f| = |h\in E(S):h\leq e_2 \up{ and }h\D f|. \] %In particular, if $E$ is a finite meet-semilattice and $P$ is a set partition of $E$ for which there exists a finite inve... | \begin{theorem}[D-Partition via Inverse Semigroup Restriction]
Let $E$ be a meet-semilattice and let $P$ be a partition of $E$ with induced equivalence relation $\sim$. Suppose there exists an inverse semigroup $S$ with $E=E(S)$ such that for any $e,f\in E$ the $\mathcal{D}$-relation on $S$ restricts to $P$, i.e.
for $... | Given the following results and definitions write a theorem that would be useful in proving the paper. You do not need to use all of these examples, but select the ones that you feel are the most relevant. Definitions: ["Let $E$ be a meet-semilattice and $P$ a set partition of $E$. Write $\\sim$ for the equivalence rel... |
Martin Malandro | %\label{ThmPosetIsom1} %If $S$ and $T$ are isomorphic inverse semigroups and $\phi: S \rightarrow T$ is an isomorphism, then $\phi$ is also an isomorphism of posets $\phi: (S\leq) \rightarrow (T,\leq)$. % | \begin{theorem}[Fundamental Structure of Inverse Semigroups]
Let $S$ be an inverse semigroup with idempotent set $E(S)$ and let the relation $\le$ on $S$ be defined by any of the following equivalent conditions:
\begin{enumerate}
\item[(i)] $s\le t$ if and only if $s=et$ for some $e\in E(S)$,
\item[(ii)] $s=tf$ for... | Given the following notation, write a theorem that will be useful in proving the paper. Notation: ["An {\\\\em inverse semigroup} is a nonempty semigroup $S$ where, for each $x\\in S$, there exists a unique $y\\in S$ such that $xyx=x$ and $yxy=y$. We call $y$ the {\\\\em inverse} of $x$, and we write $x^{-1}=y$.","Ther... |
Alexandra Seceleanu | \label{thm: reg} Let $G$ be a bipartite graph with adjacency matrix $A$. Then \[ \reg(R/I_G)\leq \brank(A). \] | \begin{theorem}
Let $G=(X\sqcup Y,E)$ be a bipartite graph with edge ideal $I_G\subset R$ and biadjacency matrix $A(G)$. Suppose that $\{C_1,\dots,C_r\}$ is a biclique cover of $G$ so that $bc(G)=r$, where by definition $bc(G)=\operatorname{brank}(A(G))$. Since each biclique $C_i$ has edge ideal $I_{C_i}$ satisfying $\... | Given the following results, definitions, and notation, write a theorem that would be useful in proving the paper. You do not need to use all of these examples, but select the ones that you feel are the most relevant. Definitions: ["\\begin{definition}\nThe \\textit{Booleans}, $\\Bc$, are defined to be the semiring on ... |
Alexandra Seceleanu | \label{thm: identity complement} Let $A$ be an $n\times n$ Boolean identity complement matrix and let $G$ be the corresponding bipartite graph. Then \begin{enumerate} \item The regularity of $R/I_G$ is 2. \item The isolation number of $A$ is 3. \end{enumerate} | \begin{theorem}
Let $G$ be a bipartite graph with no isolated vertices and let $A$ be its Boolean (i.e., $0$--$1$) adjacency matrix. Suppose that
\begin{enumerate}
\item the edge ideal $I_G$ of $G$ may be decomposed as a sum of biclique edge ideals---that is, there exists a cover $\{C_1,\dots,C_m\}$ of $G$ by bicliques... | Given the following previous results, write a theorem that will be useful in proving the paper. Previous results: ["Proposition{\\cite{Herzog}}\\label{thm: reg sum} Let $I$ and $J$ be monomial ideals. Then \\[ \\reg(I+J)\\leq reg(I)+reg(J) \\]","Corollary\\label{2.9} For a finite collection $\\{I_\\... |
Alexandra Seceleanu | \label{thm: isolation number = reg} Let $A$ be a Boolean matrix and let $S=k[A]/J_A$ be the quotient ring of the isolation ideal. Then \[ \reg(S) = \iota(A) = \dim(\Delta_A) +1 \leq \brank(A). \] | \begin{theorem}[Regularity Equals the Isolation Number]
Let $A$ be an $m\times n$ Boolean matrix and let $J_A\subset k[A]$ be its isolation ideal as defined in (\ref{eq: J}). Suppose that $S= k[A]/J_A$ is the corresponding Artinian quotient. Then the Castelnuovo--Mumford regularity of $S$ is given by
\[
\operatorname{r... | Given the following previous results, write a theorem that will be useful in proving the paper. Previous results: ["{\\cite{Herzog}}\\label{thm: reg sum} Let $I$ and $J$ be monomial ideals. Then \\[ \\reg(I+J)\\leq reg(I)+reg(J) \\]","\\begin{corollary}\\label{2.9} For a finite collection $\\{I_\\la... |
Jordan Bounds | \label{central} Let $f:N_r\rightarrow N_r$ be a linear map such that $[f(x),x]\in Z(N_r)$ for all $x\in N_r$. Then there exist $\lambda\in F$ and additive $\mu:N_r\rightarrow\Omega$ such that $f(x)=\lambda x+\mu(x)$ for all $x\in N_r$. | \begin{theorem}
Let $F$ be a field of characteristic zero and let $N_n=N_n(F)$ denote the ring of $n\times n$ strictly upper triangular matrices (with $n\ge 4$) over $F$, whose center is $Z(N_n)$. Suppose that $f\colon N_n\to N_n$ is a linear map satisfying the commuting property
\[
[f(A),A]=0\quad\text{for all }A\in N... | Given the following results, definitions, and notation, write a theorem that would be useful in proving the paper. You do not need to use all of these examples, but select the ones that you feel are the most relevant. Definitions: ["Let F be a field of characteristic zero and N_r=N_r(F) the ring of strictly upper trian... |
Elizabeth Gross | [The Maximum Likelihood Degree of $\beta$-SBMs] \label{thm:RecursionTheorem} Given positive integers $n,k,n_1,n_2,\dots, n_k$ such that $n,k>1$, \begin{enumerate} \item $\MLdeg(n)= \begin{cases} 1 & n=2\\ 2^{n-1}-n & n>2 \end{cases}$, \item $\MLdeg(n_1... | \begin{theorem}[ML Degree Invariance and Quadratic Structure]
Let $k$ be a positive integer and let $n_1,n_2,\dots,n_k$ be positive integers. Consider the $\beta$-Stochastic Block Model $M=\mathcal{M}(n_1,\dots,n_k)$ with design matrix $A$, and let $V(A)$ be the Zariski closure of its associated log-linear model with d... | Given the following previous results, write a theorem that will be useful in proving the paper. Previous results: ["For a log-linear model given by a design matrix $A$, we denote by $V(A)$ the Zariski closure of the model $\\mathcal{M}_A$, and we denote by $I_A:= I(V(A))$ the defining ideal of $V(A)$. For log-linear m... |
Christopher O’Neill | \label{thm: computing aw in fixed directions} Let $P\subset\RR^d$ be a rational polytope and let $L$ be the input bit-size of $P$. In fixed dimension, given a nonzero integer vector $c\in\ZZ^d\setminus\{0\}$, there is an algorithm to compute $\aw_c(P)$ that is polynomial time in~$L$ and the input bit-size of $c$. | \begin{theorem}[Structure and Efficient Enumeration of the Arithmetic Range]
Let $K\subset\mathbb{R}^d$ be a convex body with $K\cap\mathbb{Z}^d\neq\emptyset$ and let $c\in\mathbb{Z}^d\setminus\{0\}$. Define the arithmetic range of $K$ in direction $c$ by
\[
\mathrm{AR}_c(K)=\{ n\in\mathbb{Z}:\ \exists x\in K\cap\mathb... | Given the following definitions and notation, write a theorem that would be useful in proving the paper. You do not need to use all of these examples, but select the ones that you feel are the most relevant. Definitions: ["Fix a nonzero $c \\in \\ZZ^d$. The \\emph{hyperplane cover of $K\\cap\\ZZ^d$ in direction $c$} is... |
Christopher O’Neill | \label{thm: eql in fixed directions} Let $P$ be a rational polytope in $\mathbb{R}^d$. Then, for a fixed direction $c \in \mathbb{Z}^d$, the arithmetic width $\aw_c(nP)$ is an eventually quasilinear function of $n \in \mathbb{Z}_{\geq 0}$. | \begin{theorem}
Let \(K\subset\mathbb{R}^d\) be a convex body whose affine span contains a lattice point, and let \(c\in\mathbb{Z}^d\setminus\{0\}\) be an integer vector so that the linear functional \(\ell_c:\mathbb{Z}^d\to\mathbb{Z}\) is defined by \(\ell_c(x)=c^T x\). Assume further that \(K\) is a rational polytope... | Given the following previous results, write a theorem that will be useful in proving the paper. Previous results: ["\\begin{definition}\n Let $K\\subseteq \\mathbb{R}^d$ be a convex body and let $\\ell_c: \\mathbb{Z}^d\\to \\mathbb{Z}$ be a linear functional given by $\\ell_c(x)=c^Tx$. The \\emph{arithmetic range} o... |
Dwight Anderson Williams II | \label{thm: tower} Let $x=(x_0,x_1,\ldots,x_n)\in\{0,1,2,\ldots,n\}^{n+1}$. Then $x\in \ToH_{n+1}$ if and only if $x$ satisfies: \begin{enumerate} \setcounter{enumi}{-1} \item $x_n=0$, \item there exists $j\in[n-1]$ and distinct $k,k' \in [n-1]$ with $x_k = x_{k'} = j$, and \item if $i \neq k, k'$, then... | \begin{theorem}
Let $n\in\mathbb{N}$ and let $\alpha=(a_1,a_2,\dots,a_n)\in\mathrm{PF}(n)$ be a parking function with total displacement
\[
d(\alpha)=\sum_{i=1}^n k_i,
\]
where $k_i$ is the number of spots by which car $i$ is bumped from its preferred spot. For $d\ge0$ denote by $\mathrm{dPF}(n,d)$ the subset of $\ma... | Given the following definitions, write a theorem that will be useful in proving the paper. Definitions: ["Fix $n\\in\\mathbb{N}=\\{1,2,3,\\ldots\\}$ and consider $n$ cars entering sequentially to park on a one-way street consisting of $n$ spots numbered consecutively from 1 to $n$. \tThe cars enter the street one at a ... |
Dwight Anderson Williams II | {\label{thm: pf}} Let $\alpha=(a_1,\ldots,a_n)\in[n]^n$. Then $\alpha\in PF_n(1)$ if and only if $\alpha$ satisfies: \begin{enumerate} \item there exists $j\in[ n-1]$ and distinct $k,k'\in [n]$ such that $a_k=a_{k'}=j$ and $a_i\neq j$ for any other index $i\neq k, k'$ and \item $ \{a_i:i\in[n] \mbox{... | \begin{theorem}
Let $n\in\mathbb{N}$ and consider any parking function $\alpha\in\mathrm{PF}(n)$ with displacement
\[
d(\alpha):=\sum_{i=1}^n k_i,
\]
where $k_i$ is the number of spots car $i$ is bumped from its preferred spot. Then the following statements hold:
\begin{enumerate}
\item The displacement is bounded by $... | Given the following results, definitions, and notation, write a theorem that would be useful in proving the paper. You do not need to use all of these examples, but select the ones that you feel are the most relevant. Definitions: ["Fix $n\\in\\mathbb{N}=\\{1,2,3,\\ldots\\}$ and consider $n$ cars entering sequentially ... |
Sara Jamshidi | [The Spark Randomizer] \label{thm:main} Let $I=\ideal{f_1,\dots,f_s}\subset K[x_1,\dots,x_n]$ be an ideal and fix a monomial ordering $\prec$. Let $\mH\subset I$ be any finite set guaranteed to contain a minimal Gr\"obner basis of $I$. % \marginpar{\sonja{SEE margin note at thm\ref{thm:grobner violator basis is a ... | \begin{theorem}
Let $K$ be a field and let $R=K[x_1,\dots,x_n]$ be a polynomial ring equipped with a fixed monomial order $\prec$. Let $f_1,\dots,f_s\in R$ and denote by $I=\langle f_1,\dots,f_s\rangle$ the ideal they generate. Suppose that $H\subseteq I$ is a finite set of polynomials (for example, $H$ may be taken to... | Given the following definitions and notation, write a theorem that would be useful in proving the paper. You do not need to use all of these examples, but select the ones that you feel are the most relevant. Definitions: ["Denote by $K$ a field, for example the reader may keep $K=\\mathbb C$ in mind. Let $f_1=0,\... |
Sara Jamshidi | \label{thm:grobner violator basis is a GB} Fix a finite set of polynomials $\mH\subset R$ and a monomial order $\prec$. Then a largest basis of the violator space $(\mH,\vi_\prec)$ contains all the elements in the minimal Gr\"obner basis of $\ideal{\mH}$ which live in $\mH$. %%$\mH$-truncated (minimal) Gr\"obner ... | \begin{theorem}[Gr\"obner Bases via Violator Spaces]
Let $\mathcal{H}$ be a finite set of polynomials in $R=K[x_1,\dots,x_n]$ and fix a monomial order $\prec$. Define the mapping $\mathsf{V}_{\prec}$ on subsets $S\subseteq\mathcal{H}$ by
\[
\mathsf{V}_{\prec}(S):=\{h\in\mathcal{H}:\ \init(S)\subsetneq\init(S\cup\{h\})\... | Given the following results and notation, write a theorem that would be useful in proving the paper. You do not need to use all of these examples, but select the ones that you feel are the most relevant. Results: ["Fix a monomial order $\\prec$ on $R$ and an ideal $I$. Denote by $\\init(h)$ the initial term of $h\\... |
Craig Larson | \label{bracelets-Eger} Every bracelet with a perfect matching is Egerv\'{a}ry. | \begin{theorem}[Bracelet Egerv\'ary]
Let $G$ be a bracelet, that is, a graph consisting of three distinct nodes $u$, $v$, and $w$ together with three weak bananas joining $u$ to $v$, $v$ to $w$, and $w$ to $u$. If $G$ has a perfect matching, then $G$ is Egerv\'ary.
\end{theorem} | Given the following previous results, write a theorem that will be useful in proving the paper. Previous results: ["The graph $K_4$ is the smallest instance of what we call a \\textit{weak wheel}, i.e., a cycle $C$ with three or more nodes, a node $w$ not in $C$, and a path (\\textit{spoke}) from each node of $C$ to $w... |
End of preview.
Supplementary Material: Which Mathematical Context Supports Target-Aligned Conjecture Generation?
This repository contains supplementary material for the MathNLP 2026 paper “Which Mathematical Context Supports Target-Aligned Conjecture Generation? A Paired Ablation Study.”
The study examines which parts of theorem-local mathematical context help a language model generate a conjecture aligned with a hidden source theorem. It uses 61 main theorem targets from 12 papers spanning four arXiv mathematics domains (math.AC, math.AG, math.CO, and math.ST). Each theorem is evaluated under 13 context conditions: one full-context condition, six leave-one-out ablations, and six single-component conditions, for a total of 793 generated conjectures.
Repository contents
.
├── README.md
├── requirements.txt
├── data/
│ ├── ablation_data_analysis.xlsx
│ └── theorem_runs/
│ ├── math.AC/
│ ├── math.AG/
│ ├── math.CO/
│ └── math.ST/
├── evaluation/
│ ├── cross_model_agreement/
│ └── expert_validation/
├── prompts/
│ ├── classification_task.md
│ ├── redaction_task.md
│ ├── generation_task.md
│ └── codex_prompts.txt
├── code/
│ ├── eval_deepeval_alignment.py
│ └── eval_deepeval_quality.py
└── sources/
└── README.md
data/
ablation_data_analysis.xlsxcontains the aggregate ablation data and analysis workbook.theorem_runs/contains theorem-specific artifacts organized by mathematical domain and source-paper arXiv identifier. These materials include theorem maps, redacted seeds, generated conjectures, evaluation outputs, and run summaries.- The originally nested
theorem_objects.zip,redacted_seeds.zip, andgenerated_conjectures.ziparchives have been expanded into ordinary directories so that their Markdown contents can be browsed and version-controlled directly on GitHub.
The condition-directory names inside each paper folder are preserved from the original experiment. All Inputs is the full-context condition; the R... directories contain the corresponding ablation-condition outputs.
evaluation/
cross_model_agreement/contains repeated-judge and cross-model materials used to assess evaluator reliability and agreement, including Gemma-4-31B, GPT-5-mini, and exploratory GLM-5.1-FP8 outputs.expert_validation/contains the independent expert-validation dataset and associated automated evaluation outputs.
prompts/
This directory contains the task instructions used for:
- main-theorem classification and theorem-specific object-map construction;
- theorem-specific redaction; and
- blind conjecture generation.
codex_prompts.txt records the orchestration instructions associated with running these stages and the evaluation workflow.
code/
The evaluation scripts implement the two theorem-conditioned metrics used in the paper:
- Expected-Theorem Alignment
- Theorem-Conditioned Conjecture Quality
The scripts use DeepEval and the model endpoint used in the study. No API credentials are included in this repository. They read the credential from the RCD_LLM_API_KEY environment variable and are currently configured for the Clemson RCD endpoint and gemma-4-31b. Users reproducing the evaluation on different infrastructure should update the endpoint/model configuration as needed.
Install the reported DeepEval version with:
pip install -r requirements.txt
Mathematical-context components
Each redacted theorem seed is represented using six context components:
- Ambient Setting
- Core Definitions and Notation
- Local Mathematical Objects
- Structural Constraints and Conditions
- Auxiliary Objects and Constructions
- Potential Generalization Axes
For every theorem target, the study compares the full six-component context with six leave-one-out conditions and six single-component conditions.
Models and evaluation
The experimental pipeline used DeepSeek-V4-Pro for theorem identification, object-map construction, redaction, and conjecture generation. Gemma-4-31B served as the primary DeepEval judge. Additional materials in this repository support repeated-Gemma evaluation, cross-model comparison with GPT-5-mini, an exploratory GLM-5.1-FP8 judge analysis, and expert validation of theorem-conditioned conjecture quality.
The reported evaluation scores are target-conditioned estimates. They should not be interpreted as independent proofs of mathematical correctness, truth, or novelty.
Source papers
The experiments use publicly available mathematics papers identified by arXiv ID. Copies of the third-party source PDFs and LaTeX files have been removed from this public package. See sources/README.md for the exact arXiv versions and citations used in the study.
Reproducibility notes
This repository is intended to make the experimental artifacts, prompts, generated outputs, evaluation data, and analysis materials inspectable. Some components of the original computational environment require access to the same or equivalent model-serving infrastructure.
For the evaluation scripts, requirements.txt pins the DeepEval version reported in the paper (4.1.5).
Citation
If you use these materials, please cite the associated paper:
Which Mathematical Context Supports Target-Aligned Conjecture Generation? A Paired Ablation Study.
Full bibliographic information can be added here after publication.
Dataset card
| Paper | Which Mathematical Context Supports Target-Aligned Conjecture Generation? A Paired Ablation Study |
| Venue | 4th Workshop on Mathematical Natural Language Processing (MathNLP 2026), EMNLP 2026 |
| Authors | Austin LaHue, Michael Burr, Luis David Garcia Puente, Vinita Gangaram Jansari, Benjamin E. Nye, Carlos Toxtli |
| License | Creative Commons Attribution 4.0 International (cc-by-4.0); code: MIT License |
| Dataset DOI | 10.5281/zenodo.24179081 |
| Version | 1.0.0 |
| Code repository | https://github.com/alahue/conjecture-ablation |
| Maintainer | Human-AI Empowerment Lab, Clemson University |
Dataset structure
| Path | Format | Size | Records |
|---|---|---|---|
requirements.txt |
txt | 16 B | |
code/ (2 files) |
py | 22.9 KB | |
data/ (1 files) |
xlsx | 52.9 KB | |
data/theorem_runs/ (1694 files) |
json, md, xlsx | 9.8 MB | 1,553 |
evaluation/cross_model_agreement/ (42 files) |
json, xlsx | 1.9 MB | 943 |
evaluation/expert_validation/ (4 files) |
json, xlsx | 339.4 KB | 45 |
prompts/ (4 files) |
md, txt | 26.0 KB | |
sources/ (1 files) |
md | 2.6 KB |
1,749 files, 12.2 MB in total (metadata files excluded). Records are rows for CSV/TSV/JSONL files and list length for JSON arrays.
Known limitations
- Theorem maps and redacted seeds quote short excerpts of the 12 source arXiv papers; the source PDFs/LaTeX are not redistributed (see sources/README.md for citations).
Personal and sensitive information
None.
License
The data and documentation are released under the Creative Commons Attribution 4.0 International. Source code files (.ipynb, .js, .m, .py, .sh) are released under the MIT License.
Citation
If you use this dataset, please cite the paper:
@inproceedings{LaHue2026Which,
title = {Which Mathematical Context Supports Target-Aligned Conjecture Generation? A Paired Ablation Study},
author = {LaHue, Austin and Burr, Michael and Garcia Puente, Luis David and Jansari, Vinita Gangaram and Nye, Benjamin E. and Toxtli-Hernandez, Carlos},
booktitle = {Proceedings of the 4th Workshop on Mathematical Natural Language Processing (MathNLP 2026)},
address = {Budapest, Hungary},
publisher = {Association for Computational Linguistics},
year = {2026},
month = October,
note = {Forthcoming},
url = {https://sites.google.com/view/mathnlp2026}
}
and the dataset:
@dataset{LaHue2026ConjectureContextAblatioData,
author = {LaHue, Austin and Burr, Michael and Garcia Puente, Luis David and Gangaram Jansari, Vinita and Nye, Benjamin E. and Toxtli, Carlos},
title = {{Conjecture Context Ablation}: Dataset for ``Which Mathematical Context Supports Target-Aligned Conjecture Generation? A Paired Ablation Study''},
year = {2026},
version = {1.0.0},
publisher = {Zenodo},
doi = {10.5281/zenodo.24179081},
url = {https://doi.org/10.5281/zenodo.24179081}
}
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