Title: From Euler to AI: Unifying Formulas for Mathematical Constants
Venue: NeurIPS 2025
Accepted: yes
Keywords: AI for Science

AI for Math

LLM-Tool Integration

Mathematical Constants

Continued Fractions

Recurrences

Number Theory

Pi
Abstract: The constant $\large \pi$ has fascinated scholars throughout the centuries, inspiring numerous formulas for its evaluation, such as infinite sums and continued fractions. Despite their individual significance, many of the underlying connections among formulas remain unknown, missing unifying theories that could unveil deeper understanding. The absence of a unifying theory reflects a broader challenge across math and science: knowledge is typically accumulated through isolated discoveries, while deeper connections often remain hidden. In this work, we present an automated framework for the unification of mathematical formulas. Our system combines large language models (LLMs) for systematic formula harvesting, an LLM-code feedback loop for validation, and a novel symbolic algorithm for clustering and eventual unification. We demonstrate this methodology on the hallmark case of $\large \pi$, an ideal testing ground for symbolic unification. Applying this approach to 455,050 arXiv papers, we validate 385 distinct formulas for $\large \pi$ and prove relations between 360 (94\%) of them, of which 166 (43\%) can be derived from a single mathematical object—linking canonical formulas by Euler, Gauss, Brouncker, and newer ones from algorithmic discoveries by the Ramanujan Machine.
  Our method generalizes to other constants, including $e$, $\zeta(3)$, and Catalan’s constant, demonstrating the potential of AI-assisted mathematics to uncover hidden structures and unify knowledge across domains.


Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper presents a new AI methodology to unify different formulas which correspond to the same mathematical object. In particular, the paper is focused on one use case where the object is the constant $\\pi$, but also authors extended to $e$ or $\\zeta(3)$. \n\nThe methodology can be described by the following pipeline:\n- Extraction: regex and an LLM is used to extract mathematical expressions from ArXiV papers and make it more standardized\n- Validation: use PSLQ to check if the new formula actually calculates $\\pi$ or if the LLM makes some mistakes\n- Unification: all formulas are converted into the canonical form using RISC and using UMAPS algorithm and finding a matrix $U$ which satisfies the equation (8).\nThe unification step is the final goal of this work, where it's possible to prove equivalence of different formulas.\n\nSTRENGTHS AND WEAKNESSES:\nFor the strengths component\n\nS1. The paper is very well written, easy to follow and, although I don't have expertise in this area, seems pretty novel as well.\n\nS2. A good benchmarking exercise has been done, although it's very focused on $\\pi$ only.\n\nI see two main weaknesses\n\nW1. I don't see how this paper is interesting for an AI conference. There is no advancement in AI, nor AI has been used in a clever or interesting way. The AI part is just for filtering and generating equations which needs to be the same as the TeX formulas. Then, there is no data/training/evaluation using AI. \n\nW2. The contributions part from this work is quite limited. Showing that formula $x_1$ is the same as formula $x_2$ cannot be seen as a novel contribution (as the authors acknowledged, there are already specialized systems) and it's not bringing any meaningful and additional knowledge to the scientific community.\n\nQUESTIONS:\nQ1. I would be more interested if we can develop a system which, based on this work, is able to calculate some constants with faster convergence rate, which is an active research area. \n\nQ2. Can you elaborate more on why you believe unification is very important for unifying knowledge across different subjects? In particular, most of these objects are generated from pure math and then used in other fields, but the source is the pure math.\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 3, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 2, "ORIGINALITY": 3, "DATE": "2025-06-28"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis work searches a large mathematical literature (drawn from arXiv) and extracts convergent sequence representations of standard mathematical constants such as $\\pi$ and $e$, then uses LLMs to bring them to a standard form.  Then, several mathematical techniques are used to find relations between superficially different sequence representations: computation of invariants (\"dynamical metrics\") such as convergence rate and irrationality measure, clustering, then an algorithm for locating recurrence relations within a more general structure called a \"conservative matrix field\".  The search step uses LLMs and several evaluations and comparisons between LLMs are given.  Many relations between sequence representations are found.  For $\\pi$ the fraction of representations satisfying relations is quantified and many canonical forms listed.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths: the paper asks and studies an interesting \"meta-mathematical\" question about the many solutions to a well known problem (convergent sequence representations of standard constants) by automated literature search and canonicalization.  Most automated literature searches are fairly superficial and I know of few if any other works which automate such a mathematically interesting analysis.\n\nWeaknesses: I find the mathematical part of the paper poorly written, explaining the concepts and methods in a very disjointed way.  The relation to previous work other than O. David's or the authors is unclear, and the invention of new terminology for standard mathematical concepts, such as \"conservative matrix field\" for a flat connection, is not reassuring.\n\nQUESTIONS:\nIn section 3 describing the literature extraction, the fraction which lead to validated formulas, while large, seems small enough (maybe 15%) to wonder how representative the final results are of the full range of sequence representations in the literature.  For example, it is stated that there are 3389 computations of $\\pi$, which are categorized by type as series, continued fraction, or neither, resulting in 1678 formula candidates.  How many of these 1678 formula candidates were continued fractions?  What are the other formulas, are they just explicit series with no other apparent structure?  Then in 3.2 which verifies that the sequences represent $\\pi$, it is said that \"Out of the 682 candidates, 407 were validated as $\\pi$ formulas and passed on for canonicalization.\"   Where did these numbers come from?  In 3.3 it says \"Out of the 407 validated formulas, 402 are found to have 146 representations as order-2 recurrences, and 5 as order-3 recurrences.\"   Is this to say that almost all of the validated formulas were in fact continued fractions? \n\nThe biggest question the results raised in my mind was to what extent the authors believe the \"conservative matrix field\" approach unifies all sequence representations or not.  In the final table 10 many conjectures are stated about such relations.  Are there representations of $\\pi$ which the authors believe are *not* instances of the 3d CMF of D.2 ?  I suppose the CMF concept could be generalized to Z^D for higher D, is it possible that any recurrence relation could be accommodated this way, or are there constraints which prevent combining some pairs of recurrences into the same CMF ?  The simplest such constraints might be relations between the \"dynamical metrics,\" is this so and where is it mentioned in the text?\n\nLIMITATIONS:\nYes.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "CLARITY": 2, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-06-29"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper is a step towards automation of mathematical discovery and includes the use of machine learning techniques. The paper focuses on the extraction of specific recurrences that describe the mathematicl constant pi from the mathematical literature.\n\nThe method includes the use of scraping, various extraction and validation steps (confusingly some of which are called \"retrieval\") and an approach to classify and unify the formulas found. Starting from about 278m formulas scraped from arxiv, the formulas are filtered down to 402 that fit the criteria.\n\nI cannot judge whether the recurrences extracted here are interesting to mathematicians in any form. There seem to be some recent papers out there (from the list of references) where mathematicians have explore such formulas. So perhaps there is some interest, but the paper does not explain the significance of this field in mathematics.\n\nAs far as I can tell, the only machine learning techniques used are prompting (Appendix F.2 and G.5).\n\nThe paper discusses some validation of the methodology in section 4.2, but the \"ground truth\" is just an unspecified \"merging\" of the output of 3 language models. The paper does not answer the question whether any of the formulas found bear any resemblance to the original formulas found in papers.\n\nThe \"retrieval\" or \"clustering\" methods described in in the paper appear to be classical algorithms that are very specific to this kind of recurrence the authors explore. It is unclear to me how these methods generalize to other mathematical discovery methods or whether there is any interesting conclusions for AI research in these methods.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n- Automatic discovery of mathematical knowledge is a very interesting topic in AI.\n\nWeaknesses:\n- It is unclear whether the concrete output of this paper is interesting to mathematicians.\n- the paper does seem to contain new methods or insights for AI research\n- The paper does not discuss much in terms of validation - How accurate are any of the processing steps? Does the overall pipeline produce the expected formulas?\n- The code is not available so I could not dive deeper to clarify the many questions I encountered. The authors promise, however, to release some code in the future.\n\nQUESTIONS:\nPSLQ seems to play an important role. It is used to \"fix\" the formulas extracted by GPT-4o. Could you explain how the process works and how often it is required?\n\nThe example in section 5.1 states \"This equivalence demonstrates how two previously distinct mathematical expressions, discovered\nover a century apart, are now proven to be equivalent by an automated process.\" To me this reads like an overstatement, as the paper they cite did very explicitly point out the relation to Ramanujan's earlier work. Could you please point out what exactly was discovered here that hasn't been discovered before? It is also okay to state that your method automatically rediscovered some knowledge, which validates the approach.\n\nFrom how many different papers do your 402 final formulas originate?\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 2, "REVIEWER_CONFIDENCE": 4, "CLARITY": 2, "IMPACT": 1, "ORIGINALITY": 1, "DATE": "2025-07-01"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper introduces an end-to-end, LLM-based pipeline that tries to unify the many disparate formulas that mathematicians and physicists have discovered for fundamental constants such as $\\pi$, e, $\\xi(3)$, and Catalan’s constant. The work can be categorized as AI for Experimental Mathematics. \n\nWithin the study 455k arXiv papers are scraped using LLMs. The LLM filters strings that seem to compute the target constant and rewrites the LaTeX into SymPy code. A code/LLM-feedback loop plus a PSLQ check keeps only numerically correct formulas. \nThen, each formula is converted to the minimal polynomial recurrence (via the RISC guesser). Recurrences with identical minimal polynomials are collapsed; dynamical fingerprints, namely the irrationality measure and the exponential convergence rate, cluster the rest. The authors use a theoretical machinery that is called the Conservative Matrix Field (CMF), a 3D lattice of 2 × 2 rational matrices whose paths encode families of equivalent formulas. A new algorithm, UMAPS, searches for and proves coboundary equivalence between two recurrences by empirically fitting a polynomial Möbius map.\n\nAs a result,  for $\\pi$, 381 of 407 formulas are proved linked; 188 collapse into a single CMF. Famous identities of Euler, Gauss, Brouncker and several modern machine-generated series are shown to be equivalent. Limited experiments show the approach extends to other constants. An interesting obcservation from the paper is that the current LLMs very rarely find such equivalences.\n\nSTRENGTHS AND WEAKNESSES:\n# Strengths:\n**Significance:** The paper addresses an important gap in AI-guided experimental mathematics: connecting many scattered symbolic formulas. While automated theorem proving and conjecture generation have seen progress, the systematic, large-scale unification of symbolic formulas is an unexplored frontier. The work's idea is indeed of significance and its success points toward a new paradigm in how mathematical knowledge can be structured. I believe this could benefit computer-assisted mathematics, OEIS-style databases, and AI-driven discovery.\n\n**Originality:** I think the proposed framework is original. It combines state-of-the-art LLMs for knowledge extraction with deep, novel symbolic algorithms. The CMF idea along with the empirical coboundary solver are new and non-trivial. I think this is the largest automated “formula harvesting’’ study to date.\n\n**Clarity:** Complex concepts are introduced with clarity and supported by well-chosen examples and figures. The appendices are indeed providing the necessary details for a deep understanding of the main text.\n\n**Quality:** The methodology and pipeline and the claims are well-supported by the experimental results. The mathematical foundations are solid, and the appendices provide detailed proofs for the key lemmas that elaborate on the UMAPS algorithm. I have reviewed the proofs of Lemma 1, 2, and 3 and find them to be correct.\n\n# Weaknesses:\n1. Lemma 1 proves only a necessary condition for coboundary equivalence. No theorem/lemma is given that the empirical fitting procedure (steps 1-6 in Appendix B.3) will always find a coboundary when one exists, nor that a solution returned is unique. In other words, the paper does not provide a corresponding sufficiency proof or a guarantee of completeness.\n2. As far as I understood, the pipeline incorporates several ad-hoc choices, e.g. $\\delta$-clustering granularity 0.05 and similarity window 0.03, Maximum fitting depth N=40 and unspecified polynomial-degree caps in UMAPS, and the use of N=200 partial sums to guess recurrences with the RISC tool. Then, no sensitivity study is provided. So it is unclear how robust results are to these (hyper)parameters. For example, the $\\delta$ threshold controls which pairs ever reach the (costly) UMAPS stage, too tight a window could miss valid links, while too loose one inflates runtime. \n3. Converting every formula to a minimal polynomial recurrence (canonical form) is an important part of the paper. Subsequent equivalence proofs assume minimality, yet minimality is inferred empirically: RISC is run on the first 200 terms, then the candidate recurrence is “shown to obey” the series. Do we know formally that the guessed recurrence is indeed minimal (or even correct) for all n? Consequently, I think for constants whose true minimal recurrence has rational rather than polynomial coefficients, the inflation procedure may raise the order or degree (a common issue in computer algebra) and break minimality.\n4. Again, as far as I can tell, the authors extend the CMF from polynomial to rational entries ﻿but does not prove it's existence, uniqueness of rank for a given constant, or whether every direction indeed corresponds to a formula. The analogy with a conservative vector field is appealing yet informal. I think an algebraic geometric treatment is missing.\n5. I totally understand how challenging it is to write an AI for Mathematics paper, given the extensive cross-referencing between the main text and the appendices, but I think the exposition could still be made clearer. In particular, consider a short introduction of technical terms (that are heavily relied on Appendix at the moment) and making the introduction less repetitive, and avoiding tautology.\n6. The manuscript also lacks discussion on some important related studies that bear directly on its core tasks of harvesting, discovering and clustering identities and conjectures, such as:\n- Paule & Schneider (JSC 2024), Creative Telescoping for Hypergeometric Double Sums\n- Hashemi et al. (ICLR 2025) Can Transformers do Enumerative Geometry?\n- Jiang et al. (2025) Enhancing Deep Symbolic Regression via Reasoning Equivalent Expressions\n- Tian et al., (Nat. Commun. 2025) Interactive Symbolic Regression with a Co-design Mechanism\n- Romera-Paredes et al., (Nature 2024) Mathematical discoveries from program search with large language models\n- Studies on Symbolic computation on Knowledge Graph and relational databases\n\nI think commenting on these studies would situate the contribution within the established AI-guided symbolic-computation literature, and strengthen claims of soundness and completeness. I recommend discussing their relation to the current study.\n\nQUESTIONS:\nPlease refer to the six weaknesses listed above. In particular, addressing the the following convincingly would raise my score.\n1. Can you prove (or empirically demonstrate on an exhaustive stress-test) that the UMAPS coboundary solver is sufficient, i.e. it finds a transformation whenever two recurrences are truly coboundary-equivalent, and that the solution is unique?\n2. Please provide a sensitivity study for the key empirical thresholds: $\\delta$-bin size & similarity window, fitting depth N, polynomial-degree cap, and the 200-term limit in recurrence guessing. Ideally plot linkage percentage and runtime versus each parameter.\n3. Can you prove that the guessed recurrence for each formula is minimal and globally valid, or else report the empirical error rate when minimality fails? If rational rather than polynomial coefficients arise, how do you detect and handle order inflation?\n4. Please formalise CMF for rational entries. It would be great to provide a proof of existence for any d-finite constant, clarify whether rank is unique, and give conditions under which every lattice direction corresponds to a formula.\n\nLIMITATIONS:\nI believe the authors have not sufficiently addressed the study’s limitations. I recommend adding a concise section in the paper that explicitly summarizes the key methodological gaps, such as hyper-parameter sensitivity, incomplete completeness guarantees, reliance on empirically inferred minimality, and questions of scalability and domain transfer to other areas of symbolic computation and formula harvesting.", "TITLE": "Official Review", "RECOMMENDATION": 3, "REVIEWER_CONFIDENCE": 4, "CLARITY": 2, "IMPACT": 2, "ORIGINALITY": 3, "DATE": "2025-07-01"}