💻 computer science

COMPOSE: Composing Future Theorems from Citations and Formal Structure

The paper introduces COMPOSE, a dual-graph framework that generates plausible future mathematical theorems by jointly leveraging scientific citation contexts and formal theorem dependencies, demonstrating superior performance through a newly constructed dataset and benchmark.

Original authors: David Busbib, Michael Werman

Published 2026-05-29

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: David Busbib, Michael Werman

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to predict the next big hit in a specific field of mathematics. You want to guess what a future paper will say, but you don't want to just make up random math words. You want your guess to be plausible—meaning it fits the current conversation and follows the strict rules of logic that math requires.

This paper introduces a new system called COMPOSE (Composing Future Theorems from Citations and Formal Structure) designed to do exactly that.

Here is how it works, broken down into simple concepts:

The Problem: Two Different Maps

To predict the future of math, you usually need two things, but existing tools only look at one:

The "Social" Map (Citations): This is like looking at who is talking to whom at a party. If Paper A cites Paper B, they are having a conversation. This tells you what topics are popular and where the research is heading.
The "Logic" Map (Formal Structure): This is like the rulebook of a game. In math, you can't just say anything; you have to prove it using previous steps. This map shows the strict dependencies: "You can only prove Theorem X if you already know Lemma Y."

The Issue:

If you only look at the Social Map, you might guess a topic that is popular but logically impossible (like trying to build a house without a foundation).
If you only look at the Logic Map, you might find a logically perfect proof, but it might be about a topic nobody cares about anymore, or it might miss the "big picture" of where the field is going.

The Solution: COMPOSE

The authors built a system that acts like a bilingual translator who is also a detective. It looks at both maps at the same time.

The Input: You give the system a "Anchor Paper" (a current math paper).
The Dual Graph:
- It builds a Scientific Graph: It looks at the Anchor Paper and the papers it cites, creating a web of ideas and conversations.
- It builds a Formal Graph: It takes the math theorems inside those papers and finds their "official" versions in a giant digital library called Mathlib (which is like a verified encyclopedia of math proofs). It then traces the logical dependencies between those official versions.
The Fusion: The system uses a special "neural network" (a type of AI) to merge these two maps. It learns how the conversation (citations) connects to the rules (formal logic).
The Output: The system generates a new, plausible mathematical claim (a "future theorem") that fits the current conversation and obeys the logical rules.

The Analogy: Predicting the Next Move in Chess

Imagine you are trying to predict the next move in a complex chess game.

Old methods were like looking at a list of popular chess openings (the "Social Map"). They could guess you might play a "King's Gambit," but they didn't know if that move was actually legal or if it would get you checkmated immediately.
COMPOSE is like a grandmaster who looks at the history of the game (who played what before) and the strict rules of chess (how pieces move). It predicts a move that is not only a popular strategy but also a legal, sound move that fits the current board state.

How They Tested It

The researchers didn't just guess; they built a massive test set.

They collected 108,000 examples of papers and their corresponding formal logic structures.
They created a "future test" using papers published in late 2024 and 2025 (which the model hadn't seen yet).
They asked the model: "Based on this old paper, what will the next paper say?"

The Results:

Better Guesses: COMPOSE was much better at guessing the exact topic of the future paper than other AI models.
More Grounded: When human judges (and other AIs) reviewed the output, COMPOSE's guesses were rated higher for being mathematically deep, specific, and accurate.
The Secret Sauce: When they turned off either the "Social Map" or the "Logic Map," the system got worse. This proved that you really need both to make a good prediction.

A Limitation (The "Blind Spot")

The paper admits one weakness: The system relies on the "Formal Map" (Mathlib) to work. If a math topic is very new or niche and hasn't been written down in the formal library yet, the system might get confused and try to force the new topic into an old, unrelated category (like trying to explain a new type of music using the rules of a different instrument).

Summary

COMPOSE is a tool that helps predict the future of mathematics by combining what mathematicians are talking about with how their logic actually works. It ensures that future predictions aren't just fluent-sounding nonsense, but are grounded in both the history of the field and the strict rules of math.

Technical Summary: COMPOSE – Composing Future Theorems from Citations and Formal Structure

Problem Definition

The paper addresses the challenge of grounded future mathematical generation: the task of predicting a plausible future mathematical claim (a theorem-like statement) for a given "anchor" paper. The authors argue that a valid prediction must satisfy two simultaneous constraints:

Scientific Trajectory: It must extend the research direction indicated by the scientific citation graph and the evolution of ideas in the literature.
Formal Grounding: It must respect the logical dependencies inherent in formal mathematics, ensuring the claim is validly derivable from existing definitions, lemmas, and theorems.

Existing approaches typically model only one of these sources. Models trained solely on scientific text may generate fluent but logically ungrounded claims, while theorem-proving systems reason over formal structures but lack the scientific context to identify promising research directions. The paper posits that combining these complementary sources is non-trivial due to the fundamental differences in how scientific papers and formal libraries (like Mathlib) organize knowledge.

Methodology: The COMPOSE Framework

The authors propose COMPOSE, a dual-graph framework that conditions a language model on both scientific citation context and formal theorem structure. The approach consists of three main components:

1. Data Construction

The authors constructed a dataset of 108,000 paired scientific-formal graph examples derived from arXiv mathematics papers (2000–2023) and the Lean Mathlib library.

Scientific Graph ( $G_s$ ): Built from an anchor paper and its citation context (up to two hops). It includes abstract nodes for papers and theorem nodes extracted from the text. Edges represent citation links and internal paper dependencies.
Formal Graph ( $G_f$ ): Constructed by aligning extracted informal theorems to corresponding theorems in Mathlib. This alignment uses FrenzyMath (a corpus of human-written informal descriptions for Mathlib theorems) to perform informal-to-informal retrieval, avoiding the noise of direct autoformalization. The formal graph expands from the matched theorem root using dependency edges extracted via LeanDojo.
Benchmark: A temporally held-out test set of 47,000 future papers from late 2024 and 2025, used to evaluate the model's ability to predict actual published results.

2. Model Architecture

COMPOSE employs a dual-encoder architecture followed by a graph-conditioned decoder:

Dual Encoders: Two dedicated Graph Neural Networks (GNNs) process $G_s$ $G_{s}$ and $G_f$ $G_{f}$ separately.
- $G_s$ nodes are initialized with E5 embeddings (scientific text).
- $G_f$ nodes are initialized with DeepSeek-Math embeddings (formal signatures).
- Both use directed, edge-type-specific message passing with gated residual connections.
Fusion Module: The representations from both graphs are projected into a shared latent space and fused via a bidirectional cross-attention mechanism. This allows the model to integrate scientific direction with formal constraints.
Decoder: A pretrained math-specialized LLM (DeepSeek-Math-7B or Mistral-7B) adapted with LoRA. Cross-attention layers are inserted into the decoder to condition the generation on the fused graph embeddings, allowing the model to attend to the full graph structure during token generation.

3. Two-Stage Training

Stage 1 (Representation Learning): The graph encoders and fusion module are trained without the decoder using three objectives:
- Link Prediction: To ensure encoders capture graph structure.
- Contrastive Alignment: To align the fused graph representation with the target paper's abstract and main claim.
- Cross-Alignment: To explicitly align informal theorem nodes with their matched formal counterparts.
Stage 2 (Generation): The decoder is added and fine-tuned using:
- Autoregressive Loss: Standard next-token prediction.
- Graph Margin Loss: A regularization term that penalizes the model if it generates the same text for a mismatched graph pair, forcing it to rely on the specific graph context.

Key Results

Experiments were conducted on the 47K future paper benchmark, comparing COMPOSE against strong baselines (including GIANTS, FutureGen, GoAI, and various text-only or prompt-only variants).

Retrieval Performance: COMPOSE achieved the highest Gap (0.240), defined as the difference in cosine similarity between the generated text and the true future paper versus unrelated papers. It significantly outperformed baselines in H@10 (50.8%) and H@100 (80.8%), indicating its generated claims are much more likely to retrieve the correct future paper from the pool.
Novel Mathematical Claim Prediction (NMCP): COMPOSE led in Precision (0.560) and Match (0.730), demonstrating that its outputs align with future theorems at both formal and informal levels, not just topically.
LLM-Judge Evaluation: An LLM judge (GPT-4.0) evaluated outputs on Content, Depth, Novelty, Precision, and Specificity. COMPOSE achieved the highest overall score (3.36/5), particularly in Technical Depth (3.52) and Specificity (3.52), producing more concrete and mathematically rich outputs than baselines which often generated vague summaries.
Ablation Studies: Removing either the scientific or formal graph encoder caused significant performance drops, confirming that both modalities provide complementary signals. The formal graph was crucial for grounding, while the scientific graph was essential for identifying the correct research direction.

Significance and Claims

The paper claims that future mathematical generation benefits from combining scientific context with formal structure.

Grounded Generation: The authors demonstrate that relying solely on scientific text leads to claims that are topically relevant but logically ungrounded, while relying solely on formal structure limits the model to known theorems without anticipating new directions. COMPOSE successfully bridges this gap.
Data Contribution: The construction of the 108K paired dataset and the 47K future paper benchmark provides a new resource for evaluating grounded mathematical generation, addressing the lack of temporal benchmarks in this field.
Methodological Advance: The dual-graph framework with cross-attention fusion offers a novel approach to integrating heterogeneous knowledge sources (informal literature vs. formal logic) for generative tasks.

The authors note limitations, specifically that the method relies on approximate informal-formal alignment (which can introduce noise in areas with sparse Mathlib coverage) and that generated claims are not verified by a proof assistant, relying instead on retrieval and LLM-based evaluation as proxy signals. They suggest that future work could incorporate proof-aware verification to directly check the formal correctness of generated claims.