From Euler to AI: Unifying Formulas for Mathematical Constants
Authors: Tomer Raz
Michael Shalyt
Elyasheev Leibtag
Rotem Kalisch
Shachar Weinbaum
Yaron Hadad
Ido Kaminer
Venue: NeurIPS 2025 poster ()
Keywords: AI for Science
AI for Math
LLM-Tool Integration
Mathematical Constants
Continued Fractions
Recurrences
Number Theory
Pi
Primary area: machine_learning_for_sciences
TL;DR: A general framework for unifying mathematical knowledge, clustering it to prove previously unknown equivalences across scientific literature—demonstrated by connecting historic and modern formulas for 𝜋.

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.

From Euler to AI: Unifying Formulas for Mathematical Constants
Tomer Raz Michael Shalyt
Elyasheev Leibtag Rotem Kalisch Yaron Hadad Ido Kaminer
Shachar Weinbaum
Technion Israel Institute of Technology.
Haifa 3200003, Israel
Abstract
The constant ı has fascinated scholars throughout the centuries, inspiring numerous formulas for its evaluation, such as infinite sums and continued fractions. Despite their individual significance, many 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 novel symbolic algorithm for clustering and eventual unification. We demonstrate this methodology on the hallmark case ideal testing ground for symbolic unification. Applying this approach 455,050 Xiv papers, we validate 385 distinct formulas for π and prove relations between 360 (94%) of them, of which 166 (43%) can be derived from single mathematical abject-linking canonical formulas by Euler, Gauss, Brouncker, and newer ones from algorithmic discoveries by the Ramanujan Machine. Our method generalizes to other constants, including (3)), and Catalan's 's constant, demonstrating the potential of Al-assisted mathematies uncover hidden structures and unify knowledge across domains.
Introduction
The earliest rigorous approximation for т dates back to Archimedes around 250 BCE, who established
the bounds 223 < 22 [16]. Modern approximations employ more sophisticated formulas. For example, the Chudnovsky algorithm [15), derived from a formula by Ramanujan [48), remains
71
(6) is notable for enabling computation of specific и digits without requiring prior digits. Such breakthroughs inspired fundamental advances in computer science, such high-precision arithmetic 77]. evolutionary optimization (35), and elliptic curve cryptography [39], Recent efforts led to the development of computer algorithms capable of generating numerous formula hypotheses and sometimes proofs for mathematical constants
instrumental for precision records. Similarly, the BBP formula
g 17 46
"Corresponding author: kaminer@ttcchioon.acll Project repository: http://gghhbboooo//mmmmaaaaaaaaa
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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The plethora of results related to discovered over the centuries leads to persistent question: How are they all connected? This question is im- portant not only for preventing acci- dental rediscoveries (e.g.. Lange's for- mula from 1999 [37] had already been derived by Lord Brouncker in 1656 (42). Many equivalent formulas ap- pear vastly different at first glance. A simple example is Euler's contin- ued fraction that provides equivalent representations of infinite sums [23] This complex situation underscores the need for systematic approach unify these relationships.
4
341446000
16
8kc
1
84
4
8/c
1
5
84 6
Halley-Borwein- Ploufle
Mathetaaties of Computation (1997)
(2n) )
(126n 1 202m
2
(2n
1)
John Wallis
50
x
1636, England.
T
5+
3)
Priodrich Gaiss
4000
Traos 100
क
12
2h
1813, Germany
(-20ay
2n
Madlaava Sangamagrama 14th century, India.
12m)
22
8
15
222
(4R)A(1103 263902)
21
9
9801
(K9)
11+
Raayoni al.
Srinivasa Ramanujan
6 1nny 1914, India.
So far, efforts in AT for mathematics have focused on automated theorem
Narure (2021)
2n
Figure 1: Selected æ formulas across centuries.
proving [44 56 automated conjec-
ture generation [3 24 25 40 46), regression 31 32 [55] and, recently, LLM-tool integrations for mathematical discovery (27 22 50 59] However, to date, no work has addressed the problem of symbolic unification of mathematical knowledge.
In this work, we propose system for the large-scale harvesting, identification, and unification of mathematical formulas (Fig. 2). This effort leverages recent advances content understanding based on large language models (LLMs), the newly discovered concept of Conservative Matrix Fields (CMFs) [1 58], and novel mathematical algorithm that we call UMAPS for unification via mapping across symbolic structures, using the math of coboundary equivalence for finding and proving relations between formulas. To demonstrate this methodology. we selected the symbolic case study of formulas calculating æ. A total of 385 unique formulas were extracted from the literature and validated, which 43% were found to correspond to different trajectories within single CMF (Section 5}. We expect furrrfutue improvements of our algorithm to classify all the formulas into one or just few unique CMFs that unify all knowledge about π calculation.
Harvesting
Retrieval
Clustering
Canonical form
Mathernatcaa knowledge
Extraction
Dynamical metrics
Unification
Validation
Coboundory algorithi
12
1
2
irviv
2
74
2+
1813 Gauss
1855 Brouncker
1-1
2.
(46)) 26390k 1103
Ramanujan 9-
90"
396
16
Romanujan
Figure 2: Automated methodology for unifying mathematical knowledge. A large corpus of mathematical formulas is harvested, retrieving formulas that are each translated to executable code for validation. The formulas then clustered by conversion into their canonical forms, and unified using a novel symbolic computational algorithm that proves their relations.
To the best of our knowledge, this work the first LLM LLLM-smmbolcc tool integration for discovery in number theory, and possibly the first LLM integration with a proprietary research- grade computer algebra system. The success of this study highlights the prospects automated unification of vast mathematical knowledge. Beyond the example of we applied our algorithm to other mathematical constants like e, ((3), and Catalan's 's constant, and to variety of formula structures, showcasing its broad potential. Appendix A provides glossary of key terms used throughout the paper.

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2 Mathematical background
2.1 Recurrences as universal representations of formulas for mathematical constants
A wide range of formulas ionmuuis-includingg infinite sums, products, and continued fractions can be converted into recurrences, providing cohesive framework for unification. function un satisfies recurrence of order m if u,, 01, + u2
dm,n un ms which can be represented
via the associated companion matrix:
aon.c
...
dm-11,
CM(n)
4m-2,p
(1)
a1,m
By incrementally multiplying the companion matrix steps, get the matrix:
Pla-m
Pla-1
Pi.
P2
P2a-1
P2.
I1_, CM())
Pla--
Pha-1
P3.m
(2)
Pm,n-1
Pen.
Pl.n.... Pm.n solutions to the recurrence for the initial conditions Pt,j 8/. Other solutions for different initial conditions can be written as linear combinations of these.
Recurrences evaluate desired constant L either directly lim,-+<o Un L (c.g.. for infinite sums), or as ratios lim,-pcc P-, I. with Pn. and (n being two solutions for the recurrence (e.g., for continued fractions). the special case a second-order recurrence, u, a, + bn Un and any pair of solutions is associated with formula in the form of a continued fraction:
b2
01
When the functions On = a(n) and b, b(n) polynomials, the formula above known Polynomial Continued Fraction (PCF), denoted by PCF (a(n), b(n)). See details in Appendix
2.2 The dynamical metrics describing each formula
A formula of mathematical constant L provides converging sequence of rational numbers q= (known Diophantine approximation). The formula can be characterized by dynamical metrics capturing properties such as its convergence rate. A recent paper (52) proposed using such metrics for formula discovery and clustering. Here we use the metrics of convergence rate and irrationality measure. The convergence rate is defined as:
Pn
lim log L
gn.
When examining the connection of two candidate formulas, the ratio of their r values can hint whether one is a transformation of subsequence of the other (see Appendix E.5 for example). The irrationality measure of En is defined as the limit = lim 6n where
log
L log
Pn
8,
We found that two formulas sharing the same 5 is the strongest indication of a possible relation, since is invariant under many transformations and choice of subsequences. Below, our UMAPS algorithm used to derive and prove relation once pair of formulas share the same and ö.
2.3 Conservative Matrix Fields (CMFs)
The CMF is the mathematical structure that generalizes formulas of a particular constant, originally found by generalizing PCFs (21), and later realized to be more general (Appendix G). To exemplify the

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concept, we focus on the CMF of π. This CMF is 3D, i.c., consisting of three matrices Mx. My: Mz
with rational function entries in the variables (z. y, 2), satisfying:
Mx(r, yo 2) 1, 2) My =) =) ( 1,2)
(x (. 1, =)= - (x
1)
My y, W. (2.
M2, My (0, (,,
This property describes the path-independence of the transition between two points in a 3D lattice (lattice illustrated in Fig. 55)) in analogy to a conservative vector field. The CMF satisfies the properties of a discrete flat connection [.. For an explanation of how formulas reside as directions within the CMF, see Appendix G.T A notable feature of the CMF is that pairs of formulas found to be parallel trajectories with different initial points correspond to two matrices that are coboundary equivalent.
Many of the known ा formulas will be shown to reside within a single CMF (details in Appendix G.2)
(1
(
=(-x-y+z) M₂ = ( (x-3)
aay
Mx
204h-2242
My
25+2
(6)
y
(y-
(v
(x=
3 Methodology for symbolic unification of formulas
3.1 Harvesting: large-scale retrieval of formulas from the literature
The first challenge lies in the natu- ral language processing of formulas. We analyze the LTTXX source code of 455,050 arviv articles by combining regular expressions and LLMs,, ex- tracting all mathematical expressions, resulting in 276.242,506 strings. Fil- tering for expressions containing the symbol retrieves 121,662 r-related equations. The widespread use of the symbol in scientific literature means that most occurrences are unrelated to caleulating the constant itself. To ad- dress this, and keeping in mind that there a priori very little data on what successful formulas' LeTEX looks like, each potential formula is clas- sified as computing
Harvesting scheme
Example
Galtery Jancs "Elioeed sas nodeed to neae de serter" anty
455.060
articles
medern ailtatto Dosee Ow sis for flom
(a)
scraping
278.422.00
equatoro
(6)
retrieval
Seaith eptoanno capaa vinel piraatt wistie wwi.majjuu
121.002 S eqqueiiona
computes m??
(c)
series
continued fraction?
1858
equasionn
(d)
(CSPY) seceggaattraaaaaa 41 Bisligh Sat220, #isteg/adeeat0/
extract
8
not, using
GPT-4o mini [4T (chosen for cost- effectiveness), reducing the number of candidates to 3367. Next, GPT-4o cat- egorizes formulas by type: series, con- tinued fraction, or neither-resulting in 1656 formula candidates.
(e)
validate
via PSLQ
3015
< formulas
TO WEN.HIIOTTE CANIET CORRORIII CHIMENN NETERROO CELEECOREEE
to recurrence
formuras
Figure 3: Pipeline for automated harvesting of mathemat- ical formulas (left), exemplified using one of the analyzed formulas (right). (a) Equations are scraped from papers on arXiv. (b) Regular expressions trieve series and continued fraction patterns that contain æ as the only irrational number (see Appendix l.3). (c) Zero- shot classification using OpenAl's GPT-4o mini identifies formulas calculating the constant æ. Then, OpenAl's GPT-4o identifies the formula type (series, continued fraction, or nei- ther). (d) Extraction of the series' summand or the continued fraction's 's partial numerator and partial denominator, using GPT-4o. The formula then converted to code. (e) Formulas are computed and validated using the integer relation finder algorithm PSI.Q. (f) The formulas are converted to canonical recurrences using RISC's s tool for fitting recurrences 33
3.2 Harvesting:
extraction and validation
the 1ЛЛ strings re-
The extraction and validation stages rely on an LLM-code feedback loop that feeds a PSLQ algorithm. Each equation, represented as ЕТEX string, must then be parsed into a Computer Algebra System (CAS) for further manipulations (in our case, SymPy 388). Automatically extract- ing algebraic forms from 1РTE strings especially complicated due to var- ied LTTX patterns, which are difficult

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to systematically convert to executable code using a predefined logic. LLMs help us overcome these obstacles by processing text contextually and attending relevant parts of the text, solving the
natural language processing task that may have required elaborate rules [14] 47 Specifically, we
use OpenAl's GPT-4o translate relevant IaTb into executable mathematical code [2
(see
Appendix
D for the exact prompts used). To correct for (common) mistakes in the LLM LLM--generaed
formula code, we apply an LLM-code feedback loop for code validation: errors are sent back to the LLM along with the faulty code to correct it, up to three times (see Appendix (..3)).
We validate that each extracted formula computes the constant и by running the formula code to get numerical approximation and then applying PSLQ, an integer relation algorithm [26]. Limit values are not extracted directly from the 18TEX string for validation, since we found that GPT-4o got them wrong in some cases (see Table [4). Instead, the PSLQ approach fixes these critical GPT mistakes
and reproduces the intended formulas. Out of the 660 candidates, 385 were validated as a formulas and passed on for canonicalization (details in Appendix 1.5)
3.3 Clustering: using the canonical form
The first unification step is converting each formula to its canonical form: the simplest linear recurrence with polynomial coefficients (Appendix E..4T). Automated algebraic capabilities are unpredictable in solving such tasks. Thus, we use a computational method for converting the formulas to polynomial recurrences: a Mathematica package by RISC 33 that fits polynomial-coeffine
linear recurrences to each sequence of rational numbers. The resulting recurrences are validated numerically and passed to Maple package to guarantee order minimality [57] 60), thus finding the provably minimal polynomial recurrence. Out of the 385 validated formulas (Section 3.1). 380 are found to have representations as order-2 recurrences, and as order-3 recurrences, which can also be addressed as we show in Appendix B.6 and Appendix
The same canonical form captures wide range of formulas, continued fractions and infinite sums. Thus, the conversion to canonical forms automatically unifies different formulas, yielding 149 different order-2 canonical forms and 4 order-3 canonical forms for n,, 153 in total, from 385 formulas (selected examples in Table T
Table 1: Canonical form representation. Converting formulas their canonical forms shows equivalence of different-looking expressions (e.g. 1,2), leaving the less-trivial connections for the later steps of the algorithm. Additional details in Appendix C.4
Formula
Value arXiv source
Canonical form (CF)
CF value
Initial conditions
m
1806.03346
PCF(3n + 11 - -2n)
0
2
>>
2010.05610
PCH(3n + 1, - -2n))
100
(-1)-
2404.15210
PCF(2, (2n - -1)
1
0
2n+1
1
>n 1[
(-1)*+)
2206.07174
PCF(6, (2n + 172)
1
5) E
3m+4
2204.08275
*
-27--196 196
0
70
(20-1)
(
3-
-1
15
PCF((240n 164n 54n 29, -2166n
12288 + 11264x 15520n 3 764n + 3802n 714)
3.4 Clustering: using the dynamical metries
The clustering stage a heuristic to guide which formulas should be attempted be proven equal using UMAPS. Formulas with the same metrics are likely to be related to the same constant [52] The metrics also indicate a more intricate connection, enabling the unification of formulas systematic way that proves an analytical transformation between them. Canonical-form formulas are first compared each other using the irrationality measure 5 (Fig. 4p). which the most reliable indicator for potential equivalence. Every new formula is first evaluated relative to directions in the CMF corresponding to recurrences with the same 8. This search can be improved by using gradient descent the direction parameters, because is found to be continuous 21
5

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We found that ó is not sufficient to imply equivalence, and thus we complement it using the ratio of convergence rates A r- Canonical form A is folded (Appendix (E.5) by I'B and canonical form B is folded by (Fig. D)), making them converge at the same rate. The next step is finding their precise algebraic relation using UMAPS.
3.5
Unification: using the UMAPS algorithm for coboundary equivalence
Our algorithm for unification via mapping across symbolic structures (UMAPS) relies on the established concept of coboundary equíva- lence (Appendix (E.4), however, no specialized coboundary solver existed prior to this work.
PCFA
PCF
B
AA
A 0
(a)
TB
8AA 8B
No X
A(n), B(n) E PGLm (Q(n)) are coboundary equivalent if there exist a matrix (n) such that
Yes
(6)
PCF *
PCF B
A(n). UU + 1) B(r)
(7)
This definition carries to recurrences when their companion matrices (Eq. ( are
(c)
LA (LB)
coboundary equivalent (Fig. (h.d) and then:
PCF (m) = A(n)
PCF (n)) = B(n)
(II-, A(i)) Unn+ U(1) (111))(IPP,BC-0 B()).
(d)
U(n + A(n)
(nn) B(n)
Since any matrix with rational function coef- ficients can scaled to have polynomial co- efficients, we can write that A(n). B(n) E GL (Q [ml) are coboundary equivalent if there exist matrix U(n E GLm (Q[nl) and polyno- mials PA (n), (n E Q[n such that
(e)
(L..... = (x))
Wish ne
Yes
No
Pa(n) A(a) Un pn(n) U(n) B(n) (8)
Figure 4: The matching algorithm: connecting polynomial linear recurrences. This algorithm demonstrated here for polynomial continued frac- tions (PCFs) but can be generalized any linear polynomial recurrence. (a) Compute the dynam- ical metrics [52] for the two PCFs (irrationality measures d dB and the convergence rates ratio rA/TB). The 5 metrics are used to identify pos- sible connections, can be related via coboundary (in practice, we test for them to be within 0.06 of each other). (b) Fold PCF A by I'B and PCF B by vA (Appendix F.5). UMAPS (c)-(e): (c) Solve for a general Möbius transform (a x2 matrix U (1)) that once applied to the limit of PCF p equates it to the limit of PCF A. (d) Representing the PCFs in matrix form (A(n) and B(n)), propagate the coboundary matrix via the relation U(n + 1) = A(n) Unn) (n) up to UNN (N = 40 sufficient for our runs, Appendix D) (e) Assume the general form of (n) to have autntial-functon entries with polynomial degree up to N and solve for their coefficients using normalized (l....., .N. If such solution is found and validated, the PCFs are coboundary- related. See Appendix C for more details.
Finding coboundary between two polynomial matrices inherently a non-linear problem due to the product of unknown polynomials PA and рв with unknown coboundary matrix U. Morcover, the degree of cach polynomial is not known. Despite the non-linearity, we found a coboundary solver algorithm for general order m (Appendix CC3).
only if 5 8g, the PCFs
UMAPS finds the solution without solving non- linear equations, instead leveraging the recur- rence limits to compute a sequence of empirical coboundary matrices, whose elements are fitted to rational functions (53). The algorithm relies on the following lemma:
Lemma 1. (A necessary condition on the coboundary equivalence matrix.) Let lim PCF (a(n), b(n)) and LB =
LA lim PCF PCF (c(n), be converging PCFs ==+20 with associated companion matrices A(n), B(n) E PGL, (Q(n)). If A(n) is coboundary
B(n), then LA and La are re- lated through a rational Möbius transformation. Moreover, U(n) is the coboundary matrix, then LA = U()((LB) (U(1) applied to LB as a Möbius transformation).
A proof and generalization to higher-order recurrences (Lemma 4 as well as proof of the uniqueness of the coboundary matrix (Lemma53). are detailed in Appendix [ These combine to show that UMAPS
is sufficient to solve for the coboundary matrix, stated in Corollary [ (proof in Appendix C.3

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(a)
((n)
(c)
(m
B(n)
2n +
((n)
Un Un++1) 1)
(n)
Un + 1) in 17
n²
3m
Bn 1.5
A(z)
U(m)
(m)
A(n)
(
d
5)
Lu s)
7
G 9
6+
7
15
24
2)
(i
1 8)
( 7)
7 16
Figure 5: Coboundary equivalence: the mathematical framework connecting different formulas + once cast into their canonical forms. (a) The coboundary condition A(n) (n = U()) Bnn) recasts formulas as (b,c) parallel trajectories in a CMF. (d) Example of two coboundary- equivalent formulas, presenting their coboundary matrices and limits, which constitute proof of novel equivalence.
Corollary 1. (Sufficiency of UMAPS.) If coboundary matrix exists for two matrices and every rational-function entry of the coboundary matrix has polynomials of degree most d, then running UMAPS with N 2 2d + suffices to recover the coboundary matrix.
Fig. summarizes the flow of matching two canonical form formulas. Using this method, we find that formulas 1,2 and 5 from Table I are equivalent and that formulas 3,4 are also equivalent. Refer to Appendix [B. for descriptions of how the algorithm is applied to these formulas, and refer to Appendix C for listing of the algorithms involved. A study of the algorithms' sensitivity to hyperparameters is provided in Appendix [ The same procedure is applied to each canonical form formula, measuring its å value and relying on its continuity to locate worthy directions that yield potential formula pairs for the coboundary algorithm. The matching algorithm then applied between formulas and representative recurrences from the CME. Finding a match between a formula and a CMF representative proves the formula is generated by the CMF. The full list of results is provided in Appendix Selected results for π are detailed in Section
function of direction in the CMF
Benchmarking
4.1 Comparison to other methods for symbolic unification
Our work is the first address the problem of symbolic unification scale, thus there standard benchmarks for performance comparison. Leading LLMs are generally unable to address the full challenge. As an example, compare our equivalence detection and proving capabilities those of LLMs: We tasked 2 leading LLMS-GPP--R and Gemini 2.5 Pro Preview- proving 10 formula-pair equivalences proven our algorithm (Table The formulas chosen such that each pair has equal dynamical metrics (r,, 8) after folding, which is the simpler situation to prove (parallel trajectories the CMF). Even with these simpler tasks, the LLMs exhibit only limited success. We did not find cases in which any LLM succeeded in finding relations between pair of formulas without equal dynamical metrics.
identifying and
4.2 Comparison of LLM model performance
We utilize LLMs for classification and extraction in two different ways. Table B compares the performance of three choices for the extractor LLM, which we found to be the more sensitive choice, as it is used for the more advanced LLM-code feedback loop. A ground truth is established by merging the validated formulas (Section 3.2 found by the three compared LLMs.
7

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Table 2:
LLM performance when detecting and proving equivalence in a random set of 10 formula pairs of equal dynamical parameters (Appendix H). All LLM proofs were validated manually.
LLM
Successful detections
Correct proofs
GPT-4o
1/10
2/10
Gemini 2.5 Pro Preview
8/10
5/10
Performance of different extractor LLM choices in terms of successfully harvested formulas. The LLM errors are split to "faulty code" for code that did not run, and "symbolic mistake" for incorrect identification of some of the formula constituents like continued fraction polynomials. The bold row marks the choice of LLMs used for all the rest of the results in this paper.
Table 3:
LLM classifier
LLM extractor
Successful extractions
Faulty code
Symbolic mistake
GPT-4o mini
GPT-4o
289 (7.6%)
2 (0.7%)
5 (1.7%)
GPT-4o mini
Claude 3.7 Sonnet
266 (89.9%)
(7.1%)
(3.0%)
GPT-4o mini
GPT-4o mini
206 (69.6%)
70 (23.6%)
20 (6.8%)
5
Results
5.1
Example equivalences among famous formulas
Our automated system proves previously unknown equivalences between formulas. Among the formulas connected are famous examples. such as one of Ramanujan's 1914 formulas. as well as Lord Brouncker's, ss Euler's, s, and Gauss's PCFs from the 17th 18th and 19th centuries [23 28 42 For example, the following series found by Ramanujan in 1914 (8).
Ei= (- 1)) (e-1)))())). (4) (1123 + 214606)
2k+1
(9)
(D)
882
was proven equivalent (Appendix B.4 to a newer series from a paper published in 2020
[54]
341446000
k= (8+1)(26-1)()) 11 (1424799848A* 1533506502/ + 1086885699)
(2))
(10)
This equivalence demonstrates how two previously distinct mathematical expressions, discovered over a century apart, are now proven be equivalent by an automated process.
Fig. 5 proves the equivalence of another pair of famous formulas: (1) PCF n(n + 2)), one of the first computer-discovered formulas from 2021 (46). (2) PCF(2n + 1,, published by Gauss in 1813 (28] and provided at the time an especially efficient way to calculate digits of π.
5.2 Formulas unified by a Conservative Matrix Field (CMF)
The CMF of π, Eq. (6), captures most of the harvested formulas (Table 4 presented graphically in Fig. 6 allon with their corresponding trajectories.
with selected examples
Table 4: Summary of unification results, among all validated formulas (left columns) and among the canonical forms (right columns). Formulas are harvested from 140 arXiv papers (Table of which 137/140 (98%) have least one formula proved connected by UMAPS and 70/140 (50%) have a formula residing in the same CMF.
Found relation 360/385 (94%)
Same CMF 166/385 (43%)
Found relation (canonical) 136/153 (89%)
Same CMF (canonical) 81/153 (53%)
The full list of canonical forms captured by the CMF appears in Table 16 Improvements in UMAPS are likely to connect additional formulas (Table 17) to the same CMF.
5.3 Unification of formulas beyond π
Going beyond we automatically identified equivalent formulas for ((3), and Catalan's con- stant demonstratii the generality of the approach. Consider these two formulas for Apéry's constant, the Riemann zeta function value ((3):
8

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1700s
Gouss 1813
Buler 1700s
zCr (Ln(n=1))
PCP (2n a
(2n--11)
R-2
n
1,,00
4- PCP (2.1)
-0.21
(1.12))
O.L.D.
( nit
(2-1k2n+1]
Brounsker 1885
+1- (2
(0.12)
2X--4
22 11) Frees
(0.0.2)
(ALL.))
(3.L)
(o.0.0)
0045
Cozcundary cluster
Example from cluster
Ramanujan Machine
(3n+1=2(tn-1)
-0.65
StT
112
formula
2
+2
(2n ()(d))
Figure 6: Formula unification by a Conservative Matrix Field (CMF), Numerous x formulas harvested from the literature are automatically arranged trajectories in 3D CMF. These formulas include famous ones by Gauss, Euler, and Lord Brouncker. The full list of unified formulas and their canonical forms is available in Table [6] Each cluster (large dashed circles) denotes formulas connected by coboundary, representing parallel trajectories or overlapping trajectories. The number each cluster center is the of all formulas in that cluster. Arrows show trajectory directions. Note that multiple formula clusters can have the same 5 value without being coboundary, showing that sharing is necessary but not sufficient for formulas being coboundary-related.
((3)
((3)
(11)
1)
The second formula 36 has faster convergence compared to the classical definition of ((3), though both converge polynomially. Our automatic procedure proves their equivalence by the coboundary transform and unifies them in the ((3) CMF (detailed in Appendix B.2..
As another example, the following two PCF' for Catalan's constant are also proven equivalent by UMAPS (Appendix B.5
3+
(12)
2-26
20
1+
42
42
2-3
natural next step is to conduct exhaustive searches for other well-known constants, and fundamental mathematical structures in fields such physics and computer science. See Appendix B.3 for examples.
5.4 Formulas generated via a Conservative Matrix Field (CMF)
sample 1693 distinct æ formulas was generated from the T-CMF (per Appendix C.5 The CMF permits a new method of comparing between formulas, using the normalized convergence rafe, defined r/ r/e (1), where is the convergence rate from Eq. 1 a trajectory and is the E² -o-ro.. 57 of the formulas generated in runs shared the best normalized r of 1.76, such the formula

---

81
12
238
67473
(-G2n
96r
12t
52g
15)
48p0
108m2
70n
arising from trajectory (-1,-1,00 0). By comparison, the best pre-existing formula unified by our CMF has normalized convergence of 0.88 (the 1,, direction, see Table 16.. Details in Appendix G.6
Discussion
6.1 Limitations
Currently, the harvesting step relies on the LLM's ability to interpret and contextualize mathematical ыТЕΣ strings. This step likely introduces data loss and false negatives in formula classification. Improvements in prompt engineering and in validation techniques will enhance the robustness of this LLM use. As more advanced LLMs become available, this step will become increasingly reliable.
Formulas often include additional symbols other than summation indices, like variables defined in the text surrounding the formula, which should be extracted and substituted into formula evaluation. We made several such substitutions manually to test the rest of the pipeline for these special cases. Future improvements of the unification pipeline can address this limitation by more advanced use of LLMs and automated validation.
Most formulas analyzed in this work series continued fractions. However, UMAPS and all the other steps in our harvesting and clustering processes are applicable more broadly (to any formula that generates sequence of rational approximations for given constant, e.g., deeper recurrences). Expanding the system to accommodate additional cases is promising direction for future work.
The same unification pipeline shown here could apply to the vast family of constants derived from D-finite functions by finding their corresponding CMFs (58).
6.2
Outlook
Increasing the dimension and rank of the r-CM, along with further improvements to UMAPS [ is likely yield higher percentage of unified formulas in the near future. planned future study will employ the CMF to systematically search for fast-converging and inationality-provinng formulas.
Looking forward, the same approach of collection, analysis, and organization of mathematical knowledge could help establish rigorous connections between different branches of mathematics. The methodology presented this work could help develop more general frameworks for identifying connections between different scientific theories through their mathematical representations. As the volume of information grows an accelerating pace, finding automated ways unify knowledge will become increasingly essential, especially with the goal of providing more intuitive understanding on complex concepts.
Pairing LLMs with pre-existing and novel tools for symbolic and numerical mathematics enabled the automated discoveries in this paper. We believe this I.LM-tool integration scheme will continue advance Al for mathematics and science the coming years.
10

---

Acknowledgments
This research received support through Schmidt Sciences, LLC.
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Appendix
A Glossary
Canonical form The simplest polynomial linear recurrence that generates given formula. It found by running RISC's Mathematica package [33]. then validating numerically, and finally minimizing the recurrence with Maple package [57
Coboundary equivalence An equivalence between two matrices via a third matrix called the coboundary matrix. For two matrices A(n), (nn) E PGLm (Q(n)) the equivalence takes the form A(n) Un 1) (n) B(n). In this study we combine novel coboundary solver (UMAPS) with folding to create the matching algorithm (Appendix C.2). which finds equivalences between formulas.
Coboundary transformation A transformation of the form A(n) - (-) A(n) Unn + 1), for A(m), (n) E PGLm (Q(n)). Companion matrix matrix with a structure suitable for directly representing a linear recurrence, see Eq.
Conservative Matrix Field (CMF) A recently discovered mathematical structure consisting of lattice of matrices that enables unification of many formulas discovered over the years. Originally discovered in (20] as result of a large-scale experimental mathematics effort in formula generation, the structure was found to be generalizable to any D-finite function in (58).
CMF dimension The number of parameterized matrices composing the CMF (3 for the m-CMF).
CMF rank The dimension of the CMF matrices (number of rows, equal the number of columns; 2 for the x -CFF).
Convergence rate (r) (1)-The The exponential convergence rate of a formula, measured empirically in this study, see Eq. 4 Used the formula matching algorithm to ascertain the folds needed find an equivalence between two formulas.
Dynamical metries Numerical values derived from formulas, used to cluster and find relations between the formulas (see convergence rate r and irrationality measure 5) [52].
Extraction The stage in the formula harvesting pipeline translating formulas given in ETEX into their components, resulting in executable SymPy code.
Folding A fold by k is a transformation mapping a matrix (n) E PGL. (Q(n)) to the matrix (k M
- 1) + i). In the context of recurrences, the new matrix represents taking steps the original recurrence
time.
Irrationality measure (6) The irrationality measure of a formula, measured empirically in this study, see Eq. (5). d is used in the matching algorithm to find candidate pairs for matching formulas, as 8 is invariant under coboundary transformations. The result of this clustering phase 20-fold decrease in runtime, see Appendix D
Polynomial Continued Fraction (PCF) - An equivalent representation of second-order recurrences with polynomial coefficients. Any second-order recurrence generating mathematical constant can be represented as a companion matrix (Lemma (2). and thus as a continued fraction.
PSLQ numerical integer relation algorithm that finds integer coefficients (a,) such that 01 + 02 a, 0 for given real numbers (æ:), revealing algebraic relationships among them 9 (66
Retrieval The stage of the formula harvesting pipeline in which regular expressions are applied to filter for equations that may compute the constant of interest.
Scraping The first stage of the formula harvesting pipeline.
Trajectory A direction in CMF, a vector of integers with length equal to the CMF dimension.
Trajectory matrix- The parameterized matrix resulting from walking in trajectory in CMF. Every trajectory matrix corresponds to a polynomial linear recurrence, which has a canonical form. Matches between the canonical forms of trajectory matrices and the canonical forms representing formulas can thus be searched. These matrices act as representatives of the CME, enabling unification of formulas via the CME.
15

---

UMAPS Our novel coboundary-solvvae algorithm.
Unification Embedding formulas in a single CMF to show there is an underlying theory behind them, the main goal of the work.
Validation The stage of the harvesting pipeline after extraction in which the limits of formulas are recovered in terms of the constant of interest by running PSLQ on a numerical evaluation of formula. This stage has two roles: (1) recovering the limits and (2) validating convergence to the constant of interest (or a Möbius transformation thereof).
B Special results
The examples shown below were found completely automatically via the algorithms discussed in Section 3 and organized in Appendix C The actions at each step are described as being applied by human for clarity, but we emphasize that the connections were made automatically. A template for proofs could have been written once and filled in by computer as it executed the algorithms. Some interactive examples are available at the online algorithm demonstration [4]
B.1
Unification by the Conservative Matrix Field (CMF)
B.1.1 Unification example: formulas 1, 2 and from Table 1
Formulas and have the same canonical form, showing they are equivalent. We will next prove formulas 1,5 equivalent via the matching algorithm (Appendix C.2 We compute the irrationality measure (6) for formulas and 5, resulting in -0.65 for both canonical forms. Next, we compute convergence rates (r), resulting in 0.69 and 1.38 for formulas and 5, respectively. Dividing the two we get fn = 2. Folding canonical form by 2 and canonical form 5 by (meaning no change) results in
P.Fli PCF (60m² + 34n2 11m 3, 2n (-288n" + 624n -330m 225n 158n 24))
Applying UMAPS (Appendix C.3 the following coboundary matrix and polynomials connecting
PCFi,folded and PCFs are discovered:
38n - 85n + 28n 2304n - 9792n + 15440m 11100n 358666 -48n3 200n 223n + 51
408n
U(n)
(n) = 12n - 17
pr(n1) 3n +
This means that the coboundary condition (Fq. 8 holds:
(n()) CMI,folded (n 1) = pB(n) U()) CM,
Multiplying the terms, we indeed get the expected relation:
(12n 17) (i 2n 1-288n 5 + 624n 4
230n
225m + 158n -24)
60m 3 34n
11n 3
(48n² 59n + 2n 2304n + 4032n + 1040n
1180m
434n2
88n
48n + 56n + 33n 20
(88 85n2 + 28n 2304n 9792n
15440m 11100n 3586n2 -48n
408m
(3n + 2)
-1
200n - 223n + 51
9216n (0)
3 + 11264n
15520n
764n2 3802m 714
240n
164n. 54n
29
We have therefore found transformation from one canonical form to the other: Coboundary( Fold(PCF,, Fold(PCFs, 1). Formulas 1,2 and are equivalent- having a proof for any one formula out of these three immediately proves the other two.
16

---

Are these formulas contained in the CMF, Eq. ( We find that the recurrence of trajectory (1, 0, 0) in the CMF also has 8 -0.65. The recurrence generated by this direction is precisely the recurrence PCF,, meaning trivial coboundary equivalence (n 22x2) exists between formulas 1,2 and the CMF. Note that in the general case, showing unification requires finding nontrivial coboundary equívalence (U (n) * 2xx2) between representative canonical form of the equívalent formulas, and the canonical form of the recurrence generated by the CMF (per Appendix C.I). See Appendix B.1.3 for example.
B.1.2 Unification example: formulas 3 and from Table
The canonical forms of formulas 3 and have slowly). Given the similarity in 6, conjecture that the formulas are coboundary. Applying UMAPS yields the coboundary matrix:
and convergence rates 0 (they converge
(n) (
4n2 4n
8n
4n² 4n² + 8n +
10n + 3
2n +
and trivial external polynomials (n = pn) =. So formulas 3 and 4 are equivalent. The trajectory (1, 1, 1) of the CMF generates recurrence with = 11 and upon inspection found to be precisely the canonical form of formula 3, PCF3, unifying formulas 3,4.
The next example requires nontrivial coboundary equivalence between a recurrence of the CMF and the formula of interest.
B.1.3 Unification example: cluster (-1,3,3) (6 -0991)
The following example pertains to the d -0.91 cluster in Table 16 The canonical form generated by the CMF:
PM PCF( 2 11664n* + 6992n² 6036n 279r 162,
-2nn(nn + 1)(1n - 3)(1n - 1)(6n (6n 5)(22n² 39m 1)(22m + 49h + 9))
and the canonical form corresponding to the series
16 (22k2 17h: + 3)
(1))
(k
2T
k(4k 3) 1) (3h)
(6k)
(3k)
is
PCF PCF(33744 + 156n 1942n
261m + 45.
24n(2n 3) 3) 1)(6n - 5)(6n 1)(11n 14)(11n 8))
Computing convergence rates, we find both have 0.52, so they are not folded. Applying UMAPS results in coboundary matrix and polynomials:
11848n 7676n" 10730n 5605n? 682n 21
6690816n 504852445 15773606-1n 2860n 10680n + 13481n² 6348n 756
(m)) =
(a(n) = 22n + 61n 42
Pn(n)= = 44n 120m 2 67n +
showing that the series contained within the CMF.
B.1.4 Unification example: cluster 10 (5 = 1.00)
Here we show the four formulas listed explicitly in Fig. 6 for trajectory (0,0, 1) are all equivalent,
these correspond to indices 71, 72, and 76 Table All formulas have -1.00 of course, so they proceed to the convergence rate matching stage.
First, consider the two polynomial continued fractions (formulas 72 and 71):
PCF72 = PCF(1, + 1))
PCF71 = PCF(2, n²)
17

---

These have convergence rates O, so all combinations of folding up to 2 are passed to UMAPS. Applying UMAPS, the coboundary algorithm (Appendix C.3 each other with no folds necessary:
the two out to be coboundary to
-n?
U(n)
-1 n-1)
with trivial "external" polynomials pa (n)) pn pD( 1.
Next, consider formulas 75 and 76: Applying the same steps as above shows they have convergence rates of O. Passing all three combinations of folding by to UMAPS, we obtain a coboundary matrix relating the two formulas with no folds necessary:
(nn) = 4n² 4n +
16n² - 16n + 4n²
-1
-4n²
these too with trivial "external" polynomials.
At this point there two clusters. Can they be united? Consider the pair 72, 75. Passing all three combinations for folds to the coboundary algorithm, a coboundary matrix comes up:
[nn 4n +
16n 16n + 4n
(nn)
-1
1 4n
with external polynomials
PA (n 2n
pn ( (n) 2n
In conclusion, we have found that formulas 71, 72, 75, are equivalent each other. Only one need be proven to prove all of the others.
B.2 Unification by the ((3) Conservative Matrix Field (CMF)
The following matrices define 2D CMF that computes the constant ((3).
3
Mx (I 1)²
3 2y (2r + 1) (y 1) + (x (x+ 1)3
(13)
-xx 2n2 - 2ny² 3
x33
My
3
x² + 2n²y 2ny + y²²
Converting the two formulas for (3). Eq.
11
canonical forms, respectively yields
18
(13)
- 4[13]
648
(3)
7291
98
17
91
1006
(e
2nd 9n2 17n 12+
2nd 9n2 + 15n + 9
Applying methods, find coboundary matrix of degree with linear external polynomials, U(n) = n1 n2 +n+1
n + 5n + 10n 10n + 5n + n
_ 4n 5n
(n( =
(B(n) = n n +1
which are together certificate equivalence for the canonical forms, and hence for the original formulas too.
Both of these formulas are found in the (1, direction of the ((3) CMF (Eq. 13), which corresponds to the continued fraction:
9+
729
35
2 + 3o + 3n
18

---

B.3 Unification by the Conservative Matrix Field (CMF)
The following matrices define 2D CMF that computes the constant
Mx = (11 2
My = 0 - -y-1)
+ 1ッ
(14)
We present below the unification of all 15 formulas from 46
Table 5: Unification of all 15 e formulas from 46 by CMF Eq. (14), via UMAPS. Because these formulas converge super-exponentially the convergence rates unbounded and depend the depth to which they are computed. The folds needed to match the formulas in the (0.1) trajectory were luckily still uncovered.
Cluster
Formula
Value
Convergence rate
(1, I)
= 1.00
PCF(4n + 2,1)
14c. -I+e
17.36
(1, = 0.00
PCF(n + 2,-n)
7.30
-1
PCF(n 33-
7.30
PCF(n 3n +
4e
7.30
I+22
-2=
PCF(n 4n +
3e 2 1--e)
7.30
3.-n² - )
PCF(n 4.)
7.31
2+c
PCF(n + 5.-
7.31
6-2c
8
PCF(n + 6.-n)
7.31
PCF(n 2 + 6n +
0c
7.31
3+22
7.-n² - 3n²)
(0,
= 0.00
10
PCF(n,n)
7.30
146
11
PCF(n + 1.n)
7.30
12
PCF(n + 2,n)
7.31
-5
13
PCF(n + 3,n)
7.31
-10+6e
14
PCF(4n2 + 14n
15.98
3-c
11,-4n² 6n)
15
PCF(4m2 + 10n +
14 -2+
15.98
5.-4n² 2n + 2)
For example, consider these polynomial continued fractions:
6c
4e
3
7
3
2e
3
14
74
n² [- 3)
n <
ain + + 7+
n(n + +3+
Applying UMAPS, we find coboundary matrix
nn3 + 4n + 6n +
n² + 4n + 4n n²
U(n)
-
2
19

---

and external polynomials
PA (n) n n
PB(n) n 3
proving that the two formulas equivalent.
In similar fashion, we arrive the other equivalences summarized in Table
B.4 Equivalence example: notable formulas for
Eq.
48 and Eq. T0 S4] are converted to recurrences, both of order 2, after which they are
converted to canonical form PCFs, respectively:
239018472
PCF PCFRamanujan Ramanujan
(C)(aa(n),bl) (n))
-3528 + 1123
1047212167162854400
PCF = PCF((2(n)) b2(n)
+
01 (n) = 01101222222200021211111111111111
16865127823288'
27955409115
n /tt.:///5............0.000000--0
+
5601153837771755556m
= S50000/8857722441117772222222222
+ +228088880000004444--88222222220
by(n) n h|tp://::.......100.030000011111
11397799773092277555551771111 11512222111122222727777777722222
159968822007913316445566
The canonical forms share = -0.29 and = 13.56, so the recurrences are not folded and UMAPS
applied, resulting in a coboundary matrix of degree 10, coupled with external polynomials of degree 4, rendering Eqs. and I0 equivalent:
(a(n)= = 11 11398398784n' 4 19077544640m + 9321191372n 1315967464n 20955
(B(n) = 171680n 3 + 395264n + 290210n + 67749
(n) U12(n)
Unn)
U2i(n) (22(n))
(n( 1
1678568399663150002 -39921177739777000 o1g.:/00.........11...0.........
(12(n) 02222255005111555000000002222220
+ 203554884033731119007774111444 705526607733444475550066683
597708278114005383557777777727 o1g.0//0777777001...11.333022200 133268825544777729666444422""
U2i (m) 42050n 29337
(n)(n)
14914441975033307712200 4
92674869920888993316667
1122077700773389444441112222 98865518879991444927777
1173766699766003770 201337763211702670008 143463055820335
B.5
Equivalence example: formulas for Catalan's constant G
The two formulas for Catalan's 's constant in Eq. 12
are equivalent via coboundary matrix
(nx) (4n² 2 + 2n
16n
-1
4n + 2n
20

---

and trivial external volynomials-PAte = and (2(n) 1. Note that the formulas in Eq. 12 are not polynomial continued fractions in their current form due to a periodicity of 2 in the an. bn functions. To convert them into polynomial form, they are first inflated to make them integer continued fractions, then folded by 2 to make them polynomial, resulting the canonical forms:
16
2
32
=7+
2-26
256
2G
381
23
25
1296
1728
55
+
16n
61
+
16x
1)
Su + 8n + 7+
8n
5++
B.6
Example polynomial recurrences for formulas, some of order greater than
One of Ramanujan's 1914 formulas (shown in Fig. 3) is represented by the following order-2 polynomial recurrence using RISC's tool for finding minimal recurrences 33
18426117
1603004319
186504787
605532897
22086037
831 33295277
2072547
720
f(n)
3162112
G3242240
38626600
12648488
7M0G28
7906280
D8N1GO
20FG
568G20671
1927 D6306
1076852418
3570680181
122831663
424008847
10066461
3367
/01 +
108105601
128488448
3962640
12618188
798122
7905250
568160
3096
40984
171704
61G40
G380S
30000
G19UT
23709
565
965
193
155
153
965
1500
Some formulas generated by recurrences of higher order. The methods presented in this work can be generalized to higher degree recurrences (see Lemma of Appendix C..) For example, these two series for Catalan's 's constant
[K
G = En 2
Σ k=0
1
2G = Σn
2 (2n+1)(
Ek k 2k+1
12k+1112
(2))
are given by the same recurrence of order 3:
5m
x²
21
29n
5n²
f(n)
f(1 n)
2
4
2
4
85
13a- 2n 1(2 + n)
+
49
Tn + n²)
f(3 + m)
meaning their recurrence matrices, which are in general companion matrices (Eq. ( are trivially coboundary, with the identity coboundary matrix. Other cases in which the recurrence is not precisely the same require a generalization of the coboundary algorithm Appendix C to solve for the coboundary matrix. Additional examples of high-order recurrences, for n::
2
20
k
2h
2i
22
k
π
k -
i
32
O=
i=0
is given by the order-3 recurrence
n
81
173n.
65x
21m
5m
= -4 8n
6n
2nd
[(x)
+
f(n + 1)
4
137
297n
111n²
35n
2n
f(n 2) + (18 39m + 29n
9n 2) f(n +3)
4
2
2
2
Σ
80
1)n
(3n 1)
n
2nn
2k
2k
2
π
32n
12
k
k=
is given by the order-3 recurrence
26n
23n
12
35m
3 16
365
181n
1879n²
1589n
55m
9n2
[(n)
+
[(u + 1)
3
36
48
9
2
48
16
356
5031
1633n²
3
81md
3n
[i + 2) + (84 + 183n + 154x
568
38n'
+ n
f(n + 3)
24
48
16
21

---

C
Algorithms
This section contains an in-depth description of the algorithms discussed in Section 3 The algo- rithms are ordered top-down, from the highest level algorithm to the lowest. A hyperparameter sensitivity study supporting the hyperparameter choices below is included in Appendix [D] All algorithms used in the pipeline were run on a 13th Gen i5-13500H Intel Core and are available Runs required for the sensitivity study were
at https://gittub.. comp/ com//oommaaaaaeeeee conducted on the Technion High Performance Computing Zeus Cluster.
Table 6: Algorithms enabling unification by Conservaitve Matrix Field (CMF). The matching algorithm and UMAPS are depicted also in Fig.
Name
Depeadencies
Inpat canonical-form formulas and CMF-derived representative recurrences
Output
Coboundary graph growing algorithm
coboundary graph-a forest of equivalent formulas with
oquivalence-proving transformations stored in the edges
Matching algorithm
two polynomial recurrences A(xx) B(n)
triple of three transfornation:: a fold transform for each recurencce (FA, Fn)
and a coboundary transfomm
(U (U()), pa(x))
U(a) - PatA(a)) U[++1] (u(())))
C3
UMAPS: The
two polynomial recurrences A(R), B(n)
coboundary transform
coboundary solving
(U (a()).
algorithm
U(n)-1 A(a) U/x B(a)
Conversion to canonical form
RISC's tool
Diophantine foruula
minimal polynomial recurrence (PCF second-order recurrences)
C.5
Recurrence
CMF matrices, start point, trajectory
polynomial trajectory matrix (polynomial recurrence)
generation CMF
C.1
The coboundary graph growing algorithm
Input: (I) Initialized graph with no edges, where each node is canonical-fori recurrence for formula. Each node has precomputed attributes: irrationality measure (8) and convergence rate (r)
(Section 2). (2) Dataframe of canonical forms generated by the CMF (see Appendix C.5 with
precomputed attributes as in (I).
Output: The graph as a forest, where each edge contains the rigorous transformation between the two nodes it connects. This forest is termed a coboundary graph. Every tree groups formulas that are iigorously-equivllee
together-the trees found by the algorithm are actually subgraphs of cliques; not all clique edges are computed during the matching phase to make the algorithm more efficient, but they all exist. If a tree contains CMF-generated canonical form, all of the formulas within are unified.
Steps:
1. Cluster nodes according to 5: Initialize an empty list for each value between -1.00 and 0.05 (or higher upper bound, depending on the PCF with highest 8) at intervals of 0.05 (5 "gramularity""
including the edge values. For every cluster value de, insert every node with 5 obeying (6 18-66 6g < 0.03 (8 "similarity threshold") into the cluster's ' corresponding list. Note that a node can appear in two clusters initially. The maximum distance between two nodes in the same cluster is 2 0.03 0.06. This is intentional to prevent missing matches. Additionally, initialize an empty list "nonhubs."
2. For every cluster of nodes indexed by d.:
For each node, attempt to match it to all nodes appearing at higher indices that are not in "nonhubs," using Appendix C.2, Store successful matches as new graph edges containing the transformations. Every formula matched to the current node is placed in the global "nonhubs" list.
At the end of this process, nodes not in the "nonhubs" list are deemed final non-CMF "hubs." They are currently the roots a forest since a node is added to "nonhubs" as soon as
matched with the current node during the loop, and it is subsequently skipped.
22

---

3. Now, for each "hub," attempt to match it to all CMF-generated canonical forms with 8CMF that obey 18 & < 0.03. When an edge is found, break for the current hub-the hub and all nodes connected to it have been unified in the CMF
4. Output the resulting graph.
C.2 The matching algorithm
Equivalence a binary relation between two formulas. To find one, formulas represented as canonical form recurrences (Appendix [.4.1) are clustered based dynamical metrics (Section Promising pairs, those with similar irrationality measure (Fq. (5)), are folded (Appendix E..5 according the ratio of their convergence rates (Eq. (4)). then sent UMAPS the coboundary solving algorithm (Appendix C.3). The result is triple: the fold transform needed to be applied to each of
the recurrences, and the coboundary matrix and polynomials outputted by UMAPS. This algorithm is also summarized in Fig.
Input: Two linear polynomial coofffccen recurrences with matrices A(n) and (nn).
Output: list of three transformations connecting the recurrences (if found)
fold transformation for each of the recurrences (FA, Fg)
coboundary transformation linking the two recurrences after they are folded PA)))))((())Pp(())
(n(n) U(n)- FA(A(n)) (A((()))) + 1)= Pa
Fa((())
Steps:
If unknown, compute the convergence rates of each of the recurrences using Eq. at n 2000.
If the convergence rate is less than 5 10-2, set it to O.
2. Compute the ratio of the convergence rates of the recurrences, when defined, as R rral Assume this number be approximated rational with low-enough denominator to good degree. Set R = if either of the convergence rates (this is normally the case when 5 = -1, such PCFs converge polynomially).
Fold recurrence A by r'B and recurrence by T if R 0 If = all combinations of folding the recurrences by up 2 are passed to the next phase for total of 3 options: fold neither (one option) or fold one them by (two options). This is because the convergence rate does not contain fold information for PCFs with slow convergence.
Apply the coboundary solving algorithm Appendix C.3 between the recurrences. a coboundary transform is successfully found, output fold transforms from 4 and the coboundary transform.
C.3 UMAPS: The coboundary solving algorithm
We aim to find coboundary relation between two recurrences, yet as mentioned in Section 3.5| this constitutes nonlinear problem. To see why, let us write the equations for the coboundary matrix and polynomials full for order m 2. We have, given two polynomial matrices A(n) and B(n)
(n(n) A(n) Unn 1) pn(n) U(n) B(n)
dpa
days I days Li=
Ani.n Azi.n
Σ dayz Ar2,e Edam Amz,nn)
Ene dun
dun Un, + 1)
duia (12,(n + 1) Ei
Σ PA
Un1 + 1)
Edn Un2 + 1)5
duza
dpp
Σ Iden Uni
dun
ddig U12 Di Una,an
dun i= don Bui,"
B1I.mn
dmz B1aan
Σ Pn.in 1=0
duva Uni
Edem Bra,nn
The unknowns of this equation are the coefficients of pa(n). Pn()) and the coefficients of the four polynomials of U(n): (PA,a(k: O,... .....,dp,y dp (PB 0...., ..... (Ug,,(h 0, N....,d.iiin i,j E {1,2]). From the equation written above, it is clear that the coefficients
23

---

coupled (e.g. the product PA.0 U21,0 is in the expansion), making the equations resulting from equating powers of n nonlinear.
An empirical method that takes care of nonlinearity and the ambiguity in unknown polynomial degrees would be much preferred. Luckily, such method is possible: below), a necessary condition a coboundary matrix between two matrices must obey, to solve for (1) and then reconstruct the coboundary matrix in full. Lemma 4 generalizes this condition to general matrix order m, enabling a coboundary-solving algorithm for recurrences of any depth.
Lemma (proven
Two order-m recurrences that converge to the same irrational constants (m constants most) up to Möbius transformations (Eq. (I7) and a requested depth up to which a coboundary matrix will be fit to "measurements" of the coboundary matrix. Denote the recurrence matrices by A(n) and (n), their limits by LA and La (integer Möbius transformations of the irrational constant for = 2 and projective vectors for higher order recurrences the setup of Lemma and the requested depth as N.
Input:
Output: A polynomial matrix (n) and two additional polynomials pa PB)))) if found, satisfy- ing the coboundary condition Eq. (8)
Pa(n) A(n) (n + 1) pp(n)) Un)) B(n).
Steps:
1. Solve for the first coboundary matrix (1) up to multiplicative factor using:
(order m 2) From Lemma [ write the equation LA (L)) where ((L)(-) the
Möbius transformation, for the four unknowns of the first coboundary matrix
U(1) U11
112
u21
U22
Writing the Möbius transformation explicitly, the equation reads L Mi tin Without loss of generality, the above equation can be written as oxp+ uas Lnt
where a, b, c, d 7043 By equating coefficients of powers of o in the numerator and denominator independently, we obtain four equations for the four unknowns of U. (This must hold assuming the irrational number does not solve a quadratic equation, a condition met by all non-algebraic constants like π and ((3).) The solution is the four rational numbers of (1) (up multiplicatiie factor).
opt epta
7048
depend on {mi (Je])))()12)) and LB and LA = a
(general order m) From Lemma 4 write the equation LA U()) Every row of this equation can be written as LA. = U(1; LB where U(1); is the 7th row of U(1). Since
solving over the rationals, PSLQ (26) can be used to find solution: a; LA, - Ey bi,L = 0 (where a, (by ) were found over the integers via PSLQ) =
(11gg = by If the symbolic limits are known in terms a set of algebraically independent constants, (1) can also be found by equating coefficients.
2. Propagate the coboundary matrix to the requested depth N using the necessary condition for coboundary equivalence:
A(n) Un + 1) x Un.) B(n) - Unn 1) & A(n)-1 Un)) (nn)
The resulting matrices again the rational matrices (U:) f known only up to independent multiplicative factors.
3. Divide each of the "measured" (U()) by. c.g., the Un clement of each matrix.
Take care to pick an element U11 U12 U21. U22 (etc. for orders > 2) that does not zero out 1,2, ... N. If all do then pick the element whose last 0 arrives at the earliest index and toss out all measurements preceeding this index.
The result is new list of matrices. If polynomial coboundary relation exists between the two recurrences, (n) A(n) Ü(n + 1) Pы(n). Ü(n) Bn)) for some polynomials PA (n) and (n), then the matrices we have found are precisely the result of dividing (()) by its (c.g.) first element. Meaning our measurements should be of rational matrix.
24

---

4. Fit rational functions to each of the elements of the measured U:
Writing general m x m rational matrix requires 2m² polynomials (pt (ny(n) Hij
Equating the rational function of each element Pu(n) to the measurements that element (n),,), yields system of linear equations for the coefficients of the numerator polynomial and the denominator polynomial: Bsy(a) Uiz(n) = (i)(m) - (n))n) = for
(=)(n)
[n)
1 ,,...,N. N.
because this is The result is a rational coboundary matrix hypothesis Un (n). Hypothesis- because
rational fit our empirical coboundary matrices.
5. The final stage of the algorithm is to verify the hypothesis.
Multiply out A(n) (n + 1) and (n) B(n). If the resulting matrices differ by a multiplicative factor (not matrix, rational function), meaning the condition A(n) Un (n 1) x (n) Bnn holds, then the coboundary matrix hypothesis (n) is a valid, though still rational, coboundary matrix.
Multiply both sides of A(n)- Uni(n + 1) x (n) B(n) by the least common multiple of
the denominators of Uz. (n + 1). Un. to convert the coboundary relation into polynomial one of the form PA(n) A(n) Un + +1 = pn(n) U(n). B(n), where p. (n),, ( are
polynomials and (n) is polynomial matrix.
Output (n). (n) and (n) (which are defined only if U,, was valid).
This algorithm (for m 2) is also available on our interactive, online algorithm demonstration
C.3.1 Sufficiency of UMAPS
We can show that UMAPS is sufficient to recover the coboundary matrix, provided an upper bound on the degrees of the polynomials its rational entries. We now proceed prove Corollary According to this corollary, the sensitivity study (Appendix D) indicates that any coboundary matrix among the harvested formulas that has not been recovered must involve rational entries with polynomials of degree at least 60. Let us restate the corollary:
Corollary 1 (Sufficiency of UMAPS.) If coboundary matrix exists for two matrices and every rational ffiinta-function entry of the coboundary matrix has polynomials of degree at most d, then running UMAPS with N > 2d suffices recover the coboundary matrix.
Proof. By Lemmal [ (and the generalization to any recurrence order, Lemma 4 two matrices that are coboundary have limits related by the transformation U (1). Therefore, during the execution of UMAPS, the quantities U(1),..., U (N) correspond to correct measurements of (()).
Assume that each polynomials of degree most d. This gives total of 2d + 2 coefficients. Since multiplying both the numerator and denominator by common nonzero factor yields the same rational function, there are effectively 2d + independent degrees of freedom.
entry of the coboundary matrix has numerator and denominator
Choosing N = 2d++1 produces a linear homogeneous system of 2d equations in 2d+ unknowns for each element of the coboundary matrix. If coboundary matrix exists, then it is unique up to scale by Lemma 55] So this system has a one-dimensional null space, corresponding to unique solution up to overall scale. Increasing N beyond 2d + introduces linearly dependent equations, leaving the null space unchanged.
Hence, rational fit to the empirical measurements of ((n) will recover the rational-fun -functinnnnnnnnnn entries whenever N 2d + 1.
C.3.2 Rational function fitting details
A rational function fit is attempted in step 4 of UMAPS, detailed above. Consult 33 for thorough treatment of rational fitting. The rational conducted by finding the null space of a linear system of equations:
25

---

When fitting rational functions, the equations are of the form P()
20 = Ri where P.Q are the polynomials of the rational function and Ei is the empirical value at i. This leads to equations of q the form qi Ddeor P&t - -p Lk= Qu - 0, for the degp + dego + polynomial coefficients.
These coefficients are determined by finding the null space of the system of equations.
In practice, the system is a SymPy matrix of integers (e.g. q,th for Pt). SymPy does this via the linear system's 's reduced row-echelon form, and uses rational number objects (not floats), so precision guaranteed.
Since the rational fit is not a least-squares fit, noise and finite precision are not sources of error. Analytical correctness of the is guaranteed by subsequent substitution into the coboundary condition. Once a matrix successfully recovered, Lemma guarantees this the coboundary matrix, due to (nn)s projective uniqueness.
C.4
Conversion of formulas to canonical form
We use RISC's Guess tool 33 to convert formulas into recurrences, after which correctness is validated numerically. Minimality is then guaranteed using Maple's Minimal/Recurrence package 57 (additional details available in (60). Finally, formulas yielding second-order recurrences are represented as continued fractions. The proper initial conditions needed for the continued fractions to generate the formulas are found (Appendix(C.4.) CC..T).
Our use of RISC's Guess algorithm and the following steps can be summarized by:
Input: First N approximants of formula.
Output: Minimal polynomial recurrence generating the formula.
As more terms of a sequence are considered, the likelihood that guessed recurrence fits only the initial terms but fails for later ones decreases. In experiments, when converting each series into a corresponding recurrence. The hyperparameter sensitivity study (Appendix D) supports this choice.
N 200 partial sums
Once a recurrence is identified for a formula with a known closed form, we verify by substituting the formula back into the recurrence. If R is the recurrence operator and ((n) is the formula, then the goal is to show Vn (n) = 0. This verification is done numerically, since substituting full series expressions complex closed forms into the recurrence does not always allow algebraic simplification. We perform 30 random numerical substitutions into the expression ([f[(n), all of which must agree to within 10 10 for the recurrence to be accepted. All recurrences were successfully validated under this criterion.
Formulas given by polynomial recurrence of order 2 can be represented as polynomial 2 x 2 companion matrices (polynomial continued fractions): given a recurrence In
c(n)z, a(n))nn111111 b(n)nn-2
the corresponding canonical form
PCF (a(n), b(n)c(n - 1))
Which is achieved using inflation by c(n) (see E.4).
Converting general series to continued fraction can be done using technique devised by Euler, however, this technique relies on algebraic manipulation of the term of the series and also does necessarily yield polynomial recurrences, let alone recurrences of order 2. A formula can have recurrence of order that not polynomial, but have minimal polynomial recurrence of order (see for example Appendix B.6).
C.4.1 Conversion of series limit to Polynomial Continued Fraction (PCF) limit
Input: Series, with partial sums (not summand) index given by S(n), and continued fraction C'F(an, bn) from the recurrence to the series using RISC's tool.
Output: E Q such that the initial conditions
So
S:S1
1
26

---

generate the partial sums of the entire series when the continued fraction recurrence is applied from index 2. By demanding the second convergent of the continued fraction is equal to the second partial sum of the series (recall our notation for the Möbius transform defined by matrix U U(-)):
((1)
xS1 x
b2 22 )
(0) S2
we arrive at the equation
Sob2 xS102 S2
b2 202
Solving for x, we obtain
b2 S2 So
02
So S1
Note that only the first three partial sums of the series Sp. S and S₂ (more generally ng. nip + 1, No + 2, where no is the start index of the series summation) and a single nartial-numerator and partial-denominator pair of the continued fraction are needed. This is because the continued fraction values Pn. satisfy the recurrence, hence when equating the first two values to the values of the 4n
get the exact same series.
The algorithm's 's result can be converted to initial conditions for when the continued fraction's 's recurrence is applied from any other index by multiplying by recurrence matrices of appropriate indices and their inverses, i.c the initial condition that calculates the series starting from n = 1 (using the same notation Eq. 2 is
So
xS x
0
61 a1
b1 a1
0
by
(
by a3
02
initial conditions
This method is convenient since there is no need to keep track of transformations applied to series' recurrence in the process of conversion to continued fraction. In summary, the correct initial conditions recreating the series from the recurrence are found using only the recurrence and first few partial sums of the series.
This algorithm is also available on our online algorithm demonstration
C.5 Recurrence generation by Conservative Matrix Field (CMF)
recurrence is generated from a CMF by selecting a starting point and trajectory in the CMF. We calculate their corresponding trajectory matrix (nn) (see G. and finally convert it to the canonical companion form (see in order to generate recurrence.
Input: A CMF, starting point and a trajectory i, both vectors of dimension equal to that of the CMF. Denote the number of matrices in the CMF (the CMF's dimension) by d and the number of rows and columns in these square matrices (the CMF's rank) by r.
Output:
Polynomial recurrence of order
Steps:
Compute a single trajectory step from + (n 1)t to p nt (see Appendix G.I for a more rigorous definition):
initialize M = I,xr and cur_pos (n 1)t.
for i in I,...,d: d: for in range((4):
= M Ma, (cc)r_pos)
cur ___pppos- pur_pos__ + e; (where is at index and zero elsewhere)
27

---

By the end of this loop cur = nt and M is the work matrix from (n 1)t to + nt.
2. If interested in companion form (Eq. ( which corresponds to PCF), bring the resulting rational matrix M into polynomial companion form per Lemma
3. Return M.
is worth mentioning that there degrees of freedom in the calculation of the trajectory matrix, due to the conserving property of the CMF. The order of matrix multiplication can be selected, which advantageous in preventing matrix singularities.
Our search space for formulas in the CMP defined in Eqs. (6) and (25) consisted of the trajectory
subspace T3 {(a, b, c) 3 mox [a], 16|, |c]} I 3} and the initial positions P1 2 {(a, b, c) E Z max [a], [b], |c]]
In cases where (n) had singularity some point along
the trajectory p + kt (where t E T3 and Pi). we shifted the initial position to p' = + (A: + 1)t avoid it.
As can seen from definition of P1. starting points are actually non-integer. The CMF starting points appearing in dark blue in Fig. 6 are shown in terms of the difference from (1) : 1 which is the true (0, 0, 0) point.
1
The full dataset of recurrences generated by the CMF for this study is included in the supplementary material (described in Appendix D.
D Hyperparameter sensitivity study
Below provide sensitivity study showing the linkage percentage and runtime for different ö- clustering granularities and similarity thresholds, values of UMAPS's fit depth, as well as an analysis of the sensitivity of RISC's Guess algorithm its fit depth. Parameters used in the paper denoted in bold. Runtimes are normalized by the runtimes for the parameters used in the paper. Runs required for the sensitivity study were conducted the Technion High Performance Computing Zeus Cluster.
Table 7: Linkage rate via UMAPS for different & clustering granularities and similarity thresholds in graph growing algorithm (Appendix C.Tp. The numbers listed are % of max linkage achieved (136) between canonical form formulas. The linkage percentage with no clustering criterion is found by setting similarity threshold and 100%, as expected.
Clustering granularity
Similarity threshold
0.01
0.05
0.1
0.2
0.3
0.5
0.001
85
76
68
58
54
64
0.003
95
79
64
60
70
0.005
100
80
75
64
60
70
0.01
100
64
62
0.02
100
100
86
64
65
73
0.03
100
64
65
0.05
100
100
100
73
71
73
0.1
100
100
100
100
90
0.2
100
100
100
100
100
87
0.3
100
100
100
100
100
100
0.5
100
100
100
100
100
100
28

---

Table 8: Runtime for different å clustering granularities and similarity thresholds. The algorithm (Appendix C.T) first matches formulas to find hubs (stage 1), then matches each hub with CMF representatives (stage 2). For coarser granularities (> 0.1), the runtime shows three phases across similarity thresholds (regular text, bold, regular). This can be explained by the number of hubs from stage (decrease similarity threshold increases) vs. the number of CMF representatives that meet similarity criterion stage (increase as similarity threshold increases). The runtime value results from their product, which is largest for medium similarity threshold values. Runtime without clustering (similarity threshold = 2): 20.0.
Clustering granularity
Similarity threshold
0.01
0.05
0.1
0.2
0.3
0.5
0.001
0.8
0.6
0.8
0.8
0.7
0.7
0.003
0.9
0.8
1.1
1.0
0,005
1.1
0.9
1.3
1.2
1.3
1.2
0.01
1.0
0.7
1.2
2.0
2.0
1.3
0.02
1.0
1.0
1.5
3,4
3.5
1.6
0.03
1.0
1.0
1.4
2.4
0.05
1.0
1.0
1.0
4.0
3.9
3.2
0.1
1.0
1.0
0.9
1.0
2.9
5.4
0.2
0.9
0.9
0.9
0.9
0.9
6.1
0.3
1.0
0.9
0.9
1.0
0.9
1.0
0.5
1.0
0.9
1.1
1.0
0.9
0.9
Table 9: Linkage rate and runtime for different maximum fitting depths in UMAPS. The total runtime of the graph growing algorithm (Appendix CC.I has clear minimum. The two forces at play: the average UMAPS time increases with N, while the number of hubs from stage decreases with N;
the runtime (their product) is smallest intermediate N. Note the maximum polynomial degree found as a function of N closely follows the upper bound stipulated by Corollary [
UMAPS N
Found relation (canonical) (v/153)
Same CMP (camonical) (x/153)
Max poly deg found
Aug UMAPS time
Total runtime
6.7
12
80
67
0.7
3.1
20
130
0.9
1.2
24
132
81
11
0.9
L.0
136
81
0.9
40
136
81
14
1.0
1.0
65
136
81
14
1.1
2.9
80
136
81
14
9.7
120
136
81
14
18.3
137.3
Table 10: Success rate and runtime of RISC's Guess 33 for different fit depths over 100 random formulas. These findings indicate that Guess generates robust fits that generalize well.
Guess N
Recurrences found (x/100)
Of which correct
Avg runtime
0
0
0.02
10
24
24
0.05
15
30
30
0.08
20
57
57
0.18
25
76
76
0.26
30
92
92
0.31
50
99
99
0.46
100
100
100
0.67
200
100
100
1.00
500
100
100
2.79
29

---

E Recurrences and limit-preserving transformations
E.1 On linear recurrences
Recurrences, also known "difference equations," 34 are the discrete analog of differential equations. They play a prominent role various areas of mathematics and science, including Newton's s approximation algorithms, counting problems (combinatorics), special functions, and the modeling of economic and biological systems. In this section, we revisit the notion of a linear recurrence that introduced in Section
function Uy satisfies a recurrence order m if is a solution to the equation:
Un d1, + 02,n 2n-2 2
Om,n Un-
is customary represent the recurrence via the associated companion matrix:
0 0
am,n
1 0
0m-1,71
CM(n)= = 0
Qm-
(15)
a1,m
Observe that [un-m..... 2, - CM(n) [un-(m (mm--(m-1))) 1)...........................1.. Un- Un]. Thus, the companion matrix represents single step the recurrence time By incrementally multiplying the companion matrix over n steps, we get the matrix:
Pi,--
P1n-1
P1,0
P2,n-m
P2n-1
P2,n
M II CM())
P3,n-m
P3.--
P3,n
(16)
Pm.n-m
Pm,.-1
Pm.
which call the n-th step matrix. The functions Pl,a. Pm.n are solutions to the recurrence equation with initial conditions Ps, = 8¹, and any other solution is linear combination of these. Hence this matrix encapsulated all the information about the recurrence. Explicitly get different set of solutions we multiply the step matrix M, from left by the initial condition matrix Unit where each represents the initial condition for solution.
Alternatively, let A(n) E PGLm (Q(n) be a matrix with coefficient functions. Similar the process done for the companion matrix, we incrementally multiply the matrix:
An = II A(i) 1=1
We regard An as "cocycle" (see E.4.2 and E.4.3) We are fundamentally interested in the situation
where the matrix A(n) coboundary equivalent (Eq.
(7) to the companion matrix of a recurrence.
In the context Diophantine approximation formulas for constants, we examine the ratios of elements in the last column of the step matrix M,. Diophantine approximation formulas also be derived by taking ratios of elements in A.,
We will now provide detailed analysis for the case of second-order recurrences. In this case, we utilize the fact that a 2-by-2 matrix acts by Möbius transformation (linear fractional transformation) on R and provide complete characterization of coboundary equivalence between matrices.
E.2 Second-order recurrences
Recall that any 2-by-2 matrix M can perform a fractional linear transformation. The action of M on the number is defined as:
a
(1
b+b
(17)
c.l+d
30

---

This is known as the Möbius transformation of the matrix acting on I. It customary to extend this action to oo by defining
a c )00)==
Let an dn (n) be rational functions. and consider the recurrence relation u,, On 2n- 11 4n-2- By incrementally multiplying the companion matrix CM(n) and examining the ratio of the elements in the first (or second) column, effectively apply Möbius transformation. Explicitly, have:
My
Pn qr
MN(xx)
Pa-1
4n-1
Alternatively, we can start with general matrix A(n) PGT22 (Q(n)) and incrementally multiply to obtain the matrix:
An = II A(1
We can then examine the ratios of elements in the first second column of this matrix, or even apply it as Möbius transformation at value E Q This can potentially provide Diophantine approximation to some constant, similar to how a recurrence relation provides a formula via Möbius transformation of the step matrices.
We call such matrix ((n) formula generating mafrix if, for any I QU (oo), the sequence AI) provides an approximation formula for a constant L.
Indeed as the following lemma asserts any such matrix is coboundary equivalent (see Definition to companion matrix of recurrence equation.
Lemma 2. A matrix A(n) E PGL2 (Q(n)) is a formula-genee generating matrix and only if there exist matrix (n) E PGL2 (Q(n) and a companion matrix CM(n) associated to a recurrence uns such that
U(n): A(n) CM(n) U(n + +1)
Proof. First, assume
A(n) = (a(n) 3(n)
(n))
8(n)
Yormula-generating matrix, and denote by A,, -1" A(i). yn) 0 then the bottom left entry in A,, will remain 0 for any resulting in lim An. cc. In this case it must be that (n) nonzero since it is zero then the top right entry of A,, resulting in lim An.0 * co. Hence we may assume that either (n)) or 3(n).
If (n)) # 0 then define
(n)
aa(n)
U(n)
0
And get that
0
-a())(n) + ((n)((n) a(n+ ()((m) + 8(n) *
U(n) A(m) U- U-'('--nn (nn+1) 1)
v(n)
7(n+1)
And if 71)) = 0 define
0 U(n) = (a(n- 1)) 8(n- 1))
to get that A(n)) (n-
1))) a(n 1)a(n) A(*)
a(n-1))(() + 1)8(n)
A(n)
Where these matrices projectively equivalent to matrix in companion form 49
31

---

To conclude the proof, it is left to show that companion form matrices are formula-generatinn
Let CM(2) =(1 bn be a companion form matrix with lim an
Pr
n 1=000
On
Pn- 1111-44 pv 9n-1 ++ qn
Pn
Pr. 1+1
(1)
9n
11.+1
And since the ratio of Ph. converges, the growth rate of these solutions is the same, implying that
qn
Pa-1
1
+++ 1
lim n=+00
Pn
= 1 making lim (I) = L. 2
9a-1 1 1+11
On
E.3 Proof of expected limit value
We call companion matrix A(n) Shincar--Peroott type if the recurrence relation represented by A(n) is of Poincaré-Perron type, that is
Un = (innun-1 + a2, un-2 + ... + Gm,n umm
where the limit
lim din E R
exists and is finite, and the characteristic roots, i.e., the solutions ) to the equation
Σ 05 X X
are all with different absolute values
A1
Am
In this case, the theorem of Poincaré-Perre m-linearly independent solutions P; to the recurrence u, such that
19 Chapter 8, Theorem 10] states that there are
Pn(n +1) > P.(n))
lim
Lemma 3. Given a companion matrix A(n) with the step matrix:
Pla-m
P pl
Pl.
P2,a-m
p2
122m
A
A(I)
P3,n-
P3.t-1
P3.n
Pm.n-
Pm---
Pm.
Assume A(n) represents a --incarr--Perete companion matrix with fundamental roots
Dal
pmml-
Then any column of the limit matrix
À lim A,
n=000
X1
X²
projectively equivalent the vector X
X
where X is the initial condition for the
Xm
"slowest" growing solution and X, O, i.e, the one that grows like Amml". l" In particular
X
Av
X
32

---

Proof. Let be solution for the recurrence defined by A(n), setting
Yn
(Un
Un++1
Un+2 mn+m-11
We get that
YoAn
Yn.
Hence
yn
Un++1
An
yr++2
1/2
Un+m-1
Um-1
Each row gives the equation
m
(A
(AA))).) v2m)
vm)
ynti-1
Di
(Am i,
yn
Assuming yn is the "slowest" solution the recurrence, we get that
lim
yn+1
Am
Yn
Note that each row of A is general solution to the system defined by A (=)(n) so by Poincare-Perron
theorem (Am 1,111
Axm is independent of i, so we get that
4,1
(A,7) 1.s Nn.
ンド=1 (A )a,
lim 7-000
(An "osk,
Do4g-1
j= (An "
Hence
(A77)) Vn)
Em
(A5') Dis
===1
(Am ),,
Un
V/i
lim
(A "es Mutts
(An))) vn)
E
(An"))
As needed.
E.4
Coboundary transform
Given recurrence relation the interested in analyzing the dynamics of its solutions, which translates to analyzing the behavior of the companion matrix CM(n) increments, i.e., the behavior of the step matrix M,, approaches infinity.
are interested in the ratio between solutions to the recurrence. Consequently, we look at these ma- trices as elements in the projective group PGL. (Q(n)) We will recall the definition of a coboundary cquivalence:
Two matrices A(n), B(n) E PGLm (Q(n)) are said to be coboundary equivalent if there exist matrix (n) such that
A(n) (n 1 Un)) (mn)
(18)
From the equation above we see that A(n) = ((n) ((n) U (n+1) U thus exhibiting a "telescoping effect" the product, resulting the equation:
I A Unn 1) U(1)
B(n))
1=1
This suggests that these matrices share similar dynamics.
33

---

We denote A(n) ~ B()), to indicate that they are coboundary equivalent, and we often call the matrix (n) a coboundary transform from A(n) B(n).
Index shift:
Notice that the matrices A(n) and 1()+1) are coboundary equivalent by simply taking U(n) A(n). This implies that an index shift a coboundary operation. Since coboundary an equivalence, by transitivity, any integer index shift is also coboundary operation.
Inflation:
Given a function Un satisfying the relation
Un un-1 + 02,n 2 +
Om,,
n
We can define U, Un
II
c for some function Cn. The function U, satisfied the relation
Un = 01,n un Cn 11n Cm (m-(m-1))) uy
Let (M denote the companion form un and CM(n) denote the companion form of U, then the matrix
cn i)
0
U(n)
II c(n i)
0
c(n 1) 0
a coboundary transform between CM(n) and CM(n). This process is called inflation and allows one to construct a Polynomial Continued Fraction (PCF) from recurrence given by rational function coefficients. In [52), this is done by taking en to be the LCM of the denominators. In the other direction starting from PCF (a(n), b(n)) and taking C, n deg((a) the inflation by on called factorial reduction [10).
F.4.1 Canonical form
have seen that recurrence relation with rational function coefficients can be transformed inflation to a recurrence with polynomial coefficients. We say that a recurrence
Uh = (1i,n4n-11 + a2,n Up-2 + Om,, Um-m
in canonical form if the coefficients a,, polynomials, and for any other recurrence with polynomial coefficients a', a' which is coboundary equivalent to it, the degree of each as,n is less than equal to the degree of the corresponding a' i,.. Appendix C.4 C.4explains how the minimal recurrence found for each formula.
Lemma 2 asserts that in the case of second-order recurrences, any formula-generating matrix is equivalent to a companion matrix. In turn, this is equivalent to a companion matrix of a recurrence Un an - b(n)un-2 with polynomial coefficients through the process of inflation. We denote such recurrences by PCF (a(n), b(n)) as they are associated with a polynomial continued fraction
b(1)
(m)
Pn
6(2)
gn
a(1) +
a(2)
+
b(n)
a(n)
34

---

The canonical form of PCF (n² + 1, + n² + 1). for example, is equal to PCF (1, the simplest continued fraction for the Golden Ratio.
E.4.2 Group cocycle and coboundary
We now describe the mathematical context that motivates our definition of coboundary equivalence.
Let be a group acting by automorphisms on a group G via the map:
:p r Aut(G)
A I l'-cocycle with respect to y is map M: 1 - G satisfying the cocycle condition:
M(7172) M(m) ifn (M((2))
(19)
We declare two cocycles, M and M', be coboundary equivalent there exists an element g E G such that:
.(1)) = _M (g)
(20)
For further details, refer the interested reader to Chapter of (51]
Here, we focus on the case where For any E r, and A(n) E G, the action map is defined by 5m pm (A(n)) = A(n + k).. Since 2 is
and = PGLm (Q(n)).
generated by single element, any cocycle M: + G is determined by the value M1 (n) E G. For k > 0 the cocycle condition implies the
k-1
M&(n) II
Mi(n+ i)
Moreover since Mu(n)) Idm, have that Mz(n) (m
From the definition of the coboundary equivalence in the context of group cocycles, we get that two r-cocycles, M and M', coboundary equivalent there exists U(n) G such that:
U(n) (1 = (m) (n +1)
When regarding a matrix A(m) E PGT (Q(n)) as the matrix M1(n) generating the cocycle, this equivalence is precisely the definition given in Definition
E.4.3 Cocyles in dynamical systems
Let T be homeomorphism T: X X of some topological space. In dynamical systems, one studies the evolution of the system under the repeated application of the transformation T.
A function
A:
GL,ml (R)
satisfying
A(ki + k2.x) Alk,,
is called continuous linear cocycle [1]
Also in this context, single matrix A(n) E GLn (Q(n)) defines cocycle. This is done by taking X R. and constructing cocycle by defining
A(k II A(x
Typically, it is assumed that the topological space X compact space or probability measure space. (18) Our definition of coboundary equivalence 7 is not only mathematically natural but also captures the essence our dynamic framework. The vast data collected on related formulas suggest that this notion is effective for equating formulas, as it preserves the measure of irrationality.
35

---

E,5
Fold transform
Consider the matrix A(n) encoding process defined by incrementally multiplying the matrix. We might wish to associate matrix with an accelerated process, one that goes "k steps at time." That is, matrix (n) such that taking one step (n)) is equivalent to taking k-steps A. This requirement can be expressed as:
II
A(I)
II 1=1
A
2=1
And the k-fold matrix is given by the product:
(n) == II A ( 1 1) +)
When the matrix A(n) is a trajectory matrix within CMF (as described in Appendix G.1 ) the k-folding of A(n) corresponds to the trajectory matrix associated with a direction that is k-times the original direction.
E.5.1 An example of folding per convergence rate
The ratio of r (Eq. (4) between formulas, when they exist and are non-zero, hints that one may be transformation of subsequence of the other. To see why, consider two series; one with summand s(n) and the other with summand s(3n 2) + s(3n - 1) + s(3n). These series have the same limit, as the latter produces subseries of the former, but its convergence rate may be 3 times higher. Evidently, their summands do not appear directly equivalent. Our algorithm thus applies folding (Appendix E.5) to equate the r values of each pair of formulas that are candidates for equivalence. following is an example of two such series for (120 and 121 from Table 17
2
Gn Σ (-1))
(21)
4n +
4n +
4n
featured in 1
[030] with a convergence rate of 1.39, and
1
1
2
2
by
16 (
) (22)
4(8k + 7)
288 + 6) 2(8h
8k
3
8k
+
8k
k=
featured in [8. with a convergence rate of 2.77, are not directly coboundary equivalent. However, examine that when summing over Eq. (2 in pairs, meaning:
au
02k + 42k++
it is easily verified that by = a2k 02k+1. Thus, these formulas ought to be considered equivalent, and indeed were matched by our algorithm (Appendix C.2b by first folding the sum Eq. (2 by
2.77 =
1.39
F
Coboundary matrix properties
F.1
A necessary condition on the coboundary matrix
Below we include proof of the necessary condition coboundary matrix must obey that was leveraged to create UMAPS Appendix C.3 We start with the case of x 2 matrices (second-order recurrences), then generalize any order.
Lemma I (A necessary condition on the coboundary equivalence matrix.) Let LA = lim PCF (a(n), Un)) and Ln limm--oo PCF (c(n), d(n)) be converging PCFs with asso- n-550
ciated companion matrices A(n), B(n) E PGL2 (Q(n)). If A(n) is coboundary to B(n), then LA and L.B are related through a rational Möbius transformation, moreover, if (m) ) the coboundary matrix then LA (L)) (U) applied to Lg as a Möbius transformation).
36

---

Proof. Let An and B be the step matrices of the PCF recurrence.
An
II i=1
A(i)
Pn-1
pn
B.
II B(i)
8n-1
Sn tn tn
9n-1
9n
1=1
Let
a(n)
B(n) 5(n)
U(n +
(n))
with a, 1,,,6 polynomials. Since A(n) and (n) are coboundary equivalent
An (n 1) (1) B,
This implics equality when applying a factional linear transformation to O.
( ((p-1 Pn pn) Unn 1)) (0) (1))
8n-1 1-
8R
(0)
(23)
9n-1
to
Taking the limit, The left hand side of equation 23 above yields
lim 3m)) + pr,(()
lim Pn-1
A(n)
8(m)
+ Pn.
Pn-1
4n (m)) + (m)
n On-1
8(n)
+ 9m
8(n)
While the right-hand side:
(U(I)
8n-1
( U(1) t.
8n
8n-1
(0) U(I)(B)) tn
8n
lim 10000
n
f-
The matrix A(n) is companion matrix for second-order linear recurrence, any solution for the recurrence is composed of two prime solutions 2n Yn. where the growth of on is dominant over the growth of yn. Since converge, know that pn and 9n both have 2n, component to them, qu
f(n)
making Pa-1
and dn both grow asymptotically the same, this renders lim On-1
8(m) Pn-
(n)
+
d(n)
Qa-1
and we may conclude that
lim Pa- Bnn1 pn,
lim
Pn-1
8(m)
[f(n)
Pa
LA
n2 11000 In 3(n) + ōn)(
n-+00
Oa-1
B(n)
9n
6(m)
Hence La (L) needed.
Let us generalize Lemma 1 for matrices of higher dimensions.
Lemma 4. (Generalized necessary condition on the coboundary equivalence matrix.) Given two coboundary equivalent companion matrices A(n), B(n) E PGLm (Q(n)) (n + 1) B(n) with m and the step matrices:
A I A(i)
B, IT= B(i)
P1n-m
Pl,-1
P1.n
Q1,n-m
Q1n-1
91.1
p2nn m
p2.--
P2,n
42,n-m
42
42,m
Pa.u-
P3.
P3,n
83
43,m
Pm.n-m
Pm.n-1
Pm,n
4m.n-m m
4m,n-1
4m.
s.t the right most columns of Ans B, both converge projectively:
Pi.a
LA, La E PR, LA = lim
La
lim
n0000
100
Pm.
qm.
and all entries of LA, Lnn are non-zero, we have:
LA U(1LBB
37

---

Proof. Let
(1,1
u1mm- (n)
U1m (n)
u2 (2)
m (n)
u2, (7)
U(n + +)=
=
ug,1 (n)
43
U3, (7)
Um (n)
Um 4m.m-1(1)) (12)
um (n)
With uh (n) rational functions. Given A(n)U(n+1) U())B(n), we have AnU(1+1) (
This implies the following equality:
u1m (n2
q1,0
0
U2 m (n)
22,b
A,U(n+1)
(()B,,,
An
U(I
(24)
Ummm (m)
mm.
Taking the projective limit on both sides of this equation yields on the right:
91.m
92.n
lim (1)
= U(1LLB
72+000
:
Gm.
On the left, we first need to claim a few things regarding the growth rate of solutions to the recurrence represented by A(n). Using Poincare Perron asymptotics, we can claim that there are m different canonical solutions øn with different growth rates. Thus any solution is linear combination of
these canonical solutions. For solution Tn Σ agan
(4) we shall call the fastest growing canonical
P1.n
(i) s.t + O the dominating solution. The projective convergence of solution Zn
to nonzero
Pm,n
limits, implies that Pi,a have the same dominating solution. Otherwise, the dominating solution of Pj.n grows faster than the dominating solution of Pk,ns then lim Py. our assumption of non-zero entries in LA- Finally, let return to the left-hand side of (24)
Pk.n
O, in contradiction to
u1 (n)
Pl,n-i+ Mi (n)
12 (n)
P2. Maam (n)
An
Mm (n)
=i= Pm,n- Wi,m (n))
m
As we are interested in the projective limit, let us consider the limit of the ratio of the kth and jth elements in the left-hand side:
En Ph.. (m)
Pk.n Em
Pk-n-it1 Ui (n)
1i Pjn--it1 Mm (n)
Pj.n
Eme
Py,n-i41 Us (n)
Pi.
Now, given that both nk. Pjnn have the same dominating solution, we have that grow asymptotically the same in n, making:
Pe i11 Pk.n
Pii 411 Py.m
T7 pain
Ui (n)
Σn Ai Pk.n 14kn-41144. (n)
lim
Ph
lim
lim Pk. Pj.n
Pjo-it] Ui
Pjn it Ui,m (1))
Ps.n
Which implies, by definition of projective convergence:
u1, (n) um (n)
lim An
LA
Um (n))
Which finally gives
LA = U(1LBB
38

---

F.2 Uniqueness
Lemma 5. (Projective uniqueness of coboundary matrices.) Let A(n). B(n) PGLm (Q())) be two rational matrices as in Lemma Suppose further that the projective limit La of B(n) has no algebraic relations over the rationals. Then, if A(n); B(n) are coboundary equivalent, there exists unique coboundary matrix realizing this equivalence.
Proof. Let Uj (n), U2(n) be two coboundary matrices between A(r), B(n). We therefore have B(n) - U2(n + +1)-1 So B(n) is coboundary to We will show that (2(n) is the identity in PGLm (Q(n)), Im. ic. that there are no non-trivial coboundary matrices between (nn) and itself. By construction
+ A(n) = (n) B(n)
itself with Uz(n)
B(k) = U2(k) B(k) (s(k++))* +1
+
Therefore if U3(1)= Im then (a( = Im for any E N. Assuming LB has no algebraic relations over the field of rational numbers, follows that no non-trivial matrix U with rational coefficients satisfies
LB = LB
By Lemma 4 we have that LB = (1) hence (1(1) = Im as desired.
G The Conservative Matrix Field (CMF)
The CMF defined in 21 has polynomial coefficients, which is not enough for our purposes since the T-CMF (Eq. (25) has rational function as coefficients. We redefine the CMF in way that allows one to insert rational functions as coefficients.
Definition A d-dimensional CMF of rank m is defined by a collection of d-matrices M Mx in PGLm (Q(I1,.....) xa)) satisfying the conserving property, that is, for any pair i +
(xI, :i, -i......
Fjr- .
F.) Mx, (X), .........
Ijj : Fa)
Mx, (I1,..., Pi....... Mx. (I1,..., ,,...., I, xa)
Envision d-dimensional lattice where each edge has displacement function representing the "work" of moving between vertices. The conserving property is that: the work is path-independent.
G.1 Trajectory matrices in a CMF
Let M, be d-dimensional CMF of rank m. Given point x E Qdd and a direction v 2d the evaluation map v + (x) describes the displacement from point to point x + v. By the conserving property, the total work for a displacement along broken path, say, first from x to x + and then from x + to (x + v) + is equal to the total displacement from z to x + + w, in terms of the evaluation map this translated to:
Mc+w (. +
We can construct general matrix representing the work going from the point (n 1)u to the point æ + no, this matrix PGL2 (Q(n)) and is given by
(n) = M. + (n 12-11 1)e)
We call this matrix the trajectory matrix associated with the point i in direction v.
Note that by the conserving property Id = M ( = M-)(x) - v), this provides us with the identity
M M..(r) 2x v)
Taking point x' in the lattice + dd with the same direction yields the trajectory matrix (e,) M. + (n 1)v)
39

---

Since z' is in the same lattice, the difference w = x'' z is in Zd The direction w represents the direction from x x'. By the conserving property taking Un) 2. (n), the trajectory matrix from x in the direction w: is coboundary transform between (n) and (n).
In terms of the evaluation functions we explicitly:
Mon (x + (n 1)v) Mo+ (x + (n
(n + Ta,o (n),(n)Tz, 1)
and
Mo+w( (n 1)0) = Mw + (n 11--1)))MMM2 (n 1nn-1y-- (a' - x)) TT, (
Appendix
C.5 descrihes how construct trajectory matrix given CMF and trajectory.
G.2 The R-CMF
The following three matrices describe 3D rank 2 CMF (same as Eq. (6
1
1/
Mx
2z
2
2
1
My =
1
2z+2
(25)
z(-x-pp y+z)
ARV
M2
(v-3)(0-2)
(8- 2)
())=)
(x==)(a-=)
The cocycle M: 2 + PGL2 (Q(n)) defined on the generators of 23 as
Me, Mxx Mez My. Mes M2
G.2.1
Fundamental properties the c-CMF
Let 2 (a 02, 03; 2) represent the Gauss hypergeometric function
(b) nn (c) n!
2 (a1 a2 0z; 2)
Denoting the Euler operator by θ = contiguous relations that are encoded by the Me. matrices as detailed [58]. In particular, we have the following property:
de the Gauss hypergeometric function has well-known
1 1 ) Me, [2 + +1 a3: / 02Fi(a1 + +1, a2 a3:
02, azi 5) O2Fi 02 azi
2
(a1, 02 azi
O₂ Fi (al, 02 azi 2 )] Mez = [[2F) 02 + 1, 43:
(a), 02 + +1, as: 2
[eFi 02 asi 7) O,Fi (a), a3i
] Meg = (21 a2 42, 11 7 O,Fi
02 03 + 1; 2
G.2.2 Example a formula arising as a trajectory in the CMF
Let Letx= = 2 be an point in space defining the lattice æ + 23, Directions between points in this lattice correspond vast collection of formulas for
in Fig 6
For completeness of the exposition, we take the simplest direction es = (0, 0, 1) and show how the famous Euler formula PCF (1, nn + 1)) 2 sits as trajectory matrix over the point in the direction é3-
(
nn + 1))
CM(n)=
1
is the companion form of Euler's PCF
40

---

The trajectory matrix is equal to
(z()) Mes ( 2 2
1 -1 3
(n 1))
4n(x+11
(2n+1)²
8m(24-13
-2n-1
2n+1
(2x+11)
2m(x++1)
4m(n+1]]
Following the process described in Lemma define
2n-1
12n+112
U(n) =
2m(n+11
4n(++1]
0
and get that
(
n +n +3 n+
2n + 5m +
E(n) = (m)))
n22n + 3)
n 13433+43
Is in companion form. We let [(n) be the inflation by Cn n²² + 3n + (see relation
E.4 and we get the
[(n) E(N) CM(n) [n +
Concluding T(n) ((n) CM(n), which states the equivalence of the trajectory with Euler's PCF
G.3
The CMF proves the convergence formula
Recall that the π T-CMF records the data of the contiguous relations of the Gauss hypergeometric function as shown in G.211
For a trajectory u E Z we denote
Fn = 2Fi( +
n. + U2 n.
U3 n:
2
1
1
1
Gn = 02A u1 + n1
+ nI;
2
2
Under this notation, we get that the trajectory matrix T (n) at the initial point = (2) direction is the matrix that represents forward shift the parameter n, i.
2: 1) in
(n) [F. G.. + [Fr+1, Gn+1]
Following the procedure in Lemma 2 we get that (n), the companion form of (n), ) is the matrix that encodes the step going from [F F2+1] to [Fn+1, F242)
(a.u(a)
[F Gml
[Fn+1, Gntil
U(a)
(-+(1)1)
[F,, Fati] Canuln)
[Fa+1, Fat2]
In the case (n)) is Poincaré- Perroon type (see Appendix projective limit
B.3 it follows by Lemma that the
F2 Fr
lim Ci) [1, 22
1=1
where F2 is the solution growing like D2 th where Dua >
Ddel the characteristics roots of the
recurrence equation.
Notice that
[ C2. C,, Id
Hence, the first column of I C., (i)-t is perpendicular to the second column of II' (i). So in the limit, the second column is projectively equal 1- 1). hence the limit is equal to Fz
In conclusion, given trajectory matrix T(n) with companion form matrix C(n) that is Poincaré- Perron type, get that the limit of C(n) is Fx and consequently the limit of the trajectory matrix T(n) is (11. (- for a solution F, of the recurrence with the smaller Ag growth rate.
41

---

G.4 Proof for the trajectory (1,1,2)
Taking the trajectory (1, we get Fn = n. + +n, 2n; 2). The trajectory matrix and companion form are
(4n + 5) 7)
(4n + (4n + 7)2
2(n + 1)2
4(n + 1)2
T(n)=
(16n + 28) + 5)2
(4n 54n++) + 7)(00 + 104)
4(n + 1)2 + +)2
(n + 1)2 48)
(4n + 14n + (4 9)2
0
2n+ + 5)²
C(n)
(4n + 7) 9)(8n (6n² + 21n + 17)
(n + 2) + 5) 5)
With the limit matrix ((oo) being Poincaré-Perron type
C(oo) 0 64
48
Having the characteristic roots (eigenvalues) {24 16 24 (2)] All we need to show is that F2 does not grow exponentially like 24 + (2) 46.627.
24
By the Euler identity for the hypergeometric function
T(c)
1/' -1-001-0)0---11
2(((a,b,,z))
1)0-011(1 - xt) dt.
T(a)[(c a)
We get that
r(] + 2m)
l' (1) 111 {) -(a+ dl.
F(n) r + n2) l'n1)
Setting
(1(1-1)
((1) = log
(+1)= = 111 - 1) -1 - (/) -1/2
1-
so that
r + 2n)
1' ( (1)) dt.
2 F [[ + n) 1((n)
Using Stirling's formula
['xx) 2 t (2)
we find
r(2 2n) r + n) I(n) (V2r n (v 2T m
2TT (2n) 2m 22
4"
(1 (1+0(L)))).
/2T
Hence, the Gamma prefactor contributes an exponential growth factor 4n The function f(t) has a maximal value at t* E 10
= 22,
fmux = log(6 4v2) 21og(2 22).
Thus
max =f(t) -6 - tE(0.1)
Note that although (t) diverges at the endpoints, the full integrand
fー (1 t) n (1 1) -(++ (()))((()
42

---

is integrable for every n, therefore by the Laplace method, we get:
J )()) b(t) dt S (6--)V)))
for some constant > 0. Combining the two parts gives
4 n1
Fn S
(6 - 4v2)" C = poly(n) (24 16v2)",
2T
Fn S (24 - 16v/)" poly(n)
The exponential growth rate is thus governed by (24 16v2)" proving the convergence of the companion form matrix ((n) is to
G.5 Proof for the entire T-CMF and related hypergeometric constants
To prove the convergence for the entire CMF we need to estimate F, for each trajectory (a, 3, 7) e Z We will estimate the exponential growth in way similar to the process done for direction 11) but keep it general and symbolic. This will give us the exponential growth rate of F, in each trajectory (a, (3,7 7). Next, we will analyze the CMF to estimate the possible growth rates from the eigenvalues and show that the smallest one is equal to the estimated rate of Fn we had before.
G.5.1 The growth rates for given directions in the CMF
Recall that the CMF is constructed as a D-finite CMF with respect to bases {2Fi, θ 22 Fi We can conduct change basis to get an equivalent CMF with bases (2 x0rl Fi.. This CMF is defined by the matrices:
v0
M₂
2nt
2x+2
5
1
My
1
a 2y 2x+2
(26)
2 2- y +2) [w-
11++1)
M₂-
29
((++))
[ [r
(--
The matrices here now have coefficients balanced degree 0: When going to infinity in x, y, z, only the dominant factors from the denominator and numerator contribute, at infinity get these three commuting matrices:
1
= M2 y
2x 2z 2
2
1
My
2+ 22
(27)
=(-2-9+2
M2
(xr=)
()
2y
(m-=)
(0- (x--))
These matrices can be mutually diagonalized, set
A x + 6ry
4rz y² 4yz 42² (x + y 22)² 4xy
S = 3r + 22, S2 + 3y 2z
43

---

We have the eigenvalues:
S1
A
Mx
+
{{i,-1 A1,
2x
2x
2
V
My
-
{>2.-1 22,+1:
2y
2y
2(x + y)
2 V
M2
+
-
= =f,,.-, 23,t
2(x 23
2(x 2) 2)
Hence
A1 S A1
A or 20,
12 S 12
= 0 or S2 2 0,
1 Da.t
A = or 2 = 0 or y
The growth rates a trajectory matrix in direction v = (a, 3, are exactly the eigenvalues T, which is the limit of the trajectory matrix (n) with going to infinity. By our construction of the "infinity matrices" M, we have that
(xc) = 3,,0, -,) M,(a,,,7))
If we assume, for example, that
Si((a, () s., S2((a, 8,7) 8+320
We get that the minimal growth is given by
(,,) (a, (, (a, B, 7) (a,
(28)
G.5.2 The growth rates for the F, in each direction
We consider the Gauss hypergeometric function in the scaling form
F(n;x) = (a, b;;,;;;;;))) with a= ag + om, = by + 3n,
c= co + yn,
for fixed real slopes (ax, 7) and a fixed x E 1). As n governed by a Laplace-tyr type integral representation, and takes the general form
the asymptotic growth of F, is
[(n; x ((a,8,n,x)
where B(a, 3, x) > is the exponential base.
The base B depends on which integral representation (Euler-b, Euler-a, or Kummer at x Rs 1) is valid, and whether the dominant contribution comes from an interior saddle or an endpoint. For = these regimes can be described explicitly as follows.
Assume
7 a.
42B20,
In this regime, the integral
(a, [c b) [' (1-) xt) -a dt
[(c)
is convergent and admits an interior stationary point t* E (0, 1).
The phase function is and for the saddle equation 4 (t) has the exact solution
i (t) = 3 log t + (y 3) log(1 - t) - a log(1 - xxx)
-aa B 27) A
= 2 6a.3 4ay + 32 487 + 472
2(7 (K)
44

---

Substituting t" into the integrand and adding the Stirling rate coming from the Gamma prefactors
S = y log 1 - 3 log ||| (7 3) log 7 - 3]
yields the simplified closed form
30 + 3- -27 27 - VA
a + 38 27 VA
(a, (,,8,))
20
28
- + VA)
2(a - 7)1B -7)
Thus, in this regime,
1yv (m; b)1
Bint (a, ",
Which is equal the function Ba,, 3, To see this, recall that
-aa 3 27) VA
2( - a)
Then
S
1sa L3
1 1- = 2(7- a0) a)
2 4(7 a))
where we introduce the linear forms
(s (s= -V
L 3a B +2
12 33 2- +
Lg (27 117--a--1 3) +8
Step 1: LHS in linear forms. From the definitions,
(01) (t" (t*) 11-11 -ン [- B 27) s) 5
(4(7 - a))
(1
LT
(2(7 a
Multiplying by the Gamma prefactor gives
(- 3 27) - $ 57-8
(4(7 a))
771
LHS
Lg
(2(7 a7) 38(7-3)77-
Step 2: RHS in linear forms. By definition of the A's,
L (23) [2(7 - a) 72
RHS (22,
Step 3: Reduce to two linear identities. Cancel 72 and the common factor 8 We need show
| (a - 3 13-2- 2-) - 8) (4(7 aa
L 5
L3
(2(r a))> 3 - 8) (2a)a a)(r)))))) -3
After clearing constants this is equivalent to
(a - 3 -7) s] (8a(7 - ()) "))))22)))))) 2 - 11 Lg.
It suffices to establish the following two identities (both linear quadratic in s):
(I) [-(a - - 27) - 2(- L2
(II) Ln La = aa)
Verification of (I). Expand the right-hand side:
Lg S = (a 33 2y s)( a-8))
B) = 5² (a () 33 - 22) +s(a+32 - -2y 3).
45

---

Using s² = = (a+ 8- 2y)² + 44,3, short simplification yields
L B) (-a + 8 5) = 2(7 - 3) [-(a - - 3 27-) $],
which is exactly (I).
Verification of (II). Note L3 (2 3a B] + -(3a + -27) + Hence
L, (s + 30 27) (s (3a 3 2>)) =²² 3-(3a 3 -27)²²
But
(3a 3 = (a 3 + 2a) = (a + + - 27)² + 4a(a + + - 22) + +
Therefore,
L, = [(a + 8 27)) + 40 - [(a 8 - 27)² 40(a 3 - + 4a =8a(7 a).
which is (II).
Example. For (a, 3, = 21 one obtains B Bint = 24- 16v2, matching direct evaluation.
And for example over the point (,, = (1, 3, one obtains B = Bint (- 297 + 135 L
G.5.3 Proof method
For each regime of the parameters a, 3, 7, one can establish the growth rates of the hypergeometric function and the possible eigenvalues of the trajectory matrix at infinity. When the growth rate of the hypergeometric match the value of the slowest eigenvalue we have by Lemma that the limit is ratio of the Gauss hypergeometric function.
This procedure is can be dome for 2Fi (ao a.-,, by 3 Bn, Co 7. and the correspondent values of with depend only on a, 3 and 7 so using the same CMF for another initial point (4g, bp, (n) one get unified proof that the trajectory converges to the ratio of the hypergeometric functions. In our case, with initial points being half integers, the ratio is always rational function of π but we infinitely many new proving formulas for any other constant that is ratio of Gauss Hypergeometries (such as l2g(2) if we take integer initial points), some values even have a positive irrationality measure!
G.5.4 General D-finite CMF
This above process can be done for general D-finite CMF (see (38)). We can base-change the CMF have balanced degree coefficients and symbolically find the eigenvalues in each trajectory. We can also apply analysis to the D-finite function the CMF is built from in order to estimate the growth rate; all we need to do is see that the smallest eigenvalue matches the growth rate of the D-finite function.
G.6
Scanning the dynamical metries of the T-CMF
Fig. 7 and Fig. g Vvisualize the dynamical metrics of the T-CMF on unit sphere.
The normalized convergence rate is defined (t), where is the convergence rate from Eq. 4 t a trajectory and é! is the El -norm. It allows an unbiased comparison between formulas arising from the CMF. The need for such unbiased convergence parameter for CMF formulas stems from the artificial convergence acceleration that taking multiples trajectories can cause: if trajectory (1, 3) has (exponential) convergence rate then trajectory (2, 4, 6) will have convergence rate 2r. The convergence performance of these parallel trajectories, though, remains constant when measured by the normalized convergence rate. This enables well-defined convergence measurement for each direction.
46

---

8
32
-00
77
Figure 7: The irrationality measure 5, Eq. (5) for different directions in the x-CMF.
Figure 8: The normalized convergence rate r r/L (t) for different trajectories t in the CCMM
47

---

H LLM formula equivalence detection and proving
LLMs were independently asked whether formula pairs proven equivalent by our system equivalent (equivalence detection) and subsequently tasked with proving equivalence (equivalence prooving), regardless of the answer the first question.
H.1 Test formula pairs
The following equivalent formula pairs. proven by our system, were given to GPT-4o and Gemini 2.5 Pro Preview for detection and proof: (1,2), (3,4), (1.5) from Table (9,10), (63,66), (75.76) from T and (82,84), (101,103), (120,121). (140,142) from Table [7 Detection is binary and easily checked, while proofs were checked manually (recall Table 2) The full results are shown in Table
Table 11: Detailed outcomes of LLM equivalence detection and proving attempts. The three segments of the table respectively contain formulas from Tables 16 Gemini 2.5 Pro Preview remarkable proving 50%, sometimes by finding Wilf-Zeilberger pairs [45]. but still falls short of the full challenge.
GPT-4o
GPT-4o prove
Gemini 2.5 Pro Preview
Gemini 2.5 Pro Preview prove
Formula pair
Note
detect
detect
(1,2)
equal term by term partial fraction
+
(3,4)
decomposition
(1,5)
(9,10)
(63,66)
equal term by term up to added constant
(75.76)
(82,84)
Ramanujan- Sun pair from Section S
(101,103)
(120,121)
direct fold by (see Appendix E55
(140,142)
H.2
Equivalence detection and proving prompts
Detection system message:
You are an equivalence detector. Your task is to decide whether two given formulas are mathematically equivalent.
Two formulas are equivalent one formula implies the other, and vice versa. other words, you need to determine whether the following statement is true: Formula A converges to Value and only if Formula converges Value B.
You not required to prove the equivalence- only to determine whether the equivalence appears to hold. Base your decision on mathematical structure, known identities, or other relevant patterns. Respond with clear answer: Yes (equivalent) No (not equivalent).
Detection user message:
Two formulas are said be equivalent one holds if and only if the other does. Specifically, Formula A converges to and equals value A if and only if Formula B converges to and equals value
In other words, the truth of one formula guarantees the truth of the other, and vice versa. Given the following two formulas:
FORMULAA VALUEA
FORMULAB VALUEB BALUUEBB
Are these formulas equivalent?
48

---

Proving system message:
You equivalence prover. Your task is to determine whether two given formulas mathematically equivalent. That is, proving one formula should be sufficient to establish the other.
In other words, your goal is to rigorously assess and demonstrate that: If formula A converges value A, then formula must converge to value B, and vice versa.
Do assume equivalence--jusity it. Use mathematical reasoning, transformations, or known identities to support your argument.
The proof of equivalence should be rigorous and detailed.
Proving user message:
Two formulas are said to be equivalent one holds if and only if the other does. Specifically, Formula A converges and equals value A if and only if Formula B converges to and equals value B.
In other words, proving the validity of one formula must be sufficient establish the validity of the other.
Given the following formulas:
FORMULAA_ AALUUAAAAAAAAAAAAAAAAAAAA
FORMULAB VALUEB
These formulas are equivalent. Please provide rigorous and detailed justification, complete proof of equivalence.
I Formula harvesting details
L.1 Article retrieval
arXiv's search API not reliable for retrieving papers with formulas. Some simple queries such as "formula for pi" or "æ formula" returned few results (and if the search method not set word-for-word results mostly irrelevant). Not knowing all patterns in which tends to be calculated in, we went with a more exhaustive approach.
455,050 articles from the following categories which were indexed in the arXiv metadata dataset [5] as 24 November, 2024, were scraped: math.CA, math.NT, PR, math.CO, math.GM math.HO, es.Al, cs.LG and cC.
1.2 WEX equation patterns and preprocessing
The following unnumbered inline ITTE math environments were scraped from all articles:
$ $.
$$ $$
A[ AJ
\( V)
math
The following L8TEX equation environments scraped from all articles:
equation
align
gather
multline
alignat
49

---

equarray
Starred (+) versions of the latter equation environments were also collected, for a total of 17 environ- ments.
Equation environments were kept so strings with multiple equations could be split into distinct formulas during preprocessing. Preprocessing was mainly aimed at removing text within equations and setting uniform symbols for objects like \ddots, Acdots.
1.3 Formula retrieval patterns
Table 12: ETEX formula patterns. Each pattern paired with both "Api =" and "= \pi," and the setrac-based variants of the \frac-containing regular expressions were also included, resulting in total of 10 patterns.
Pattern
Python re pattern
Scries
\ sum \s*_[(s::)))) ((** char
Sum of frac
(C2A) s*M fract (ttpcc/shhh...//////////////////
Nested \ frac
(Mracc) s (1 MM*"""ss"0"
((?A) frach (I (M (Mss 19****)**)))
In addition to positive regular expressions (Table [2 presence of any of the following strings in preprocessed data halted processing: sqrt, tan, cos, sin, log. In, zeta, pi-
1.4 Formula classification and extraction
LLMs were run via API- GPT-4o with the OpenAl API, Claude 3.7 Sonnet with Anthropic's API and Gemini 2.5 Pro Preview through Google AI Studio.
As formula is extracted, information collected from previous prompts is appended to the following prompts to reinforce context. See Appendix [.6 for the prompts used. All prompts to the L.MM use a temperature of promote consistency. The OpenAl GPT API supports a return format called Structured Outputs, which guarantees a json schema of choice is returned by the LLM. Formula candidates are passed to the LLM for zero-shot binary classification, into classes: formulas that compute the constant π and formulas that do not. Candidates are then passed again for trinary classification into classes: series, continued fraction (the targets of this study), and other formulas (which are either not formulas are but not series continued fractions). The third class is intentionally redundant, as some strings which GPT-4o mini gets wrong may be correctly filtered out by GPT-4o: out of 3367 strings classified s--ooppting series or continued fractions by GPT-4o mini, 1711 were discarded by GPT-4o, leaving 1656. All examples which were checked manually false positives, discarded when double checked by GPT-4o. Example: g + j=1 2n (arXiv 2206.11256
The vast majority of formulas classified π formulas by GPT-4o were series rather than nested fractions (1,591 vs 65). In some instances the LLM successfully generalized sequences of numbers to polynomial expressions, see Table The cost for API calls for the entire analysis classis--lassiiicatiin and extraction-was under $50.
In the future, additional formulas could be gathered by allowing the L.M to decide whether call polynomial fitting function on such sequences.
1.5 Formula validation
Validating the collected formulas presents a challenge. The safest way to validate that a formula con- verges to the expected constant is by computing the formula (attempting to automatically reconstruct
50

---

Table 13: OpenAl's GPT-4o extracts the partial numerator and partial denominator of a continued
fraction. Explanation by the LLM. 1TЕΣ taken from 46 (arXiv 1907.0022 900.00005).
K
\frac(1 3)55 \frac(2 4)(7 +
\frac(3 5H9 + \fracf4 * 6)11 + ....I
(2 {1} ( 1raa(1111) \frac(5H(2; -
\frac(4)\pi 2}
Rendered
2 (1.1)--11)
Prompt
Identify the partial denominator an partial numerator bn. Extract each of them and write them as proper SymPy expressions as function of depth n.
Output
a, = 2 + 3, b, = (a 2)
Explanation
The partial denominator an the sequence 5, 7, 9, 11,.. which can be expressed as 2 + The partial numerator bn is given by the sequence 1*3, 2"4, 3"5, 4"6,.. which can be expressed n² 2).
and verify a proof in formal language is beyond the scope of this study). As exemplified in Table 14 when prompted for the value of formula, GPT-4o sometimes omitted free constants, multipliers, simple arithmetic adjustments required for the proposed value to match the true value of the formula.
Table 14: A typical extraction by OpenAl's GPT-4o: The LLM extracts the partial numerator and partial denominator of a continued fraction, but fails to find the correct value the formula it collects,
which needs to be isolated through simple arithmetie; the correct value
-1)
2446
75x+ 256 - 75x
ETTX taken from arXiv 1806.03346.
MIX
\fr \frac(2){3} \frac(4)(3)
(4)
\6rac(6)(55 [1 \cfrac(2) + \efrac(1 3}(24 +
\efrac(3 * 6)(24 \efrae(5 * 7)224 }}}} ]
2
Rendered
1-
25
3-5 5
21
5-
24
24
Prompt
Extraction pipeline (Appendix 1.6] with an added prompt for identifying value.
Output
a, 24, (2n-1))(2n+1), value pi/
addresses this by finding the correct Möbius transformation between the constant of interest and the formula's value, relying solely on the algebraic expression extracted by the LLM. Given the prior belief that formula computes constant and that nontrivial integer relation exists between the formula's 's empirical value and the constant of interest, the integer relation found is likely the correct one. The validity of collected formulas is strengthened further when equivalence proofs are found between them.
PSLQ 26
After classifying and extracting, GPT-4o still produces some expressions that do not compute л. The PSL.Q-validatio stage removes these false positives by finding the precise limit of each formula in terms of æ. Of the 660 candidates, 147 not Diophantine formulas (e.g. they contain æ itself in the series term), leaving 535 formulas. 150 of these did not pass validation, and many of these
were formulas that do not compute like
(-1)"
4G (arXiv 1906,0, and
(2n+1)=
51

---

formulas with mistakes like Et=
(arXiv 1811.05831, under proofs.tex). Example for (n 2 " ( (arXiv 2001.081000 (04), as
3x can be computed only when given the irrational denominator.
non-Diophantine formula for ~ (1)
I.6 Prompts for harvesting formulas
We utilized the Structured Output feature of OpenAI's models and used temperature of 0 for all prompts. The return schemas are included with the prompts. LTTX from prompts is rendered for readability.
I.6.1
Classification prompts
Initial classification by GPT-4o mini:
System message:
You are a model that classifies whether 15TEX string is a formula that can be rearranged to calculate the constant
{constant}, Specifically, we are interested in continued fractions and series.
User message:
Is this continued fraction or series that can be rearranged calculate the constant {constant]? (latex_string)
Structured output: boolean.
I.6.2
Extraction prompts
For added context during extraction, the second classification query shown Fig. Be is actually conducted during the extraction stage. Values in source strings collected (see Appendix L.5
System message:
You are a model that extracts formula information from a ВTTX string.
Your task is to:
Classify the type of formula: series, continued fraction, neither.
Extract its components and identify the variable.
This information will be used to compute the formula later, so it is critical that the extracted value and components are accurate ensure correctness.
You will be asked separately about each of the following steps: Step 1. Classify the formula: Determine whether the ЕТЕX string represents a series or continued fraction that can be rearranged calculate the constant {constant).
Step 2. Identify the formula type: Specify whether it is series or continued fraction.
Step 3. Extract the formula components:
For series:
Identify the term and the start value.
For continued fractions: Identify the partial numerator and partial denominator.
Step 4. Identify the variable of the formula: Clearly state the variable used in the formula.
User message: step 1
52

---

Step 1:
this formula a series or a continued fraction that can be rearranged to calculate the constant {constant}?
(latex_string)
Structured output: boolean.
User message: step 2
Step 2:
Determine the type of formula.
Is this formula continued fraction series?
(latex_string)
Structured output: 'series' "ef" (for continued fraction).
User message: step 3
The prompt in step 3 depends on the classification result.
If formula formula_type type == 'cf' (continued fraction)
(ITTEX for one-shot example taken from 46 (arXiv 1907.00205).)
Step 3:
The formula a continued fraction. Identify the following components:
1. The partial denominator (an) as a function of depth (n).
The partial numerator (bn) as function of depth (n).
3.
Any unknown variables or expressions (other than the depth n).
Write each component proper SymPy expression. For example:
The string
2. 3) 1.
22-++1
Vz V
2:2
18
has the following:
an: *3*n 1
bn: 'n* (2*z (2*n 1))
unknowns: ['2]]
The continued fraction is:
{latex_string)
Structured output: ('an': str, 'bn': str, 'unknowns': list[str]]
If formula_te tapee 'series?
(ITTEX for one-shot example adapted from arXiv 1806.03346. Note the adapted formula is incorrect the unknown was inserted at random.)
53

---

Step 3:
The formula is a series. Identify the following components:
1. The term SymPy expression.
The dummy variable.
3. The start value of the dummy variable.
Any unknown variables (other than the dummy variable).
For example: The string
22
(-1)"
2-
24
7
(2n 1+ 22n +2 2) 3)(2n + 4)(2n 5)
n=2
has the following:
Term: '(-1) **n ((2*n + 1) * (2#n + 2) (2#n + *2*h 4)* 5))*
Dummy variable: 'n'
Start: '2'
Unknowns: ['z']
Pay attention to special symbols like _symbol (c.g., \frac12) _n)__n which often indicate SymPy Risingfactorial. Another symbol to look out for is H_, which often means SymPy harmonic.
The series is: (latex_string)
Structured output: ('term': str, 'dummy_var" str, 'start': int, 'unknowns": list[str)).
User message: step 4
(L5TEX for one-shot example taken from arXiv 1806.03346)
Step 4:
Identify the variable used in the formula. the formula is a series, focus on the variable used in the outermost summation. If the formula contains nested summations other variables, ensure you extract only the variable from the outermost summation and exclude all others.
For example: The string
(-1)" (2n + 1)(2n + 2)(2n + (2n 4)(2n + 5)
24
n=2
has the outermost summation variable: 'n'.
Extract the variable from the formula: (latex_string).
Structured output:
L6.3
Code-correcting prompts
After extraction, each of the code components extracted in steps 3, 4 from the above prompts went through an execution test. Faulty code was sent back to GPT-4o for correction to SymPy code that runs properly. A total of only 16 code-correction iterations were needed during a on 847 classified formulas, a testament to the L.LM's s ability write executable code. are confident that this stage could be removed in future runs with minimal consequences to the size of the formula dataset. Since
54

---

the number of corrections is so low, the cost of leaving this stage in is minimal. In short, the correction stage is largely insignificant when using GPT-4o to derive executable code from ИTT
The following prompts were used in the loop for up to three iterations:
System message:
You helpful assistant tasked with extracting mathematical expressions from strings and rewriting them proper SymPy format.
Your output must be valid Python code that can be executed without errors.
Always focus on processing the original string provided and ensure the response contains only the corrected SymPy expression, formatted executable Python code.
User message:
(Includes the Python error message from a failed execution attempt e.)
The last attempt was invalid SymPy code: {str (e) [:400] }.
Last attempt: {string)
Task:
1. Extract the expression from the original string below.
Rewrite in proper SymPy format as valid, executable Python code.
3. Only return the corrected SymPy expression, formatted valid Python code.
Original string:
Process the original string and provide the corrected SymPy expression.
Structured output: str.
55

---

Some results also discussed in Tables of full results
This section contains the full list of order-2 canonical forms (PCFs) derived by our system, grouped in equivalence classes discovered by running the matching algorithm (per Appendix C.1 Formula sources are shown Table [5] Tables T6 and list the formulas that have been unified by the CMF and those that have not yet been unified, respectively. We believe wider scans of the CMF (Appendix CC5). with deeper, more complicated trajectories, will unify additional formulas from latter. Enlarging the T-CMF to higher dimensions and rank will also likely increase the unification percentage.
We invite the reader to explore the formulas and algorithms leading to these tables, available in our project repository httpps://ithuub.com wwwoo///aaaaaaaaaanneeeeeile the online algorithm demonstration [4].
The supplementary material accompanying this paper contains two json files:
pcfs. json: The full dataset of order-2 canonical forms (PCFs) harvested, along with their sources. 149 PCFs in total.
Columns:
ab: list containing on, b, of the PCF, both strings.
a: just an-
b: just bn.
limit: string of the symbolic limit of the PCF terms of n.
delta: the empirical irrationality measure of the formula. Eq. 5 as float.
convergence_ratee the empirical convergence rate of the formula, Eq. 4 a float.
sources: list of dictionaries, each describing harvested formula that led to the PCF when canonicalized. Keys:
* type: 'cf" or 'series".
* formula: as a string.
* formula_lime limit: in terms of æ, as string.
* id: the arXiv id of the source paper, string.
file: TEX file in which the formula was located, string.
* line: exact line number in the TEX file where the formula was found, int.
* equation: the source equation string.
cmf _pcfs. pc.pc.sssoon A dataset of order-2 canonical forms (PCFs) sampled from the 7-CME 1693 PCFs in total.
Columns:
same as pcfs.jsoo except for
sources: list of 2-tuples where each tuple contains a trajectory and starting point in the CMF, both lists. Generating trajectory matrix using these (trajectory, starting point) pairs and converting to companion form yields the PCF (Appendix (.5). Trajectories are lists of ints. Starting points are lists of strings for easy storage of rational numbers ( stored as 'a_b').
56

---

Table 15: Formula sources. Summary of papers from the literature containing formulas, showcas- ing the success rate of UMAPS (colored indices). Canonical forms are shown in gold if connected to least one other canonical form by UMAPS and in cyan if unified by the CMP: 98% papers with a formulas had at least one formula connected via canonicalization and UMAPS (gold or cyan) and 50% included a formula unified by the CMF (cyan). Black indices represent formulas that remain unconnected to to other formulas via UMAPS. See Table 4 for statistics in terms of formula counts. Formula indices are consistent with Tables [6]17 Canonical forms 150-153 are not included in this table as they are order-3 recurrences (see appendix(B.6
* Tabulated formulas from the data provided in this paper were added manually.
Xixiv source
Canonical forms
arXiv source
Canonical forms
arXiv source
Canonical forms
1907.00205*
2, 13, 20, 21, 22, 28,29, 30, 31, 48
2305.14999 14995
44
1807.07394
96
2308.
2308.11829 11829
2307 05607
44
1906.07384
96
2307.03086
6, 96, 107,
2307 080063
44
1908.05123
96, 126
137 7
1407 8465
2312.174012
44
03058
96, 126, 137
2204.08275
8,9, 10, 11, 38, 39,
2409 06658
44
2310.04642
96
40, 41, 86, 87
2405.02776
12, 130, 144
math/000614]
44
1008.3171
106, 113
2212.12361
14, 17, 19, 24,
math/0206177 779
44
2001.08100
108, 137, 140
25, 26, 27, 34,35,
36, 37
0707.2124
16
math/0402462
44
2048
113
1507.01708
16, 118
1433774367
45
1302. 2892
113
2112.00622
16
2105. 11771
49
1910.04328
113
2211.11484
16, 126 42
2206.085 0888408844
49
2103.0 07837
115, 116
2310.03699
1601.03 001180030000
51
2203. 2203.02631
113
1806.03346
43, 45, 46, 48,
1806008411
51
2205.08617
50, 51, 52, 59, 60. 61,
62, 64, 65, 66, 67,
68, 69, 70, 71, 72
0707.2122
1209.5739
2208.07698
0707. 2500
44
2411 100200
55
2305.04935
113
0708. 2564
1504.01028
73, 126
2108. 1108.12770 12796
114, 115
0806:0150
44
1804.082 084
73, 74, 78, 79
091 0911.241 1.2155
117
0807.0870
44, 108, 113, 118,
1805 06.6866688
73, 74, 78, 79, 81,
1103.3893
L17
120, 126
148
1206.3431
44
2005. 04672
73
1110.5308
117
3657
44
2111.10990
73,
1804.00794
117
1301, 2584
2204 04535
73, 75, 76
0708. 0708.3307
126
1302,0471
2403.040 .09444
77
1210.0269
126
1303, 1856
1708.00 042269
80
1330259944
126, 140, 146
1310.5610
0712.1332
82
1303.6228
126
1406. I 168
0911 5665
82,
1510.02575
126
1501, 005457
44,53
1203. 1255
1808 03213
126
1511. L08568
44
1302.0548
82, 96, 126
190107962
126
1602.00336 00336
1610 04839
82, 107
1909. 10294
126
1704.02 024.02498
44
1611 1.0217
82, 96, 126
1910.07551
126
1801.09181
1911.05456
97,
1912.00765
100, 103, 109, 111.
127. 128, 136, 138,
141, 142
1802.01473
2110.03651
89, 92, 93, 94, 95,
2003.02572
126
96, 98, 101, 102, 104,
105. 110, 112, 122,
123, 124, 126, 127.
134, 135, 139,
1802.01506
44, 126
1101 0600
87
2101.09753
126
1809 00998
96, 87
2109.09877
1812,06643
44
math/0503507
2203. 16047
126, 127, 128
1906.9662
44, ST, 106, 113,
09966
88, 99, 119, 125, 133
2210.01331
1907 04089
44
0805.27 22788
96, 126, 137
2301. 12932
126
1911.
1103.6022
96
2303.05-402
126
1912.03214
44, 51, 147
1104.0392
96, 126, 137
2310. 15207
126
1912.03527
1104.1994
96
1808.04717
128
10774
44
1104,3854
96, 126, 140
1501. 06413
129
2241.12412
44, 96
1410.5514
96, 113
2305 0002660066
131, 132, 145
2105.05809
44
1504.01976
96
0704,2438
137
2106.04517
44
1512.04608
96, 108, 126
1004
137
10,00755
44
1604,00193
96
18021
137
2203.09465
44, 53, 54, 56, 149
1604.01 106
96, 126
0909 2387
149
2206.07174
44, 51,
1609.07270
96
mach/0502582
149
1312.13687
1804.02699
96, 140
57

---

Table 16: Formulas unified by the Conservative Matrix Field (CMF) as shown in Fig. 6 Clusters of formulas harvested from the literature are given in terms their corresponding trajectory the CMF. Dashes indicate a formula's ss canonical form (CF) is the same as the formula collected. Each canonical form has numbered row and some are followed by additional source formulas.
(CF)
Convergence riee
1,2)
-0.21
+ 1,0²)
1.76
2
PCR(2n + 3,0" 4. 2m)
PCP 3,x² 201
1,76
3
PCR(2 + 6,n² 4m)
PCF + 6,* 4m) PCF(-18H 1080 70o - 6dd 96n 12r 32n 15x²)
1,76
4 12, 26-6 96m + 12n'
PCF(-18u" 108m 70m
30
3.53
32n 15m²)
(3. I)
0.45
(-4) (2
5
6T 10
PCF229120m?
3.29
110616n 132106n + 198 338888800
1739440480m
822863792N
252606312r
62970828H
9173682
2336310)
10tov- 2011
(2))
15x
PCF(99008 +
12876099--119411
225
3.29
3094 427970n
2379613504m
5231 138152
48004214444>
1334845540rr
61614540)
Ee 11
7
PCF/728 + 2822n
3.30
M669n 1360,21168p
89208h 130380 +
78570a 15162n 1458n
390)
I)
-0.18
Σ
(-2 12000022000
2w
PCR(1761 + 8596G
2.07
2053,5080132n*
48521
16365000m
29955532,
29379250n'
188858610
3159708m 203361n)
9
2
(-1)
164,1152,
PCT(8442 328. 1 41le
10
2.08
6968% 4230n
838 950 33)
(2)
2
PCI(420 202.
-1472-08m
12 - 44,28800m
960n 573352 8096
27706 + 4097m 2503)
11
2CF 1004 4
2.08
126 11
1321 591. 16368
58176A 116024n
10088445 384580
5813
12
PCF(13977601
4.16
126- TOUS
111047688
647/00400
(3)
366577924
(4),
650070104
00085140."
394378122, 201001231
24573901116n
308930056832
1005500611000
282238639101
AG6850093444 4n
775T3868371
1178920116
203961241724
19923
10648 1638,
18151699l 8419350)
58

---

Cluster
Formula
Value
Canonical form (CF)
CF value
Convergence
0..0.0)
-0.65
13 PCF(3a + 1,-2n² +)))
0.69
PCF(3 2,-2
PCH(3 4.-21
Eis1 7py
9
PCF(3 + 4,-26
0.69
E A(A))
Ecso 1p
17
PCF(3a + 5.-2x 2 5a
Sr+30
0.69
Ee [-)
2'
PCF(30 7.-2n²
12-45
PCF(3n' 9n 5,-2n" 9n
On + 3)
PCF(3a,-2 30)
PCF(3n.- 20 3-1)
PCF(3 +3, 3,-2n
PCH(3 3.-2h+
PCF(3 4,-2n=
0,70
24
PCF(3a 4. mn)
0.70
25
PCF(3 +. +))
-Sals
0.70
26
PCF(3 + 5.-2nn
0.70
27
PCF(a + 5.-2n -3x) 3m0
0.70
PCF(3 + 2. a +)
0.70
PCHi3
30
0.70
31
PCF(3a +5.-2n² +
0.70
32
PCR(a+
+e+
3
0.70
33
PCH(3 +. +6.-2n 0006--5.. -5n+3
0.70
34
PCF(3 + +,-2nn -3n
0.70
PCF(3a + 7.-2n² 5n++3)
0.70
PCH(3
PCF(3n
9, 5e
0.70
Σ 4%
PCF(20 86n
123a
+:24
1.38
59, 64n 416p
1004, 1090 504n
(1111++1)
39
Een
(1)
3
PCR240m 1320m
2792n²
16112
1.38
9216,-9216. 78336.
205020m 450096
113725 196792e²
1764)
40
PCRGDA + 214x
30
1.39
237m 80,- 80,667
2208 286th
1330
+37
411122 Een
(2k DN
+
984n 321,-9216m
PCR240m 884.
9:2
1.39
43008 65536n.
31904n 4900m²
691 In + 1140)
3,1,3,3)
-091
42 Σ
ksl 3)(4 (1
16 (207-177--)
2T
PCR37884' + 15292.
0.52
21230 + 1198In
2304 3345408m
13220568r,
16690200r,
191 2008 + 4-1111111 +
2792250m² 27384n
173304n 20620)
59

---

Cluster
Fomula
Value
Canonical (CF)
CF value
Convergnnee rale
(1.1.1)
1.00
43
PCR(A, -1
01.00
44
Eis 2043
PCR22,n" + do + 17
4-
0.00
+ So
46
PC 4
0.00
47
48
PCR 1)
0.00
49
PCR2,44 + 12n
PCFREA4 +
51
PCR(0. +
2
Em
-0+-
6-
52
m
PCHWAA + 16n + T) PCR33 640
01.00
S
In-olttn-1
38.2250n
68n
PCRSSD 800
258n"
55 2
PCHAEE 90c
01.00
25--2 3026n
1504 9000 2353
56 E Ion MiSEEty
PCHBEW 96c
01000
D10, 110-2000r 206 1124
1504 900 225)
57
-3+4
PCRG2 128n
0.00
134, 134-2066" 1536
B424 03000 1225)
58
-2
PCR 128n
0.00
186, 2562" 15366
3424. 1225)
PCR(8,47
2nd
PCRRAN -1)
60
PCF 6-8 5)
0.01
PCR(10.12
62
PCT
0.01
Emsl 02
PCR10.6 +
100
64
PCH10 12n 7)
0.D1
65
PCR1000= + 20n +9
0.01
66
Em
PCR10.4 + 29n 45)
0.01
67
[m-1l) (ine (inaay (lnkay
PCR + 80,16e
128 +322" 22th 49)
0.01
68
>
[m--1) (2n+ (2053) (Dnesy (D447) (2443)
PC 16816e -25-000----5--------
001
PCR(2
70
PC 288
0.02
256m L1G4 960n 225)
60

---

Cluster
Value
(CF)
Convergence
-1.00
PCR2..))
PCR +
0.00
73 E
4
PCT(Sn" | 16c 19. 16n
0.00
32n
20n
E
(())
$
Σ
4
((H)...)
EN
En
4
(
PCF(S
+-20
32
13. 16m
48m 36m?
8n)
En (1+3)*
-4+ 16
PCF(Sn +
20
0.00
17. 166n
48n
52m
4)
4.
[- 211
-m
-4
PCR
19.-16a 64m 92n
56p - 12)
2 (613122 21l
-ite
PCR(8 28n 27.
14
72
0.00
140g 100n 24)
(())
2m
-5 18
PCP
36m
16n 112n
2926 336n 144)
(())..) 2
Em (n+1)
250-8im
PCR(8
-18
57, 16
128
372 468n 216)
(G) 1W12 10H
2i
25
PCF 44n +
0.00
(Bee 17 12 (29)"
65, 18n
144g
484n
(G))
101--511 100r
PCP(S 44m
0.00
733-166m 16n 144n
MEM 2000
324)
61

---

Table 17: Additional formulas for that were automatically harvested from Xrv,,, clustered, and proven equivalent within the clusters, but not unified. We expect these formulas to be unified in the future, by enlarging the T-CMF (Eq. (25 to new dimensions, i.e. adding matrices and directions to walk in, and by improving the matching algorithm. This table complements Table [6 the formulas are mutually exclusive and were collected using the same harvesting pipeline, These formulas were then connected among themselves as described Appendix C.I resulting in clusters.
Claster
Formula
Canonical (CF)
value
Convergence ralo
6 = -0.29
11155660 (310000 H. (l..))...
82
PCFF63421 62825600
13.56
16906116319666 196
168051 27873128
81808 183
27955 4091 15.3968767994229601000
59868003557 50055333335500355000
3422186-4211 DOKG) (584
(-1)7s52 (a (H (1.
(-1pssarttt H), (3),,,
mobor partee"(aaann (2100011120 (+),.. (1)
2
(-1pasee 11 (3)
[-19m18tas) Yaratang -oo----------- -0000 200 (C
83
[00 110
13.56
206 0886631 96i
2600901 122
6208537411978 197732
1554771695156
0851272114451300 117002211770008888888888888 12...
101 1015589809999-55 885596r
28
176807 1582009872
13072527756 (92477918888
76 90788832222 "
716951 3765967766339000
411 281-220042229900322
[- 1-1801886411 14447795480 -at.ae-------00 -15100000000 10scassoo) (1))
84
11-
13.56
19904733639658222
5967677271 625633322
5-111 466 103761115 1,111,1111
1501 11
578122 3621 909137777-455
17777777
10 809927897889-22 71
190228888 262 779077883111 2500
658926 2042965550 900Sm
1219
155908
1599881200799
1TA.:/220007908816656568010 (560m))
85
De (-199148500)
1961 1.861740880 (0017700019666 9654 71167
7258021)
13.56
PCRT 072
46238 180025509922022
4230037022 41 1209857504
801 BOTU
365
156579 211.22291222..1
4409957 7788002198 98199991990 1299 17 1718399500 100
1203ST 2285828829622553 2122222627777777777 12
40T BUTTI 1278
2233673440 4007 4D0SS ASD 0000-----00000-6636666 36
235T 228722833677 4 965066227 47720000
104803--6
0.29
86 E
PCF(1450* 517W"
2.60
572a 188, 1330g"
4995m2 -8186n" 81866'1-1-20000 2505m
+
(2)
PCF(725H 713
160n + LL-33700
22r
8T76m> + 635664
150576" 9007262
4844k 3696)
193
62

---

Formula
Canonical (CF)
Convergence rale
0.34
88
Σ
22-1 Ma
1-4
PCR21810M + 261512 +
4,16
1520976n
3480988n
5103230b
1255500 7226344n
1281638 Alm 10
475 37510
11575171161v
160228 15088
12901220
1--556391022
605 138 1960
1147502250n
94394500)
-0.0
Ees
82944) (2428400K*
50449688
2584521) (2) [
PCR225004224000
5.77
210000 1121
68107991504m"
16780867552xe
1692088262c
0600881521 19669220011 5705680000
8502 221 389122 197900m
177182247755
1308U889 1803000
241612211 U22G5 1552250
19972882961 1930m
1367 (0672044552122522441
G21 63 110-1000---60-0000000000
(9)
PCP211224351
51380 -00
5.78
34-737922n +
3221
81170044 1648852n
2945321633444
8678002208488
5388 5008NITION
12966689792
500626557440m
235884433/200
13011151228
(12040) Capzne" + -a00 ( (2)
91
2000
PCR331220444
2000-RITE
5.78
340624196m
222122284
116510516
1833705724211000
119-18709981 li
299066266222666
129696000
45.2908 141952n
2024778-44 192ce
2 82044)
(24757408004
2475031 103) (2 (20)
2238840
PCTV255995600SS00I
5.78
307024 1422803552
368257 16295115111 512gg
46008005619 194x
33728 775973
01925-2488 28220444 2013322
3761 986G 3220377
308 152311 100 009778222222
14760288 S5122SS55A500 S000 26122
4663876629 07ENG
9636 1151119999222222 2000022331112000 102567 72
19620 222 1907 77377777333
61 9292666 11 12221222251 109 (367 7701
2979947 45877777777 7007 00702 400 60066m
1802 1312233112570000 2750003611
63

---

Cluster
Formula
Value
Canonical form (CF)
CF value
Convergence rate
= -0.50
Z 1090-Aga (1986* 4252 210)
(2)
PCT(0296002
2101216 512 307
4.15
2760000
61356n 22843n
4600. 100579584m 19
67550908
026060400n
1278275584m2
21288622188
212229376'
106499044-0n
441 2622880e))))
Eis 4096
(554412
119002
58500)
(t))
Ei-o 1096
Agt
(50422
3146
11)
PCF22225002 4
4.15
1419280m" 2902704h,
2018-1577
28144166"" 1 1289208p
10
8587837410ma
28095-1911221
443923251 120m"
21396893996
172765002061
6864182784
209392128
95
Σr 1096 (AP + (12630442
921331k
109205) (26)
1068412
PCF00782080
12202 1003412 A2TGL
4.15
279217360m
1292128016H
18169260-8mz
1980776344w
1616809667
57721030, 636-66636111111111111113-
35647:5166152081
181961111115225
5920083395661850
7800 33774976x 10
69010738 10324480
582619
232 22011223311000000
599 9506740 9328
130007 508852222
65377554561552266
96
Me 1096 (126 + (3))
PCR21840: 67952.
10
4.16
73008N" 295080
2007, 28077-722550 7225344w"
12558336G 676544
168
890368
EN
, 2-120 (42n (20)
D
13 (12n (aa)
En-D
4096
+ B) (2n)
D
2000 3mmm 2i
Pa 19
((E))))
15
(42m (1))
(
18
(())))
(n+
+) (1))
ve
come (260 12- 120
97
PCF(1-1560; 41261H
4.16
30272, 6488%
d-
3578n
221124222 10
2523136x + 69301322
4874240m" 39772166
2521088 + 1328
36541661
64

---

Cluster
Foemula
Vilue
Cunonical borm
(cOOL)
131-17
PCR2218000 + 631200" +
S6976or' + 153d8n
-27-43
1922
4.16
2065
590 75
118701208
77306822 21G102886
5676632,
703545667
19.5
4.16
12802 + 121776'' +
311766 6834n
1305, 2500
11008177
0794670H
30
32111344n'
10003:20n.))
000 (1ea 1
100
20
4,36
206288. 507964m'
7084 4111000
195165, 126-15056m
5007172000"
4797664500
28072265880
7862 0211088n*
$58400n
120520m)
2.4-0
1998
PCF221800m
4,16
566370m
155800 72236m
603832226
21061 8016
3500613544
4470261 76
2900177
25100200
10736844 82422))
102
Exto 4096 14(22022 1) 129
4,16
20760S0m
27.77666
180033000
1888051 200
6384728522m
205 si
2031 403 4690636
24973633024:
110062201224
2316787072
103
PCF 8400
3300066
4,16
1590 d5611ddn
63377200" 42N2764
1126 92
5219450SS00
2384
5082508-4
3098021964
1071358416442
1861616in
211061766)
000 "0022205" 10000--0000000000000000 (759
312222
PCRTI0EAAN
4.36
1535018 18
889351 20n
253021 Talk
3706-18377060.1
185-65471 6259
16200 0E2097632002 7632928522222 12
2N 200611152
12092-4-677
5213461 6228000.2
700091960 16626)
65

---

Chaster
Formuts
Valoe
Canotical form (CP)
CP valse
Convergenee rite
-0000
(sorL)
(32 08-
PCIC
16
ST 1o
12504125 141
20285 3000
372500006/771
1630 552
4050187 (6) -10 00500200000007020
(2
56580411 86900544k
3001 T1S188333
338547 23005000001
38T80T
1263161021
778m
100210002557
106
2
(-1)²
10100
2-35221
5821651 5200005-
56000001 1001
3052411 22 LIS2USSSS-
2000000 GITSI 01500
71315583622
250836852500 20000777 141 AOTT2 2222216220 LOSOn, 00000220000000000000000000000000
LOLST 4ERAA22HI SIDIIOIOOOOOO DOODIICCOOO 0000000
65CS SN SISTISS MITIER 19 00000
2 TOB 235000 OI
SORS 7T177HI 295 10000000000 (1000
SGOCUCIATE 1ES2TTTTII 100.O (000000
D1 3EE-DR--AGEEEE 6A6TU120IION 1O000000000000000000000
11.5 SANS 233-- TIS PESCOCO0OO
7230 15 2U21
77543360 S ASTTOOO LAS
11551133 40G LOO SASSAAAASSEE
4177897 NO CAN
Mil (-1) 200
0.51
100
3.30
120248
27004407
1200644 2073000
72000N" 121920
SUT 12 HOT
200
532501 10p
4860044004
SUSO TOs
2500 190A
5350/900
17754H")
Zanet
66

---

Cluster
Formula
Value
Canonical form (CF)
CF value
Convergence
-0.64
108
(416n W3n)
55 + 12132n
27 Su
2.60
13017a" 5354n
620,-991150n
1738665n" 137781n + 1151091n" 6889799
+
107406x²))
Em 2- +
(20) (
10F
(160
En
1458 fim 1 76-0) (2)
-m
(26- 133
PCF289771 + 82089
4500
74487n' 17735n
4602n
872, 572.-33992051n"
33743223,
10133002
66518334n
58348674m
31006143m
5149170mm
2770200
110
Eimn 1458-$1 (90022
2097k + 929) ( (())
PCF2234900m2
131876100
683883n + 611136m
60813n 136828n
35-24.331122-000000
13804880000 "
4483826766n
00519719045
52000563327m
28416937500
192661 42005
4537514700m')
(10244 233)
111
E
PCF(397664r;
9005250
2.60
1075-1050N
1757 15x 2799174m
2058011n" 75503-1n
130501.101158000
17061598300m"
13192053522
244441 4698851
134861 45949m
126802379077"
65621 8897 In.
11305-1378m'
579595821n")
112
EW 1458-2 (560256.
112584k
56551)
(222 (2h)
RC211462552
2.60
18911
85526387 In
213672 19
19975670966
137291 !
254250924752175. 275.2
208732773 723000
00001733075 193332
3100625 192211922200 1932163223330003 4
6492528614418 41963999
9302 1331131111100011
9005 100316666655688888
0836-00524549673000
20596254 41037334336"
933663777611
67

---

Cluster
Formula
Value
Canonical form (CF)
CF value
Convergenoe
-0.68
16 (1000"445111444"
113
000
PCR1144480n +
-47+15m
7LADUN
2.77
7458304x 21619136x
32438504n +
26156230n 11070645m
1852434, 60397977600n
3935591487560m"
10397302
140627638272
97 1112205110n"
41226947481677
17011 470566 +
1280 1 4877184n" +
25005952192v
18025198724
2768834448n
391128960n
18316800)
210h 16
(-
8446
8243
8FF4
8E4F
2xmo 16
+
96+4
54+5
96++4
86+1
Em 16-m
8m-+6
5m++5
8m++
Em 16-n
8m-6
Braed
2h-02
2648
8645
8C42
8C+7
16 10-"100
"10m= 1000m-7nmm -
(1. (2)
E
PCF(34S160n
2.78
18-07
1270272n + 1612608
788632k² 85322
12295n +
67148-0010884400;
90596966100
1116104294477
18061866328m
3976336800n
11906629632n
1939392192
2448268800m
933823680m
839872n 334622722. +
12799688
391680)
10-2(1200 212062 +22306
3), ( (1), (H).., (3)...
ΣE
(8) (+)
6
PCFC1014480
286025 22-17
2.78
1009-10804 +
39436736m
792839 12
472728000
101 013-5644.603999777000
1385
1946303556640n
361 198803200
352 55244003300
1407537070080
3544994965504mm
5336091 75552
235379122120xz
20669097856m
239773233
2291 02911475166
109831680)
16 16200 (1200"=2000-000 +00
(1)
(4)
(2)
68

---

Cluster
Formula
Value
Canonical form (CF)
CP
Convergence
-0.68
Evso 16- (- 8647
116
2x
7C9(1) +
2.77
31889-108m
100376832
167914100n
159965424x
85913231n 23812137e
2611170,- 916666616066611
713501442045mn
21100999886666m
30977820721 52,
19361501 806592n
5089203585044
162965538577244
957 29861 58088
722307942400
177010890288
932890659901
197879557824n
13151503632n'
1452925152n 199825920)
== -0.71
117
PCF(20n + 111
1.39
2++1
25.-64v*
96n
18n 8n)
(
Σx
2641
118
18
-1+3 4
PCR(200" 848
72x
1.39
80, 89.-614n 352..
648n 216)
>
-1+
119
0c
4
PCR(120m + 460n
88m
514n + 230
7296n 4480.
2880n 1264xx 261n)
-0.74
120
(-1)" 1
PC + 11296R +
d1dm
1.39
10
2501 25646
11905n
1962.1638:10,,
8155840m 15609856m
116211187. 37668
5753472, 3107120
303108m 22982882
74472x 6624)
69

---

Chuser
Formula
Value
Canonccal form (CF)
value
Convergence rale
6 -0.74
(cont.)
121
De 16
2.77
21 653 680055/8m
151471 57:50225
1774561 1598 600m
1688G1
7048361
174547215 SS
300036- 10 LOGAST 2000 4a
I L6GOOT TUADDIN D4
20 4747788 2009012757 4062
4IPOSETH 20-220S-
153766440
101222001
405 BODSS 1OERR
2019 ASTIRATT DOINS LITT
A72N0S06ROO ORGTT TOON
1802285R2SAACT SONI TES
11 BETEEEEEETTTTSSSEE
127 12 7O7TE3S LTO Ti
51300- EEGOS
7UNTATURIEE DONNENN 217 (05458558-0000)
Dz 16
120
Dr 256 422 (017
(21)
PCHE/DW/ CURN
1228
198080 +
619626 186752e 74204 4147216")
60086
De 162 (1921P
103) (36
PCF12680% 1840n
1.38
11056 $4504n
105848 9000En
87822
1570 9137--9577
4738252811 15
13000092 19
1585578266
10
7538-488
647366
357325806n
27114:44226
2476019200
40146240rz2)
124
DE. 256-6 (10608ec 250508
10589)
27206
PCR5541220H
2297558000
3161 65260,
742418111m
65 208770...
3100550445n
07416244
4740638108-441
654772
840-17250H000000.
289675734
67241 3N1
GAL UST 82200'6n
AE AEANNSINNTO TAST UNDOS
1951 PAR8IIATTTS SEO
54305421 NOI B0135288
70

---

Cluster
Formula
Value
Canonical form (CF)
CF value
Convergence
0.75
(cont.)
Lo.
PCT 4 148n 166n+
72
1.39
55.-256n" 896"
832n" + 96n 288n²)
126
Er
250 (Gk
PCF(210n' 736n'
1.39
792n + 336n +
39, 29.-9216n" 16896n 2560n + 9984n
6336n' + 1120n²)
E
256
(6n (2a)3
En
8e
(6n + (
E
256
(20)
E
(Gm 111211
(-1)
(DA +
((0)+)((4)
E
(())))
m
0
(())..)
Ei
256 (111)
PCR(2400 4 + 1280
176
1.39
126-133
2504n + 2120n 651, 651,-92161 47616n7
94720n 9190dn
43840x 7712n
896n + 352m)
256 (122"-1)(77
128
Et=o
(22-1)
PCF(480n" + 1298n
2
1.39
1083 208 94n 21, 36864 20 18432h 89088n 4454hn
50944n 25472n
7744n 3872n")
-0.75
"* (1600=4))))00033
(..,,. (+).....
129
Σ
0
PCF(1365000000
125004400
4.16
6886000000
13628850000m
13283880000m
646 64661275900 2
1346342044mz
61 111000117.0. 2522
101 "00000000000000
100000000000000000
370735500000000000
100000000000000000"
545:99200SS000006
125293 19632000000
147198558499F000000
331
201
12000)))
6 01.77
130
Eg
GHDN 4), (9),)
21
PCF(15360n 121088
58500
02 21
1.39
387901x 615552
5885000 279001 +
53940. 37748736n
346030080m
1354330 440x.
3879
31258972160 66
1452979712
30270-1640m
15363632
10035756 41673778n
1802088a
175770)
71

---

Cluster
Fonuala
Value
Canonical form (CFF
CF value
Conrergence rate
-0.77
paa +400yy (1 D. (C)..)
PCE11602224
237788
191
11254433
6263 13271400
77175, 2399289771
27826 SE88th
17617 1770--880
ITS COOOSSOO
341937I L8STTSS
14T 27200002 76
67 47
34597318800
66127 688n
47034M8N0n
0.77
(saomy Lanoas 4520440 wamba (ar7 (()),.. (1t),, (C
(80g (4), (43,,
149 00r
PCEY 28877787777
2:25
56471 427072
5021 521 5750662
1128 1128L5139006166
292524675248 g0m
696835
10224555
1941997
21 4099-104
100877980 131112508,
27534813 42002944d01
248 66227 #006 241
158001 T (1356600665100
780
7:00 L 9115919779925S
270584289K0050 48 7L322094422
SO SOAGNSSGTOTO 385022 & $2577
18056737 495677770 71667722244 (368 50000
36401 LITTLEA
57361 SES1716077 1365611
STOONS
130KSSSS & 4032 150 LOG SR
MG801 7011 RETTIEIIE TOUTUE
2910022 LORSE L135 T10
250
433885000300 2966 790 6i
26-16 190525441 7895590500
9168515571 91911225577991222400
0.78
PCR TOKSSiH
12000-110000
B0000 19852862
27015660
11006-10506.,11
109961 1456
4835911688
110540892
S161 116199222
339-22800
145800)
72

---

Cluster
Formula
Value
Canonical form (CF)
CF value
Convergence rale
-0.80
134
Eiso Eiso (-1021) ( 12 (10042 1904 327) (L) (2k)
PCFF33400n2 + 99584
12056400
25408 - 156256n
68607 6353000000
3945267200m
8600 130
T602647520m
1450076328m
69039501 600 +
(109212800mz))
135
Le (-1021) (198 + 393248 + 19481) [[-]]
(20)
1000
PCF
1904640m
351079946000
1.38
10E3001 4715S5
22504064mz
108510496m
256884424m
330405838m"
2130847Thn
63326350, 161 2290446000 22
2976051 3041920
116344701 6357 888
3591 080000028-4000
748406431 895466 89596166 16
1966 12058 177770444 104m
1026555072 1549052
638 1908 1908722722200
251 4089737825744mp
371 7679482311600
136
16
PCR 1--6 634n+
1.39
261, 40966 + 10240r
6912n* + 128 610n
137
E (-1024) (20k (22
PCF(1920m* + 6368n
-5-
1.39
7288 115,1638400n. +
1910784m² 1054976n 150144n
(-1] 21922-100 10u (20u +33 (20y Pyteg
1200 (1 (1) (2) (8)
(-1)0--0/20
(3) cc
(-1722 2200 31 (t) (2)
(20m (20m (3)
+) (1) (3)
(-1091) 100 20
PCH(38400 P 192488
14608.. + 4676x.
165. 165.65536000
39321 60.
1:5000116
7626732
66980366 35681288
-086
139
Σ= 648 018-442(654822
3630k
910)
( (G0)
PCFF96502 +
6778822000
0.92
14h
16895 In
357406n
-B187-8700
1238824..
1267 17667199h 179h
461835,
1690387072
10
42057792
28244340360z
34524320580m
71756129 78
143426573097x
43072909 10n"
22847151600.
140
Ee 618 618-~ (10 (2)
PCF(791n* 237
0.93
2482n" + 1007n
105, 127008n
226800n 173346
119769 821314
11664xx)
(N) (T71))))) (B)
2r
73

---

Cluster
Formula
Value
Canonical form (CF)
CF value
Convergence rute
-0.86
(cont.)
(12"" Lak SYP7)
141
E
[2K-11(E-1]
PCF(13750n' + 35811n'
21-10
0.93
29614n 7104n 965n 385,-38679322m 4
17392320n" +
83657448m
39260700m
41524002h
201 201115133 151
2277882n'
1108080+
142
2
(110866--1164 -10 (2) (36)
348
25
720798m 810302n
PC
313-891
7US894
0.93
328875n² 34877n
$700, 9386692128n
781797339 9
23062928050m +
111939 7
19783391 1157g,
6108011 388 +
6029620560n
339444756H
3050573440c
143
Elso 648- (210747 4359k + 2249) (,24) (44
PCF2238091
12570792200 2016-415F
0.93
2007189 143539430
36586679h
50726276 362144824
10126185, 1150770111811
21 414.66248288
177 10000100
85 12112913 175033
0063233224978666
50090 1529 15077n
808276371 49008
7860332431
49539739912066m
1820958822580
296786091 42000.²)
0.86
(189++10-0)) ((H).))
86m
PCF(19410n>>
0000 2lom
(B), (1) Ehro
103464n 2082966
197430n + 89543n
16560 601661766
2088716448m2
527726016H*
8114674880
406166400m
115455024n
1562292h. 29814726M
990499882 1164570p
45150)
74

---

Canonical (CF)
Convergence
0..8
( ( (())..)
(E),
18
PCF(6698400.
83250
1.39
18-
35458566m 706968002
+
6990624, 3499812n²
+
776232n
43503, 78364164006m
300396062368n
325086162176m
76-129246466
3511 20365508m
187384679922
7603508256
290001100804
700061 76006
49150800000
146
E
2- (35- 108n (Ea) (oe) (10)
PCF
3=3P
1.39
22394885.
23029926 +
557280m
385164n
180000n
12925. 78364164000m
39182052018nn
30184715065 11151713
5592357629666
3521 115720966
1 176777786445
157361 356800
786806784008 400,
101601800000
8= -1.00
147
Ei (-1)*t)
aa ok-3)
PCF(72 + 144a
The
62,5184n + 15552n P 13104n 288
18-5x
3428n² 980n
1.00
(8)
(w+1192
618 - 1807
25
PCF(72n2 180n
33000-500x
-68-+205m
0.00
149, 1296n
3888m
4008
1656n 180)
-1.00
149
Σ minh 174m
PCR[1Gn + 52n 68n
0.00
31-6im 6in
221n
308n' 212n 77a
Anidee Span E
75

---

NeurIPS Paper Checklist
1. Claims
Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope?
Answer: [Yes]
Justification: The abstract and introduction describe the absence and need of framework for symbolic and rigorous unification of mathematical formulas, and touch upon the focus of this study: unification of и formulas as a proof-of-concept of the general scheme. Section shows that most canonical forms are unified within our framework and that the methods described generalize additional constants.
Guidelines:
The answer NA means that the abstract and introduction do not include the claims made in the paper.
The abstract and/or introduction should clearly state the claims made, including the contributions made in the paper and important assumptions and limitations. A No or
NA answer to this question will not perceived well by the reviewers.
The claims made should match theoretical and experimental results, and reflect how much the results can be expected to generalize to other settings.
It is fine to include aspirational goals as motivation as long as it is clear that these goals not attained by the paper.
2. Limitations
Question:
Does the paper discuss the limitations the work performed by the authors?
Answer: [Yes]
Justification: S emphasizzs the following limitation: Depending on the constant to which the unification algorithm tuned, the algorithm will need to be modified support the corresponding CMF. We also discuss the performance loss when changing the LLM models.
Guidelines:
The answer NA means that the paper has no limitation while the answer No means that the paper has limitations, but those are not discussed in the paper.
The authors encouraged to create separate "I "Limitations" section in their paper.
The paper should point out any strong assumptions and how robust the results to violations of these assumptions (e.g.. independence assumptions, noiseless settings, model well-specification, asymptotic approximations only holding locally). The authors should reflect on how these assumptions might be violated in practice and what the implications would be.
The authors should reflect the scope of the claims made, e.g., if the approach was only tested on a few datasets or with few runs. In general, empirical results often depend on implicit assumptions, which should be articulated.
The authors should reflect the factors that influence the performance of the approach. For example, facial recognition algorithm may perform poorly when image resolution low or images are taken in low lighting. Or speech-to-text system might not be used reliably to provide closed captions for online lectures because it fails to handle technical jargon.
The authors should discuss the computational efficiency of the proposed algorithms and how they scale with dataset size.
If applicable, the authors should discuss possible limitations of their approach to address problems of privacy and fairness.
While the authors might fear that complete honesty about limitations might be used by reviewers as grounds for rejection, worse outcome might be that reviewers discover limitations that aren't acknowledged the paper. The authors should use their best judgment and recognize that individual actions in favor of transparency play an impor- tant role in developing norms that preserve the integrity of the community. Reviewers will be specifically instructed to not penalize honesty concerning limitations.
76

---

3. Theory assumptions and proofs
Question: For each theoretical result, does the paper provide the full set of assumptions and complete (and correct) proof?
Answer: [Yes]
Justification: Lemma D and its generalization, which enable UMAPS, are proven Ap- pendix F.Il Corollary which states the sufficiency of UMAPS given a bound the polynomial degrees the coboundary matrix, is proven in Appendix C.3
Guidelines:
The answer NA means that the paper does not include theoretical results.
All the theorems, formulas, and proofs in the paper should be numbered and cross- referenced.
All assumptions should be clearly stated or referenced in the statement of any theorems.
The proofs can either appear in the main paper or the supplemental material, but if they appear in the supplemental material, the authors are encouraged to provide short proof sketch to provide intuition.
Inversely, any informal proof provided in the core of the paper should be complemented by formal proofs provided in appendix or supplemental material.
Theorems and Lemmas that the proof relies upon should be properly referenced.
4. Experimental result reproducibility
Does the paper fully disclose all the information needed to reproduce the main ex- perimental results of the paper to the extent that affects the main claims and/or conclusions of the paper (regardless of whether the code and data are provided not)?
Question:
Answer: [Yes]
Justification: All code, data, and results are available on the project repository: http:///ttubbcoom//aaajjjjaanna
including the final coboundary graph, which stores the transformations between equivalent formulas and directly leads to the result statisties in Table g All LLM prompts included also in Appendices [T.6] and H Harvested formulas are included in Appendix [ and are included in the supplementary material ZIP along with their arXiv sources. CMF-generated formulas are included in the supplemental material ZIP. The UMAPS algorithm is available also on the online algorithm demonstration
Guidelines:
The answer NA means that the paper does not include experiments.
If the paper includes experiments, a No answer to this question will not be perceived well by the reviewers: Making the paper reproducible is important, regardless of whether the code and data are provided or not.
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Reviews:
[{"review_id": "uHL0Chri3y", "reviewer": "Reviewer_xauJ", "summary": "The 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.", "questions": "Please 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.", "limitations": "I 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.", "rating": 3, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 1, "clarity": 2, "significance": 2, "originality": 3, "strengths_and_weaknesses": "# 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.", "was_revised": true, "final_justification": "I maintain my score, but would like to reduce my confidence to 4. I think the paper requires a significant rewrite of new content and a much deeper re-evaluation of the formal parts.", "rebuttal": "* *“The paper addresses an important gap in AI-guided experimental mathematics … The work's idea is indeed of significance and its success points toward a new paradigm in how mathematical knowledge can be structured.”  \n“The methodology and pipeline and the claims are well-supported by the experimental results. The mathematical foundations are solid…”*\n\nWe thank the reviewer for their interest and appreciate their positive feedback.\n\n---\n\n* *“1. Can you prove…that the UMAPS coboundary solver is sufficient...and that the solution is unique?”*\n\nLemma 1 implies UMAPS is sufficient: if two formulas are coboundary then their limits are related by $U(1)$. So during the application of UMAPS, $U(1),…,U(N)$ will be correct measurements of $U(n)$, and thus the fit will work provided N is large enough. Following this comment, sufficiency is now highlighted in Section 3.5 by renaming Lemma 1 to “sufficiency of UMAPS” and by including the short corollary. Lemma 2 generalizes Lemma 1 and shows sufficiency for general order recurrences.\n\nBesides sufficiency, we now show in a new lemma (Lemma 4) the uniqueness of the coboundary matrix for the $2\\times 2$ case for constants not algebraic of degree 2. The idea: assume $U1(n), U2(n)$ are two coboundary matrices between $A(n),B(n)$ $\\Rightarrow A(n) = U1(n) B(n) U1(n+1)^{-1} = U2(n) B(n) U2(n+1)^{-1}$. So $B(n)$ is coboundary to itself with $U3(n) = U1(n)^{-1}U2(n)$. Define $L_{B_k} = \\prod_{n=k}^\\infty B(n)$, the limit of the recurrence when started at index $k$. We have from Lemma 1 $L_{B_k} = U3(k)( L_{B_k})$. Since $L_{B_k}$ is not algebraic of degree 2 and $U3(k)$ acts as a rational mobius, we conclude $U3(k) = I_{2 \\times 2}$ for any k. Therefore $U1(n) = U2(n)$ for all $n$.\n\n---\n\n* *“…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…no sensitivity study is provided.”  \n“2. Please provide a sensitivity study…Ideally plot linkage percentage and runtime versus each parameter.”*\n\nWe now provide a sensitivity study showing the linkage percentage and runtime for different $\\delta$-clustering granularities and similarity thresholds, UMAPS’s fit depth, as well as an analysis of the sensitivity of RISC’s Guess algorithm to its fit depth. All of these have been added to the appendix and are summarized below (parameters used in the paper are denoted in bold). The polynomial-degree caps for UMAPS are derived from the fit depth $N$: the degree “budget” is split equally between the numerator and denominator polynomials, for a degree cap of $\\text{floor}(N / 2) - 1$. When the UMAPS fit depth is raised to N=120 for an allowed max degree of 59, no new connections are found.\n\n**Sensitivity study:**\n\n**Linkage percentage** (% of max - 381) **for different $\\delta$ clustering granularities and similarity thresholds.**  \n| Similarity \\ Granularity  | 0.01 | 0.05 | 0.1 | 0.2 | 0.3 | 0.5 |\n|-|-|-|-|-|-|-|\n| 0.001 | 84   | 79   | 63  | 50  | 46  | 59  |\n| 0.003 | 92   | 80   | 65  | 52  | 48  | 60  |\n| 0.005 | 100  | 83   | 67  | 52  | 48  | 60  |\n| 0.01  | 100  | 95   | 73  | 52  | 49  | 61  |\n| 0.02  | 100  | 100  | 78  | 52  | 54  | 61  |\n| **0.03**  | 100  | **100**  | 78  | 52  | 54  | 61  |\n| 0.05  | 100  | 100  | 100 | 70  | 71  | 61  |\n| 0.1   | 100  | 100  | 100 | 100 | 91  | 61  |\n| 0.2   | 100  | 100  | 100 | 100 | 100 | 78  |\n| 0.3   | 100  | 100  | 100 | 100 | 100 | 100 |\n| 0.5   | 100  | 100  | 100 | 100 | 100 | 100 |\n\nLinkage percentage with no clustering criterion (similarity threshold = 2): 100% (381).\n\n**Runtime table for different $\\delta$ clustering granularities and similarity thresholds.** The algorithm (Appendix B.1) first matches formulas to find a set of “hubs” (stage 1) and then matches each hub with CMF representatives (stage 2). For coarse granularities (>0.1), the runtime shows 3 phases in its dependence on similarity threshold, delineated via the split into regular text, bold, then regular text (right). This can be explained by the number of hubs from stage 1 (decrease as similarity threshold increases) vs. the number of CMF representatives that meet the similarity criterion in stage 2 (increase as similarity threshold increases). The runtime value results from their product, which is largest for medium similarity threshold values.\n\n| Similarity \\ Granularity | 0.01 | 0.05    | 0.1 | 0.2 | 0.3 | 0.5 |\n|-|-|-|-|-|-|-|\n| 0.001   | 0.8 | 0.6 | 0.8 | 0.8 | 0.7 | 0.7 |\n| 0.003   | 0.9 | 0.8 | 1.1 | 1.1 | 1.1 | 1.0 |\n| 0.005   | 1.1 | 0.9 | 1.3 | 1.2 | 1.3 | 1.2 |\n| 0.01     | 1.0 | 0.7  | 1.2 | **2.0** | **2.0** | 1.3 |\n| 0.02       | 1.0  | 1.0 | 1.5 | **3.4** | **3.5** | **1.6** |\n| **0.03**  | 1.0  | **1.0** | 1.4 | **4.7** | **4.9** | **2.4** |\n| 0.05    | 1.0 | 1.0 | 1.0 | **4.0** | **3.9** | **3.2** |\n| 0.1      | 1.0 | 1.0 | 0.9 | 1.0 | **2.9** | **5.4** |\n| 0.2      | 0.9 | 0.9 | 0.9 | 0.9 | 0.9 | **6.1** |\n| 0.3      | 1.0 | 0.9 | 0.9 | 1.0 | 0.9 | 1.0 |\n| 0.5      | 1.0 | 0.9 | 1.1 | 1.0 | 0.9 | 0.9 |\n\nRuntime with no clustering criterion (similarity threshold = 2): 20.0. This highlights the role of clustering in reducing runtime.\n\n**Linkage rate and runtimes for different maximum fitting depths in UMAPS.**\n| UMAPS N | Found relation (x / 407) | Same CMF (x / 407) | Max poly. deg. found | Avg. UMAPS time | Total runtime |\n|-|-|-|-|-|-|\n| 4| 96| 96 | 1| 0.5| 6.7 |\n| 8 | 141| 136| 3| 0.5| 3.7 |\n| 12| 200| 170| 5| 0.7| 3.1 |\n| 16| 301| 185| 7| 0.8| 2.5 |\n| 20| 373| 188| 9| 0.9| 1.2 |\n| 24| 375| 188| 11 | 0.9| 1.0 |\n| 29| 381| 188| 13 | 0.8| 0.8 |\n| 30| 381| 188| 14 | 1.1| 0.9 |\n| **40**| **381**| **188**| **14** | **1.0**| **1.0** |\n| 50| 381| 188| 14 | 1.0| 1.3 |\n| 65| 381| 188| 14 | 1.1| 2.9 |\n| 80| 381| 188| 14 | 1.7| 9.7 |\n| 100 | 381| 188| 14 | 7.1| 48.8|\n| 120 | 381| 188| 14 | 18.3 | 137.3 |\n\nThe total runtime has a clear minimum. The two forces at play: Avg. UMAPS time (increases as N increases) and number of hubs from stage 1 of the algorithm (decreases as N increases). \n\n**Success rate and runtime of RISC’s Guess for different fit depths N over 100 random formulas.**  \n| Guess N | Recurrences found (x / 100) | Of which correct | Avg. runtime |\n|-|-|-|-|\n|8 | 0 |0 |0.02 |\n|10 |24 | 24 |0.05 |\n|15 |30 | 30 |0.08 |\n|20 |57 | 57 |0.18 |\n|25 |76 | 76 |0.26 |\n|30 |92 | 92 |0.31 |\n|50 |99 | 99 |0.46 |\n| 100 | 100 |100 |0.67 |\n| **200** | **100** | **100** | **1.00** |\n| 500 | 100 |100 |2.79 |\n\nThese findings indicate that Guess generates robust fits that generalize well.\n\n---\n\n* *“3. Can you prove that the guessed recurrence for each formula is minimal and globally valid…”*\n\nYes. Correctness is proven automatically by substituting the formula of origin into the resulting recurrence. When symbolic simplification is too complicated, 30 random numerical substitutions validate the expression is identically zero: all substitutions must be under 1e-10 to pass this validation. All recurrences are successfully validated this way.\n\nRegarding minimality, it is not guaranteed by the Guess algorithm. However, most formulas found are of order 2 and cannot be order 1 which have a very unique structure and growth. For higher orders, the canonicalization stage is indeed only as good as RISC’s Guess. A new lemma (Lemma 5) now shows that a series must be hypergeometric to yield a polynomial recurrence of order 2, showing the 5 recurrences of order 3 are minimal as well. All instances of known minimal recurrences were successfully found by Guess.\n\n---\n\n* *“3. If rational rather than polynomial coefficients arise, how do you detect and handle order inflation?”*\n\nThe inflation procedure we conduct (described in Appendix C.2) converts the recurrence coefficients to integer polynomials without changing the recurrence order. Minimality is hence preserved.\n\n---\n\n* *“4. formalize CMF for rational entries…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\nThe CMF is formalized for rational entries in Appendix D. The $\\pi$ CMF (Eq. 6) is symbolically shown to satisfy the conserving conditions for the variables $x,y,z$ (above Eq. 6) and thus proven a CMF. This CMF is constructed from the Gauss Hypergeometric function $2F1(z)=F(a,b;c;z)$ using its contiguous relations and differential equation, a method applicable broadly to parameterized D-finite functions like pFqs (discussed in a recent arXiv paper 2507.08138). \n\nTo show the uniqueness of the rank for a pFq CMF: the number of matrices (dimension) = p+q (3 for ours) and the matrix dimension (rank) = max(p, q+1) (2 for ours). CMFs with higher rank are promising for finding deeper recurrences.\n\nRegarding correspondence of lattice directions to formulas, trajectories starting $(x=a,y=b,z=c)$ converge to the value $F(a,b;c;1/2)/F’(a,b;c;1/2)$ where $F’ = z \\cdot d/dz F$, now included as Lemma 6 in Appendix D.2. The corresponding formula does depend on the direction.\n\n---\n\n* *“consider a short introduction of technical terms…and making the introduction less repetitive”  \n“...lacks discussion on some important related studies…”  \n“I recommend adding a concise section in the paper that explicitly summarizes the key methodological gaps, such as hyper-parameter sensitivity…reliance on empirically inferred minimality, and questions of scalability and domain transfer to other areas of symbolic computation and formula harvesting.”*\n\nThank you for these suggestions. We have now included a brief summary on terms used, and compacted the intro. To the discussion section we added a paragraph about the empirically inferred minimality and how we can prove it, as well as text on scalability and domain transfer. We now also cite the works listed by the reviewer and discuss their relation to this work."}, {"review_id": "7Zxz9qFir0", "reviewer": "Reviewer_7riT", "summary": "This 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.", "questions": "PSLQ 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?", "limitations": "yes", "rating": 2, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 2, "significance": 1, "originality": 1, "strengths_and_weaknesses": "Strengths:\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.", "was_revised": false, "final_justification": "", "rebuttal": "* *“The method includes the use of scraping, various extraction and validation steps (confusingly some of which are called \"retrieval\")”*\n\nWe apologize for the confusion. The retrieval stage refers to collecting equation candidates that match $\\pi$-computing patterns from the database of scraped equations. This is now clarified in the text.\n\n---\n\n* *“I cannot judge whether the recurrences extracted here are interesting to mathematicians…the paper does not explain the significance of this field in mathematics.”  \n“It is unclear whether the concrete output of this paper is interesting to mathematicians.”*\n\nThe concrete output of this paper is of interest to mathematicians. We can generally classify these significant outputs in 4 types:\n\n1. **Novel algebraic structure.** The discovery that many historic formulas are generated by a single algebraic structure, the CMF, is mathematically significant. It opens up a new perspective on classical results, connecting them through a shared generative mechanism. The system automatically finds implications of the form FormulaA = LimitA iff FormulaB = LimitB, resulting in a streamlined theory of formula equivalences. The knowledge contained in 381 unique formulas from 151 papers is condensed into 136 canonical forms, which are reduced further into just 32 formulas, proof of which proves all the others systematically, verifiably and automatically.\n\n2. **Active research on proving formulas for special constants.** For example, Ramanujan-type formulas for $\\pi$ have occupied decades of deep mathematical research, with major contributions by Chowla (1928), the Borwein brothers (1987), the Chudnovsky brothers, Zudilin and Guillera. Moreover, a 2025 paper by Cohen and Zudilin raises interest in polynomial continued fraction formulas for constants (arXiv 2501.10090). Many such efforts (e.g., the unification of just two formulas in (Wei et al., Extensions of Ramanujanʼs two formulas for $1/\\pi$. Journal of Number Theory, 2013) are regarded as significant advances. Our work achieves results of this kind - including automated proving - on a much larger scale than anything done before.\n\n3. **A new method in automated theorem proving.** UMAPS itself is of interest for mathematicians as a novel automated proving method.  \nFor example, since submission UMAPS has proven equivalent entire infinite classes of formulas, such as generalizations of the famous Madhava-Leibniz series, the class of formulas discovered empirically by Raayoni et al. (2021), and classes from (arXiv 1104.0392, p. 5), (1805.06568, p. 3), (1806.08411, p. 6). \n\n4. **Specific equivalences connecting papers from different fields**, including quantum field theory and string theory, commutative algebra and ring theory, classical analysis and differential equations, operator theory and functional analysis, and probability theory and stochastic processes. Each of the 1021 specific discovered equivalences could be of importance, and should be analyzed independently. Equivalent formulas for constants have in the past revealed deep mathematical connections among special functions, continued fractions, lattice sums, Fourier–theta dualities, modular forms, and Feynman integrals.\n\n---\n\n* *“…the only machine learning techniques used are prompting”*\n\nIn addition to prompting, our clustering algorithm is also a novel contribution to the broader field of machine learning, unique in its combination of symbolic and numerical clustering: Converting to canonical forms unites disparate looking formulas, after which clustering according to the irrationality measure decreases runtime by a factor of 20. Utilizing the transitivity of coboundary equivalence for rigorous clustering during the graph growing algorithm (Appendix B.1) further improves runtime, for a total improvement factor of >200 compared to the no-clustering case. We now expand on this method and its value for efficient formula exploration in Section 3.5.\n\nThis work marks the first large-scale application of AI to number theory, and specifically **the first time that a LLM is used in this field** in any form. This achievement is possible due to our integration of LLMs with automated verification and proving. So even though this work looks different from most conventional works in machine learning, it is unique in its **contribution to AI for math**, for which we believe NeurIPS is an ideal platform.\n\n---\n\n* *“The paper does not answer the question whether any of the formulas found bear any resemblance to the original formulas found in papers.”  \n“How accurate are any of the processing steps? Does the overall pipeline produce the expected formulas?”*\n\nEvery result in our paper is validated using symbolic computation - without LLMs. Specifically, the formulas found are proven, so every resemblance to an original formula is verified. The LLMs are used for formula harvesting and not for validation. The benchmark used for testing the LLM performance uses as ground truth extracted formulas that passed numerical validation via PSLQ.\n\nFollowing the referee’s comment here, we revise the manuscript to clarify this important point, now explicitly saying that our methodology provides validation of absolute precision. Once two formulas are found related, they are automatically proven equivalent by our algorithm. This means that 381 of the formulas (found in 151 papers) were proven equivalent to each other in 136 distinct forms, which prove one another in 32 clusters.\n\n---\n\n* *“The code is not available...”*\n\nCode producing all the final results was available in the supplementary material. We apologize if this was not made clear in the original submission, and now explicitly mention it in the main text as well.\n\n---\n\n* *“PSLQ seems to play an important role. ...Could you explain how the process works and how often it is required?”*\n\nPSLQ is used during the extraction of every formula. Limits appearing in the equations of origin are sometimes inconsistent with the pure form of the formula collected, due to missing symbolic manipulations (see the example in Appendix G.7 Table 8), so they are not used directly.\n\nThe process starts by measuring the convergence rate numerically, then computing the formula to depth 2,000,000 (if slow convergence) or 50,000 (otherwise). This high-precision numerical value is fed into PSLQ, which returns an integer mobius of $\\pi$ that equals this numerical limit if one exists. If PSLQ fails, the formula is discarded as it does not compute pi. PSLQ thus acts as both a critical validation step and for complementing the extraction by the LLMs.\n\nThis check is critical: without PSLQ, the fraction of incorrectly extracted formulas is on the order of tens of percent.\n\n---\n\n* *“...how these methods generalize to other mathematical discovery methods or whether there are any interesting conclusions for AI research in these methods.”  \n“the paper does not seem to contain new methods or insights for AI research”*\n\nThe main insight for AI research our paper promotes is: Discovery that is verifiable and even automatically proven can be achieved by pairing LLMs with new tools from symbolic mathematics. To show this, we have successfully paired LLMs with UMAPS and with PSLQ, resulting in an automated system for equivalence discovery.\n\nMoreover, the approach shown here moves beyond proof-checking and into symbolic structure discovery, which is a frontier direction for AI in scientific reasoning.\n\nAdditional contributions to AI research:  \n1. Our work provides a novel dataset to gauge mathematical reasoning: A dataset of hundreds of equivalent formula pairs, useful for benchmarking LLMs on the ability to transfer mathematical knowledge (see Table 2 and Appendix F.2). We tested frontier models and they currently fail in the majority of cases (which are proven by UMAPS), indicating that this dataset will be useful for testing future LLM developments.\n2. Our framework can apply to other areas of AI for math: We provide an equivalence pipeline for recurrence relations and rational sequences, applicable across number theory and computational math, unmatched in handling millions of expressions in bulk. \n3. More generally, equivalence detection as developed here could become a crucial capability for certain generative AI use cases, aiding originality verification, data leak detection, and plagiarism identification.\n\n---\n\n* *“The example in section 5.1 states \"This equivalence demonstrates how two previously distinct mathematical expressions, discovered over a century apart, are now proven to be equivalent by an automated process.\" To me this reads like an overstatement...what exactly was discovered here that hasn't been discovered before?”*\n\nThe mentioned paper (Sun 2020) indeed explicitly points out the relation between their 97 new formulas to Ramanujan's earlier work (Ramanujan 1914). However, these new formulas are presented as novel. They are only mentioned as belonging to the same family (having the same overall structure); **not proven to be equivalent to Ramanujan’s**.\n\nOur work discovers and then proves the equivalence of 16 of Sun’s formulas to a total of 134 other formulas, including one of Ramanujan's, in 7 clusters - each one implies the other is correct too. This is what we meant by *“demonstrates how two previously distinct mathematical expressions…are now proven to be equivalent”*.\n\nWe revise the introduction to clarify what was known and what was discovered here, and also elaborate on this in Section 3.5 where UMAPS is discussed, and in Section 5.\n\n---\n\n* *“From how many different papers do your 402 final formulas originate?”*\n\n154 papers (see Table 4 in p. 8). This is now better clarified in Section 5.  \nNote that all 154 are included in the supplementary material."}, {"review_id": "7AtJkQSe2X", "reviewer": "Reviewer_3XHw", "summary": "This 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.", "questions": "In 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?", "limitations": "Yes.", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 2, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths: 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.", "was_revised": true, "final_justification": "The method and results of this paper are interesting to mathematicians and the quality of the writing and analysis, while not great, is adequate.  The authors addressed my most pressing concerns.  I feel the other reviewers are just not very interested in this topic and are rejecting it for this reason.  Perhaps it should go to a different conference focused on AI for mathematics.", "rebuttal": "- *“Strengths: 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…I know of few if any other works which automate such a mathematically interesting analysis.”*\n\nWe thank the reviewer for the interest and the positive feedback.\n\n***\n\n- *“I find the mathematical part of the paper poorly written, explaining the concepts and methods in a very disjointed way.”*\n\nWe are sorry to hear. In our revised manuscript, we have compacted the introduction and added a summary of terminology, aiming to create a more continuous description of the mathematical concepts needed in the rest of the paper. Section 2 is supported by this added context. We have also explained where each mathematical term fits in the different stages of unification. The mathematical part of the appendix has been organized for better readability.\n\n***\n\n- *“The relation to previous work… 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\nWe thank the reviewer for pointing this out. Please note that the Conservative Matrix Field (CMF) terminology is not new, as it was introduced in (Elimelech et al. PNAS 2024), and used in several papers since then. Part of these papers mentioned that CMFs indeed satisfy the defining properties of a discrete flat connection. But importantly, CMFs have more specific properties arising from their domain and rational entries, which are critical for their useful behaviors regarding formula discovery and mathematical convergence. This is why we choose this terminology. We now also mention the relation to flat connections when defining the CMFs for the first time.\n\nIn addition, Appendix D now includes a discussion on the broader mathematical contexts of CMFs, and their special properties that distinguish them. Specifically, the cocycle perspective plays a central role in our algorithmic framework for symbolic unification using coboundary equivalence, with the equivalences between trajectories naturally expressed in terms of coboundary transformations.\n\n***\n\n- *“In section 3 …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?”\n“What are the other formulas, are they just explicit series with no other apparent structure?”*\n\n1613 are series vs. 65 continued fractions (Appendix G.4). \nTo clarify, 3389 is the number of formulas singled out by GPT-4o mini as pi computing, before being extracted and validated. The other 3389-1678 formulas are mostly false positives from GPT-4o mini’s classification.\nChecking 20 random examples, all were false positives, discarded when double checked by GPT-4o. Example: $g ( x ) = \\frac{\\pi}{2} + \\sum_{j = 1}^{\\infty}a_nx^n$ (arXiv 2206.11256).\n\n***\n\n- *“Then in 3.2…it is said that \"Out of the 682 candidates, 407 were validated as formulas and passed on for canonicalization.\" Where did these numbers come from?”*\n\nAfter classifying and extracting, GPT-4o produces some expressions that do not compute $\\pi$. The PSLQ-validation stage removes these false positives by finding the precise limit of each formula in terms of $\\pi$. Of the 682 candidates, 147 are not diophantine formulas (e.g. they contain $\\pi$ itself in the series term), leaving 525 formulas. 118 of these did not pass validation. Many of which were formulas that do not compute $\\pi$ like $-\\frac{4}{\\pi}\\sum_{n=0}^{\\infty}\\frac{(-1)^n}{(2n+1)^{2}} = -\\frac{4 G}{\\pi}$ (arXiv 1906.04927), and formulas with mistakes like $ \\sum_{i = 1}^{\\infty} \\frac{1}{i^2} = \\frac{\\pi}{6} $ (arXiv 1811.05831, under proofs.tex).\n\nExample for a non-diophantine formula for $\\pi$: $\\sum_{n = 0}^{\\infty} \\frac{ ( \\frac12 ) _n ( \\frac13 ) _n ( \\frac23 ) _n}{ ( 1 ) _n^3} ( \\frac{2}{27} ) ^n = \\frac{3\\pi}{4 \\, \\Gamma^2 ( \\frac23 ) \\Gamma^2 ( \\frac56 ) }$ (arXiv 2001.08104), as $\\pi$ can be computed only when given the irrational denominator.\n\n***\n\n- *“In 3.3 it says \"Out of the 407 validated formulas, 402 are found to have 149 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\nAlmost all of the validated formulas are series (364 series vs. 43 continued fractions) because series are the most common forms of formulas being studied in the literature. \n\nInterestingly, a series must be hypergeometric to yield a polynomial recurrence of order 2, shown in a new lemma (Lemma 5). Order-2 recurrences can always be transformed to an equivalent continued fraction, as shown in Lemma 3. So, in total, 402 of 407 formulas were transformed to order-2 recurrences (continued fractions). The other 5 were transformed to order-3 recurrences (these were initially given in a form of series). \n\n***\n\n- *“The 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?”*\n\nWe believe that additional formulas from Table 10 may yet hide in the 3d CMF of D.2 (Eq. 6). There are two theoretical limiting factors to unification that may have prevented these from being found so far: (1) UMAPS fit depth and (2) sampling of additional CMF representative formulas. \n\nMore generally, to unify all formulas, one would need a “larger” CMF, which will contain the 3d CMF of D.2 and additional dimensions. To see this, note that order-3 recurrences, such as the examples in Appendix A.3, need higher rank CMFs (3x3 matrices). \n\nWe are glad to report that we now have a 5d rank-3 CMF that contains our previous 3d rank-2 CMF (Eq. 2). This CMF also contains order-3 recurrences converging to $\\pi$.\n\n***\n\n- *“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…”*\n\nYes! We hope to find CMFs of higher dimension, containing even more unique directions for formula search to accommodate all formulas for $\\pi$.\nOur CMF was constructed using the Gauss Hypergeometrics function 2F1(z)=F(a,b;c;z), using its contiguous relations and differential equation. The CMF object was shown in (arXiv 2507.08138) to be generalizable to D-finite functions, like generalized hypergeometric functions - pFqs, using their contiguous relations and annihilators (which serve the same role as the differential equation in our construction). For a pFq CMF, the number matrices (the CMF’s dimension) = p+q (indeed 3 for ours) and the matrix dimension (the CMF’s rank) = max(p, q+1) (2 for ours). CMFs with higher rank are promising for finding deeper recurrences.\n\nThe eventual hope is finding one of a few appropriate D-finite functions that can create CMFs that will capture all the formulas of $\\pi$, and in a similar way appropriate functions and CMFs for any other constant of interest.\n\n***\n\n- *“…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\nThe most rigid constraint which prevents combining two recurrences is computing separate irrational constants. Separate constants require different CMFs and admission of a formula to a CMF requires consistency between the constants the two compute. \n\nRelations between the dynamical metrics of a formula do not seem to impose constraints given the wide range of dynamical metric values found for formulas generated by the same CMF. To demonstrate this, we have added a dynamical metric “scan” of the CMF (Eq. 6), showing the convergence rate and irrationality measure as a function of direction (on the unit sphere). Fig. 6 and Table 9 also provide multiple dynamical metric values from this CMF, showcasing its versatility."}, {"review_id": "tLcR24nJsT", "reviewer": "Reviewer_5mQZ", "summary": "This 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.", "questions": "Q1. 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.", "limitations": "yes", "rating": 3, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 3, "significance": 2, "originality": 3, "strengths_and_weaknesses": "For 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.", "was_revised": true, "final_justification": "The two main points were solved, although I'm still struggling to see the significance of these claims.\nThe current manuscript is insufficient and newly added contents (during rebuttal and discussions) are substantial and require rigorous re-evaluation.", "rebuttal": "- *“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\nWe thank the reviewer for this positive feedback.\n\n---\n\n- *“A good benchmarking exercise has been done, although it's very focused on $\\pi$ only.”*\n\nRunning the entire pipeline for other constants like Catalan’s or zeta(3) is possible with only slight modifications. The benchmark is demonstrated on $\\pi$ because it has many known formulas that make a large-scale statistical analysis more precise.\n\n---\n\n- *“W1. …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\nWe appreciate the reviewer’s comment and welcome the opportunity to clarify how this work fits within the scope and interests of the AI community, and NeurIPS in particular. Our submission does not involve training a new model, but applies LLMs to a domain where AI has had limited reach: number theory. By combining LLM-based extraction with symbolic validation tools and algorithms like PSLQ and UMAPS, we enable the first successful application of LLMs in this space.\n\nA key component is LLM-based filtering. Because mathematical papers vary widely in definitions and notation, and no deterministic ruleset can precisely isolate expressions related to targets like $\\pi$, we rely on LLMs to interpret and filter candidate formulas across thousands of documents. This usage is essential for identifying useful mathematical content at scale.\n\nOur pipeline reflects a broader trend in AI for mathematics: combining LLMs with formal or symbolic reasoning. PAL (NeurIPS 2022), Parsel and DELI (NeurIPS 2023), and Self-Discover (NeurIPS 2024) all used existing LLMs alongside external computation, achieving strong results without model training. We now cite these works to put our contribution in this broader context of LLM integration with computer algebra systems (CAS).\n\nNeurIPS has actively encouraged this kind of interdisciplinary infrastructure work, especially as LLMs begin to interface more deeply with formal tools, numerical methods, and structured mathematical knowledge. We focused on number theory, a domain that remains underexplored in current AI pipelines.\n\n---\n\n- *“W2. …Showing that formula $x_1$ is the same as formula $x_2$ cannot be seen as a novel contribution…and it's not bringing any meaningful and additional knowledge to the scientific community.”*\n\nWe appreciate the reviewer’s comments and opportunity to explain the contributions of our work. Our algorithm is the first to unify formulas in bulk, with no prior work achieving this. This approach has multiple novel contributions:\n\n1. **Structure discovery**: The system automatically finds implications of the form FormulaA = LimitA iff FormulaB = LimitB, resulting in a streamlined theory of $\\pi$ computation. The knowledge contained in 381 formulas from 151 papers is compactified to 136 unique canonical forms which are further condensed into just 32 formulas. This technique greatly increases the efficiency of formula proving, as proof of any formula in a cluster proves all the others systematically, verifiably and automatically.\n\n2. **Novel dataset to gauge mathematical reasoning**: Our results provide a dataset of equivalent formula pairs, useful for benchmarking LLMs on the ability to transfer and unify mathematical knowledge. To demonstrate this contribution, we tested state-of-the-art LLMs, asking them to prove the equivalence of formula pairs (see Section 4), showing that this benchmark is indeed useful for gauging their math reasoning.\n\n3. **CMFs for formula generation**: The CMF generates infinitely many $\\pi$-formulas, some of which yield faster convergence than the original formulas. Such formulas can be used for irrationality proofs - for example, automatically generalizing formulas to reveal Apéry’s creative sequence along a diagonal, illustrating how unifying structures can uncover deep number-theoretic insights.\n\n4. **Exploring individually the 1021 equivalences identified in our work could lead to advances in mathematics**: Similar situations have happened in the past, where equivalent formulas for constants revealed connections among special functions, continued fractions, lattice sums, Fourier–theta dualities, modular forms, and Feynman integrals. Euler’s comparison of two formulas for $\\pi$ - the Wallis product and Brouncker’s continued fraction - revealed the hidden structure of beta and gamma functions, now central to analysis and physics. In the 1800s, efforts to compute $\\pi$ via Buffon’s needle experiments and Gauss’s lattice point counts culminated in the Poisson summation formula, connecting geometric probability and theta-function theory. In the 20th century, Ramanujan’s many rapidly converging series for $1/\\pi$ were traced to modular forms and elliptic functions, leading to both the fastest known $\\pi$-computation algorithms and the theory of Ramanujan–Sato series. In each case, the breakthrough emerged not from a single formula, but from uncovering connections among different formulas. Our work enables this kind of discovery at an unprecedented scale.\n\n---\n\n- *“Q1. 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\nAlthough this was not in our original plan for this paper (was planned for a future work), this comment made us use our developed formalism to find a faster converging formula, compared with the known ones. We are glad to report that we found such a faster converging formula and it is now added to the findings of the paper. The *normalized* convergence rate we found is 1.76, for a formula in the CMF with trajectory (-1,-1,0). In comparison, the trajectory (1,1,2) is the best pre-existing formula found in the CMF and has a *normalized* convergence rate of 0.88. We added a discussion of how this result was achieved to Appendix E.\n\n**Additional details**:\n\nTo make a fair test of convergence, one has to *normalize* the convergence rate by the “length” or “complexity” of the trajectory. This normalization avoids trivial improvements, such as artificially accelerating convergence by taking every k-th step, which can misleadingly inflate the raw rate. The normalized rate is the convergence rate divided by the number of CMF steps (i.e., the trajectory's length in the lattice). This means that trajectories along the same line such as (1,2,5) and (2,4,10) will have the same normalized convergence rate.\n\nUsing this metric, we are now able to map the convergence landscape over the CMF directions. In the revised appendix, we include:\n- A visualization (\"convergence map\") over the unit sphere of directions.\n- A detailed discussion on the convergence rate and how CMF trajectories change this rate in a continuous (in the spherical angle) manner.\n\nIn summary, the CMF framework not only unifies symbolic formulas but also enables structured exploration of convergence behavior, which may help guide the discovery of new, fast-converging formulas in future work.\n\n---\n\n- *“Q2. 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\nWe appreciate the opportunity to clarify this point.\n\nMathematical progress often arises from finding new ways to conceptualize ideas that appear across different areas. One of the most far-reaching examples is the development of category theory, which provides a language for describing patterns and relationships shared across seemingly unrelated mathematical structures. Initially, this language didn't introduce new results but offered a unifying lens that has since become foundational in many fields, enabling new questions (such as - given a functor what is its adjoint) and methods of reasoning (such as - this object is an initial object in the category).\n\nIn a similar spirit, our proposed work aims to unify the many expressions for mathematical constants that appear in varied domains. These constants, such as $\\pi$ or logarithmic values, frequently emerge in formulas tied to geometry, topology, number theory, and mathematical physics. Although these formulas originate in pure mathematics, they are scattered across different subfields and often studied in isolation.\n\nAs a concrete case, recent work by Bettin and Drapeau (Modularity and Value Distribution of Quantum Invariants of Hyperbolic Knots, 2020) demonstrates that constants related to volume and modularity—particularly $\\pi$—arise in surprisingly structured ways. They show that the asymptotic behavior of quantum invariants, such as the colored Jones polynomial of the figure-eight knot, encodes the hyperbolic volume of the knot complement. This leads to a precise numerical expression involving $\\pi$ - found in our CMF by UMAPS as coboundary to the direction (1,1,1). One can imagine a direct use of  the CMF for this problem  by identifying possible relationships between the parameters defining a knot (like its crossing number or twist pattern) and the directions within the CMF that represent the derived formula for the volume. Then, by just analyzing the CMF (which is easy to compute), one could reveal new insights into the invariants of different knot types.\n\nUnification, in this sense, is not just about organizing known results. It creates a higher-level language for recognizing deep patterns and making new connections—often across fields that traditionally develop in isolation. By framing diverse formulas within a common structure, the CMF enables a more global view of mathematical knowledge and opens new directions for inquiry grounded in pure mathematics."}]

Author rebuttal:

Meta-review: The paper solves a very interesting and important question: "How are discovered mathematical formulas connected?". The paper relies on LLMs and various mathematical techniques to extract various representations of approximations for, e.g., for $\pi$ and $e$, and convert them into standard forms, after which unification of formulas can be performed. Several relations between different representations are found. For example, for $\pi$, 381 of 407 formulas are proved linked and several well-known representations (e.g., from Euler, Gauss, and Brouncker) are shown to be equivalent. 

The paper is well motivated, and the studied problem is very novel/original. Most reviewers (and also I myself, after reading the paper and the discussions) think the work is quite impactful for mathematical research.  The decision on this paper is especially difficult, as despite its novelty and contributions, its writing and analysis could be further improved, which I believe the authors have done a reasonably good job in the discussions with the reviewers. After discussions between the reviewers/ACs and the ACs/SACs, I think this is a good paper related to using LLMs for mathematical explorations, thus a recommendation for its acceptance. I strongly recommend that the authors consider the comments during rebuttal/discussions to revise the paper accordingly in its final version.
Decision: Accept (poster)