The Problem Is the Problem: Towards Scalable Mathematical Discovery

Zeyu Zheng; Shengtong Zhang; Jeremy Avigad; Prasad Tetali; Sean Welleck

cs.AIAug 17, 2026

The Problem Is the Problem: Towards Scalable Mathematical Discovery

Authors: Zeyu Zheng, Shengtong Zhang, Jeremy Avigad, Prasad Tetali, Sean Welleck

Organizations: Carnegie Mellon University · Anysphere Co.

Abstract

AI systems are increasingly capable of contributing to mathematical research. In research practice, frontier-model reasoning is a limited resource, and expert mathematical review is even more sharply constrained. Allocating these scarce resources well is therefore central to making AI-assisted mathematical discovery efficient. In most current AI-for-math workflows, human effort is concentrated at the beginning and end, in selecting suitable research problems and later reviewing the resulting artifacts. These two stages are becoming bottlenecks for research-level mathematics. We address them by proposing a new human-AI discovery paradigm. The human input is no longer a single problem selected in advance, but a research direction in which the experts have interest and expertise. The system then searches a broad literature corpus for candidate problems in that direction. Inspired by search and recommender systems, we build Find, Attempt, and Recommend (FAR), a literature-to-review cascade that automates the search for suitable problems and focuses human attention on artifacts that have passed several stages of filtering. In a combinatorics pilot, the pipeline starts from 5,245 combinatorics papers, recovers 6,453 candidate conjectures or open problems, and filters them to 4,717 apparently well-posed and still-open conjectures. Subsequent reasoning and automated triage stages surface 598 potential resolutions and select 77 items for author-team review. Among them, we identify many interesting discoveries, including results on conjectures and questions of Davies--Jenssen--Perkins--Roberts, Erdős--Straus, Ikenmeyer--Pak--Panova, and Lund--Saraf--Wolf. These results demonstrate the effectiveness of this new mode of human-AI collaboration for mathematical discovery.

Figures & tables

Figure 1: From choosing a problem to choosing a direction. The upper part shows the problem-level interface; the lower part shows our approach. Figure 3 gives the details of FAR .

Figure 2: Schematic view of the current reachable region. The figure is illustrative, and both axes should be read qualitatively. A single pass over the pool probes which conjectures can produce artifacts worth review under the current model. As model capability improves, more conjectures may become reachable.

Figure 3: From papers to recommendations for expert review. The upper row shows a search or recommender pipeline that recalls and filters candidates from a large corpus. Its numbers indicate typical orders of magnitude. The lower row shows the analogous FAR pipeline. Numbers in the lower row are counts from our pilot run detailed in Section 4 .

Field	Value
Source paper	Ikenmeyer, Pak, and Panova, Positivity of the Symmetric Group Characters is as Hard as the Polynomial Time Hierarchy
Extracted label	Conjecture 5.3.2
Extracted statement	The problem ComputeCharBinary is $\mathsf{GapP}$ -complete under many-one reductions.
Status	Open. The check found no credible resolution, and records that the completeness question is still unsettled.

Table 1: A recovered candidate, from source text to the pool.

Panel of Figure 4 — Figure 4: Each score against the quantity it judges. Panel (a) plots $\delta(d^{-1}[a,b))$ on an axis starting at $50\%$ , panel (b) plots $\iota(i^{-1}[a,b))$ . $n$ counts attempts in (a) and accepted resolutions in (b). Candidates that a later status recheck reclassified as solved or invalid are excluded.

Figure 5: Allocation strategies against the budget. Each fit is made on four fifths of

\mathcal{P}

and applied to the remaining fifth, from which

B/5

conjectures are drawn. The five selections together make one set of

B

conjectures, and each point averages what that set returns over

1000

random partitions. Ties are broken at random, and the uniform baseline is computed exactly from its closed form. Here,

B

only counts the allocated attempts.

Appendix figures & tables14 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 6: Each score against the quantity it judges, with Wilson intervals. Panel (a) gives

\delta

against the difficulty score, panel (b) gives

\iota

against the importance score.

strategy	$B=10$	$B=25$	$B=50$	$B=75$	$B=100$	$B=200$	$B=300$
uniform random	0.17	0.43	0.86	1.29	1.72	3.44	5.17
rank on $\hat{p}$	0.40	0.97	1.64	2.29	2.95	5.42	7.92
rank on $i\,\hat{p}$	0.15	0.47	1.02	1.59	2.24	4.78	7.25
rank on $\hat{p}$ inside the top $1/10$	0.30	0.70	0.99	1.01	1.06	1.99	2.00
rank on $\hat{p}$ inside the top $1/5$	0.17	0.42	0.86	1.25	1.68	2.99	3.72

Appendix

Table 2: Expected number of artifacts, the objective

f_{1}

strategy	$B=10$	$B=25$	$B=50$	$B=75$	$B=100$	$B=200$	$B=300$
uniform random	0.08	0.21	0.42	0.63	0.84	1.68	2.51
rank on $\hat{p}$	0.16	0.40	0.73	1.05	1.38	2.62	3.87
rank on $i\,\hat{p}$	0.09	0.28	0.60	0.89	1.22	2.49	3.72
rank on $\hat{p}$ inside the top $1/10$	0.26	0.60	0.84	0.86	0.90	1.69	1.70
rank on $\hat{p}$ inside the top $1/5$	0.13	0.34	0.69	1.00	1.34	2.44	3.06

Appendix

Table 3: Expected total importance of the artifacts returned, the objective

f_{2}

strategy	$B=10$	$B=25$	$B=50$	$B=75$	$B=100$	$B=200$	$B=300$
uniform random	0.078	0.176	0.299	0.386	0.448	0.574	0.629
rank on $\hat{p}$	0.157	0.354	0.437	0.467	0.485	0.499	0.500
rank on $i\,\hat{p}$	0.090	0.280	0.546	0.597	0.600	0.600	0.600
rank on $\hat{p}$ inside the top $1/10$	0.259	0.596	0.841	0.850	0.850	0.850	0.850
rank on $\hat{p}$ inside the top $1/5$	0.126	0.294	0.540	0.690	0.779	0.850	0.850

Appendix

Table 4: Expected maximum importance among the artifacts returned, the objective

f_{3}

Panel of Figure 7 — Figure 7: On the left, the unique $4$ -uniform tree $T$ of order $7$ : its two edges meet in the single vertex $4$ . On the right, two of the fourteen affine planes of $\mathbb{F}_{2}^{3}$ , drawn as faces of the cube; they meet in the two points $000$ and $100$ , marked by the thick segment. Lemma C.2.4 says that two red edges can never meet the way $e_{1}$ and $e_{2}$ do.

Panel of Figure 8 — Figure 8: Two fibers of $G_{m}=C_{5}\,\square\,K_{m,m}$ , over adjacent vertices $x\in L$ and $x^{\prime}\in R$ . Each fiber induces a copy of $C_{5}$ , and the gray matching between them preserves the $\mathbb{Z}_{5}$ -coordinate, so the independent sets chosen in fibers on opposite sides must be disjoint. The gray halos mark a dominant splitting from Lemma C.3.6 : every fiber over $L$ takes its independent set inside the nonadjacent pair $X=\{0,2\}$ and every fiber over $R$ inside the complement $Y=\{1,3,4\}$ , and this is the splitting of $\mathbb{Z}_{5}$ that maximizes $f(X)f(Y)$ . The filled vertices show one admissible choice inside them, namely $\{0,2\}$ over $x$ and $\{1,3\}$ over $x^{\prime}$ .

	units digit	$X$ -block	$X^{\prime}$ -block	selector digit
	$Q^{0}$	$Q^{1},\ldots,Q^{k}$	$Q^{k+1},\ldots,Q^{k+k^{\prime}}$	$Q^{k+k^{\prime}+1}$
$a_{S}$ $(S\in C)$	$0$	$\mathbf{1}_{S}$	$\mathbf{0}$	$0$
$b_{S^{\prime}}$ $(S^{\prime}\in C^{\prime})$	$0$	$\mathbf{0}$	$\mathbf{1}_{S^{\prime}}$	$0$
$c_{A}$	$1$	$\mathbf{0}$	$\mathbf{1}$	$1$
$c_{B}$	$0$	$\mathbf{1}$	$\mathbf{0}$	$1$
$H$	$2$	$\mathbf{1}$	$\mathbf{1}$	$1$

Appendix

Table 5: The base-

Q

digits of the parts of

\mu

and of the two targets

s

and

s-1

. Here

\mathbf{1}

and

\mathbf{0}

denote the all-ones and all-zeros patterns on the block indicated, and

\mathbf{1}_{S}

denotes the indicator pattern of

S

on the

X

-block, with

\mathbf{1}_{S^{\prime}}

defined analogously on the

X^{\prime}

-block. No column can reach

Q

, so subset sums may be compared digit by digit: a subset of parts summing to

s

must take

c_{A}

, hence miss the

X^{\prime}

-block entirely and pick out an exact cover of

(X,C)

, while a subset summing to

s-1

must take

c_{B}

and pick out an exact cover of

(X^{\prime},C^{\prime})

Panel of Figure 9 — Figure 9: The graph $G_{5,3}$ , with its ten clique edges and its two private leaves at each clique vertex drawn in gray, and with the vertices covered by the matching drawn solid. The matching $M$ consists of the two heavy black edges $v_{1}v_{2}$ and $v_{3}y_{3,1}$ , so the set of clique vertices missed by the clique part of $M$ is $\{v_{3},v_{4},v_{5}\}$ and exactly one of them takes a leaf. The residual graph $F_{M}$ , the set of edges meeting no edge of $M$ , is drawn heavy and dashed: it is the copy of $G_{2,3}$ carried by the two clique vertices that took no leaf, and $\nu(F_{M})=2$ .

Panel of Figure 10 — Figure 10: The $t$ -stretch of an edge $uv\in S$ inside $G_{t}$ , on the left, and its contribution to the Schur complement, on the right. The $t-1$ internal vertices, drawn hollow, carry the positive definite block $T_{t-1}$ of $A(G_{t})+2I$ ; eliminating them subtracts $(t-1)/t$ from each of the diagonal entries at $u$ and at $v$ and subtracts $(-1)^{t}/t$ from the entries at $uv$ and at $vu$ , which is the content of Proposition C.9.6 .

Panel of Figure 11 — Figure 11: The two counts behind Theorem C.10.1 , drawn for $q=7$ and $\nu=3$ . On the left are the eight tangent lines to $S$ at a point $p$ , all lying in the tangent plane at $p$ ; the four solid ones are those whose direction $[u:v]$ has $Q(u,v)=u^{2}-3v^{2}$ a nonzero square, and these are exactly the lines of $L$ through $p$ . On the right are the seven points of $\mathbb{F}_{q}^{3}$ lying over a fixed $(x,y)$ , labeled by the value $s=Q(x,y)-z$ ; the four solid ones are those belonging to $P(L)$ , namely those with $s\in\{0,1,2,4\}$ .

Panel of Figure 12 — Figure 12: The two arcs making up $A_{a}$ and the two arcs making up $A_{b}$ , drawn on the cycle $\mathbb{Z}_{p}$ . Each arc has length $q$ , so $|A_{a}|=|A_{b}|=2q$ ; as in the proof of Lemma C.11.5 , the pair $\{a,b\}$ is neutral precisely when the four arcs are pairwise disjoint, which is the situation drawn.

Panel of Figure 13 — Figure 13: The three ingredients of $J_{\beta}$ . On the left, the three lifted triples equation 33 sitting over one edge $uv$ of $G$ , drawn as triangles sharing the palette vertex $w_{\beta(uv)}$ . On the upper right, the chain of copies of $S$ that forces $w_{1},\dots,w_{t}$ to receive a single common color. On the lower right, the copy of $D$ that forces the three copies of a vertex $v$ to receive three distinct colors.

Panel of Figure 14 — Figure 14: The digraph $G_{k,m}$ . The hub $x$ feeds $m$ arms, each a directed path with $k$ internal vertices running from $x$ to $p_{1}$ , and the tail $p_{1}\to p_{2}\to\cdots\to p_{k}\to y\to x$ closes the digraph up. The dashed arcs $p_{j}\to x$ , present for every $j$ , are what keep $x$ out of every $k$ -kernel.

Panel of Figure 15 — Figure 15: On the left, the hypertree $H$ on $V=[6]$ with hyperedges $e_{1}=\{1,2,3\}$ and $e_{2}=\{3,4,5,6\}$ drawn as gray ovals. On the right, the tree $T_{\Omega}$ for the choice $\tau_{e_{1}}=(1,2,3)$ , $\tau_{e_{2}}=(3,4,5,6)$ , $B_{e_{1}}=\{2\}$ and $B_{e_{2}}=\{1,3\}$ : each hyperedge is threaded along its chosen order, and the three bold arrows are the strict steps, the two thin gray arrows the weak ones. Reversing the bold arrows gives the acyclic orientation $Q_{\Omega}$ , and the labeling $\omega_{\Omega}$ that assigns $1,2,3,4,5,6$ to the vertices $4,6,1,3,5,2$ in this order is increasing along it, as required by equation 38 .

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