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From Euler to AI: Unifying Formulas for Mathematical Constants
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Builds an automated LLM-plus-symbolic pipeline that harvests pi formulas from 455k arXiv papers, validates them with PSLQ, and unifies them via canonical recurrences and a new coboundary solver (UMAPS) inside Conservative Matrix Fields.
Problem
Formulas for mathematical constants such as pi are scattered across the literature and their interconnections are unknown; no prior work performs symbolic unification of such formulas at scale.
Approach
Harvest formulas with LLMs, convert each to a minimal polynomial recurrence (canonical form), and prove equivalences via coboundary matrices computed by UMAPS, placing formulas as trajectories in a Conservative Matrix Field (CMF) for pi.
Key results
- From 455,050 arXiv papers, 385 distinct pi formulas were validated, and 360/385 (94%) were proven related to others.
- 166/385 (43%) of validated pi formulas reside in a single Conservative Matrix Field.
- On 10 formula pairs with equal dynamical metrics, Gemini 2.5 Pro detected 8/10 and correctly proved 5/10; GPT-4o detected 1/10 and proved 2/10.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| use GPT-4o rather than Claude 3.7 Sonnet as the extractor, with the classifier fixed to GPT-4o mini | formula harvesting · formula harvesting | successful extraction rate: 89.9% -> 97.6% | Helped |
| use GPT-4o rather than GPT-4o mini as the extractor, with the classifier fixed to GPT-4o mini | formula harvesting · formula harvesting | successful extraction rate: 69.6% -> 97.6% | Helped |
| use GPT-4o rather than Claude 3.7 Sonnet as the extractor, with the classifier fixed to GPT-4o mini | formula harvesting · formula harvesting | faulty code: 21 -> 2 | Helped |
| use GPT-4o rather than GPT-4o mini as the extractor, with the classifier fixed to GPT-4o mini | formula harvesting · formula harvesting | faulty code: 70 -> 2 | Helped |
| use UMAPS to find and prove formula relations | π formula unification · validated π formulas | formulas with a found relation: 360/385 (94%) | Helped |
Surprising findings
- Gemini 2.5 Pro detected 8/10 equivalences but only proved 5/10, and GPT-4o achieved only 1/10 detections, showing LLMs struggle even on equal-metric pairs.
- Sharing the irrationality measure delta is necessary but not sufficient for coboundary equivalence.
Limitations
- Harvesting relies on LLM interpretation of LaTeX, likely causing data loss and false negatives.
- Some formulas required manual variable substitutions to be evaluated.
- Only 43% of validated formulas land in the single pi CMF, so unification is incomplete.
Experimental setup
Datasets: arXiv LaTeX source (455,050 papers)
Benchmarks: pi formula harvest from arXiv (385 validated formulas) LLM equivalence detection on 10 formula pairs
Metrics: number of validated formulas fraction of formulas with found relation fraction in same CMF successful detections and correct proofs
Baselines: GPT-4o Gemini 2.5 Pro Preview Claude 3.7 Sonnet GPT-4o mini
Builds on: Polynomial continued fractions Conservative Matrix Fields PSLQ integer relation algorithm RISC recurrence fitting
Closest papers by meaning
- Unsupervised Discovery of Formulas for Mathematical Constants (NeurIPS 2024 · similarity 0.92)
- The Ramanujan Library - Automated Discovery on the Hypergraph of Integer Relations (ICLR 2025 · similarity 0.91)
- Llemma: An Open Language Model for Mathematics (ICLR 2024 · similarity 0.83)
- Efficient numeracy in language models through single-token number embeddings (ICML 2026 · similarity 0.83)
- FIND: A Framework for Discovering Formulas in Data (AAAI 2025 · similarity 0.83)
- HERMES: Towards Efficient and Verifiable Mathematical Reasoning in LLMs (ICML 2026 · similarity 0.82)
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