Unsupervised Discovery of Formulas for Mathematical Constants
Authors: Michael Shalyt
Uri Seligmann
Itay Beit Halachmi
Ofir David
Rotem Elimelech
Ido Kaminer
Venue: NeurIPS 2024 poster (2024)
Keywords: AI for Science
Automated Conjecture Generation
Experimental Mathematics
Mathematical Constants
Irrationality Measure
Unsupervised Learning
Formula Generation
Continued Fractions
Primary area: machine_learning_for_other_sciences_and_fields
TL;DR: Unsupervised learning on dynamics-based features discovers novel formulas for mathematical constants.

Abstract: Ongoing efforts that span over decades show a rise of AI methods for accelerating scientific discovery, yet accelerating discovery in mathematics remains a persistent challenge for AI.
Specifically, AI methods were not effective in creation of formulas for mathematical constants because each such formula must be correct for infinite digits of precision, with 'near-true' formulas providing no insight toward the correct ones. Consequently, formula discovery lacks a clear distance metric needed to guide automated discovery in this realm.

In this work, we propose a systematic methodology for categorization, characterization, and pattern identification of such formulas. The key to our methodology is introducing metrics based on the convergence dynamics of the formulas, rather than on the numerical value of the formula. These metrics enable the first automated clustering of mathematical formulas.
We demonstrate this methodology on Polynomial Continued Fraction formulas, which are ubiquitous in their intrinsic connections to mathematical constants, and generalize many mathematical functions and structures.
We test our methodology on a set of 1,768,900 such formulas, identifying many known formulas for mathematical constants, and discover previously unknown formulas for $\pi$, $\ln(2)$, Gauss', and Lemniscate's constants. The uncovered patterns enable a direct generalization of individual formulas to infinite families, unveiling rich mathematical structures. 
This success paves the way towards a generative model that creates formulas fulfilling specified mathematical properties, accelerating the rate of discovery of useful formulas.

Unsupervised Discovery of Formulas for Mathematical Constants
Michael Shalyt
Uri Seligmann" Itay Beit Halachmi Ofir David
Rotem Elimelech Ido Kaminer Technion Israel Institute of Technology, Haifa 3200003, Israel shalyt@techhion. ac il, uri seligmann@gmail com itaybe@campus technion ac. i1, eofirdavid@gmil.. com rotem. el eli@campus technion. ac. il, kaminer@technion.ccm ac.
Abstract
Ongoing efforts that span over decades show a rise Al methods for accelerating scientific discovery
[Fajtlowizz 1988 PPetkovsck et al. 1996 Wolfram et al. 2002 Buchberger et al., 2006 Bailey al. 2007 Raayoni et al. 2021 Davies et al., [02] I Fawzi et 2022 yet accelerating discovery in mathematics remains persistent challenge for ÁI. Specifically, AT methods were not effective in creation of formulas for mathematical constants because each such formula must be correct for infinite digits of precision, with "near-true" formulas providing insight toward the correct ones. Consequently, formula discovery lacks a clear distance metric needed to guide automated discovery in this realm. In this work, we propose systematic methodology for categorization, characterization. and pattern identification of such formulas. The key to our methodology introducing metrics based the convergence dynamics of the formulas, rather than the numerical value of the formula. These metrics enable the first automated clustering of mathematical formulas. We demonstrate this methodology Polynomial Continued Fraction formulas, which are ubiquitous in their intrinsic connections to mathematical constants [Lagarias 2013 Bowman and Mc Laughiin 2002 Laughlin and Wyshinski| ]004], and generalize many mathematical functions and structures. We test methodology on a set of 1,768,90 900 such formulas, identifying many known formulas for mathemat- ical constants, and discover previously unknown formulas for æ, (2(.), Gauss', and Lemniscate's constants. The uncovered patterns enable a direct generalization of individual formulas to infinite families, unveiling rich mathematical structures. This success paves the way towards generative model that creates formulas fulfilling specified mathematical properties, accelerating the rate of discovery of useful formulas.
2 Introduction
Historically, formulas of mathematical constants were a symbol of aesthetics and beauty, Continued fraction formulas such as those for the golden ratio & and for tan(x:)
1+
tan
enable calculating infinitely many digits for these constants. Discovering such formulas often leads profound revelations regarding the properties and underlying structure of fundamental constants. For example, the continued fraction formula for tan(x). shown in Eq. was used by Johann Heinrich Lambert in the first proof of the irrationality of Pi [Lambert, 1768) Berggren et al.) 2004]. Unfortunately, such formulas are notoriously hard to find on-demand, often relying on mathematician's profound intuition. Part of the challenge the lack of well-defined 'distance', or
"Equal contribution. Project repository: huwp://ghhcccoooommiiaamaiiiiiie
38th Conference on Neural Information Processing Systems (NcurIPS 2024).

---

metric, between a formula and given constant. i.e., there is no known way to tell whether a formula nearly accurate. The formula either works, or does not. In other fields of science, prediction accurate to 1000 digits is precise enough for any practical need. However, in mathematics, if the 1001 digit is wrong, the formula is incorrect and gives no insight regarding correct formula. This lack of a metric is substantial hurdle both for human efforts and for automated analysis, as many methods for optimization such as gradient descent become unsuitable.
Recent efforts developed computer algorithms to discover a multitude of formula hypotheses for mathematical constants [Raayoni et al., [02T], even implementing the first large-scale distributed computation for such discoveries [Elimelech et al. 2023), but they relied mostly on exhaustive search methods. These approaches complements earlier applications of algorithms for automated theorem proving (ATP) (such as computer proof's of hypergeometric identities [Petkovsek et al. 1996], Malarea [Urban 2007], and Flyspeck [Kaliszyk and Urban 2012)), and automated conjecture generation (ACG) (such as mechanical mathematics [Wang, 1960]. the Automated Mathematician
[Lenat 1982]. EURISKO [Lenat and Brown 1984 Davis and Lenat 1982]. and Graffiti (Fajtlowizz) 1988])
Here we propose a fundamentally new methodology for automated investigation and discovery of formulas for mathematical constants. We constructed a large dataset of continued fractions, and en- riched it with metries based on their convergence dynamics, which are found to embody fundamental information about each continued fraction. These dynamical metrics enable the identification and generalization of patterns within the dataset. Using the metrics, we develop a process of categorization and clustering (Fig. D of continued fractions that share similar values of their dynamical metrics. Analyzing each automatically identified cluster of formulas, we find that all its members often relate to the same mathematical constant, showing the value of the dynamical metrics for the discovery of new formulas and the internal structure of families of such formulas. This novel method of formula discovery allowed us to identify both previously known and completely new formulas for constants such as ग, (2). cot(1), the golden ratio, square roots of multiple integers, the Gauss constant, and the Lemniscate constant.
Cluster data
Label clusters
(di1) Ves
Come rite
Compare datapoints to anchor
(a)
match?
Filtering
Identical anchors?
Validesed Clustor
[b]
rate
Fexture Estract
No
Over generatized cluster
[cl
Cluster anchors?
s.
(c.3)
Anchor found
PSLO
(c)
No anchor found
Define
inwestige gatiin
Further
Dimenssion Choice S.
Come Eile
No (e)
Canii eite
anchors
(e.2)
Clustering kng
PSLTL
No connection found
Figure 1: Systematic clustering and labeling of formulas by dynamical metrics. Our methodology analyzes Polynomial Continued Fractions (PCFs) in two main stages. Clustering: (a) Filter degener- ate PCFs. (b) Evaluate PCFs and extract their dynamics- best few metrics and use them cluster the data. Labeling:
metrics (section 3}. (c) Choose the every cluster, look for PCFs known in (d) If anchors are found in the cluster, validate that they do not
the literature and use them as anchors. contradict, i.e., relate to different constants. (d.1) If all anchors are in agreement, choose a random subset of other points in the cluster and use PSLQ to validate that they also relate to the same constant. If the validation is successful, the cluster is labeled. If not, the cluster should be split. (d.2) If the anchors relate to different constants, the cluster should be split return to step c for finer clustering of the data. When focusing on a specific cluster, the best metrics could be different than those for the full dataset. (e) If no anchor is found in certain cluster, attempt to label by (e.1) choosing small subset of PCFs in the cluster and running a PSLQ search for each of them against a large set of potential constants. If connection is found, the cluster now has anchor return to step d. (c.2) If an anchor is still not found, attempt to connect a sample of data points within the cluster using PSLQ. If successful, conclude that the cluster is correct, but has no identified constant. Define new label for that cluster. PSLQ failed to connect points within the cluster, return to step e for finer clustering. If no further refinement is appropriate, flag the cluster for further analytical investigation.
2

---

As part of our analysis of metrics of continued fractions, we developed and applied the most complete classification of polynomial continued fractions known to date, detailed in Appendix [B] This classification includes the prediction of whether the continued fraction converges directly based on its defining polynomials.
Traditional clustering methods attempt to relate data points by calculating distance metries based on the parameters of these data points, c.g., the coefficients of the defining polynomials. The most common approaches (like SVM) rely on linear classification, while more advanced methods rely on non-linear kernel transformations but usually use various functions calculated directly on the data parameters. In our dataset, each point is a continued fraction formula defined by the polynomials used to construct it. However, we find that it is not sufficient to use the parameters of the polynomials, and not even the numerical limit of the continued fraction. Instead, we find that it is the dynamics of the continued fraction generated by these polynomials, rather than any direct function on their coefficients, which provides the most useful metrics for analysis. In other words, we find that the useful underlying metrics to extract from each data point are embedded within the intricate progression of the sequence created by the formula, rather than the explicit numerical value (limit) of that formula, or the coefficients defining it. Thus, in order to assess the distance between two polynomial continued fractions, and identify relations between such formulas, it is imperative to characterize the nuanced behavior of their sequences, analyzing trends in the convergence process of these sequences, spanning over numerous terms.
Some of the metries we extract, such as the irrationality measure, are well-known in the mathematical community, yet were never considered for a large-scale classification effort. The evaluation of the irrationality measure is technically challenging for formulas whose limit not known advance (which is the vast majority). This challenge made it impossible to extract the irrationality of formulas for large dataset. Consequently, we develop a new algorithm - the Blind-ó algorithm (Section 3.4 to enable the extraction of the irrationality measure of a continued fraction without prior knowledge of its limit. This algorithm allowed us to extract the irrationality measure for the entire dataset.
These advances provide the building blocks for our novel methodology for formula discovery. We cluster formulas by their 'closeness' other formulas according to these new metrics, thus identifying promising formulas regardless of their numerical value (Fig. eeft) Once a candidate formula is found, we numerically validate it by calculating its value to a large precision and then identifying its relation to a mathematical constant. The "generate = validate" approach is inspired by works in Al-driven code generation [Ridnik et al. 2024] and problem solving in geometry (Trinh et al., 2024).
3 Methodology for Data-Driven Discovery
3.1 Definitions
Polynomial Continued Fractions
In this work we chose to focus on polynomial continued fraction (PCF) formulas as our test case due the combination of their simplicity and expressive power. PCFs relate wide range of mathematical fields, represent a variety of constants, are equivalent to infinite sums [Buler [748). and cover mathematical functions such Bessel functions, trigonometric functions, integral families, widely used Taylor series, and generalized hypergeometric functions [Cuyt et al., 2008). Thus, studying PCFs can provide insight into a plethora of mathematical objects and applications.
PCF at depth n is defined as:
b
Pu
ag
+
(2)
b2
Jn
01
bn
On
where an = a(n) and b, = b(n) are evaluations of polynomials with integer coefficients. The PCF Pn (when exists). The converging sequence of rational numbers Ps.
value is the limit lim x 0o dn. provides an approximation of L,, which is known as a Diophantine approximation.
The Irrationality Measure of Number
3

---

While irrational numbers cannot be expressed using simple quotient of integers, they can be approximated by them. Moreover, some approximations are "better" than others, and one way to evaluate their quality is by a quantity called the irrationality measure [Hardy et al. 1979].
We define the irrationality measure of a sequence Pn L as the limit 8, 6, with
de
Pn
log
L
8n
Gn
Gn
4n
(3)
log (mm
ged(pn, Qn))
For every L R, the irrationality measure of L is defined the supremum of all possible 6 for which there is a sequence of distinct rational numbers Pe:
-> L; Ph. + L. that satisfies
Pn L
1
On 1+8
It is known that for irrational numbers this measure is N (Dirichlet theorem for Diophantine approximations), and for rationals it is O.
Note that the irrationality measure of is greater or equal to the irrationality measure of any specific sequence converging the same L.. While the irrationality measure of sequence can be any numher 2 21,, the irrationality measure of its limit L is always either O or > [Church] 2019).
3.2 3-Predictor Formula
The classification of large number of continued fraction formulas requires an efficient and accurate calculation of the irrationality measure for each formula. This calculation is challenging because depends on the asymptotic behavior of the converging sequence, and because 6 appears as an exponent of a large basis number. The 5-Predictor formula that we present here provides way around this challenge requiring no specific knowledge about the convergence rate and trajectory, or even about the sequence limit itself:
n log
(n) Az(n)
ppredicted
lim +00000
(5)
log 9m
ii bn
where (n) and (n) are the eigenvalues of the matrix
A1 (1) > [2(())).
an
This formula extends hypothesis made in previous work (David al. 2021 which was limited to
PCFs with deg( B) = 2deg( ()) and with a Gn that grows exponentially. As we found this work, Eq works for any converging PCF. It validated numerically and proven for the deg( (B) 2deg( A) case in Appendix [F] This formula helps estimate the irrationality measure, critical dynamical metric for our work. Specifically, the asymptotic behavior of qr, and X1/xa are still required for finding Spredicted but they are usually easier to derive.
3.3 Discovery Formulas by Unsupervised Learning
Each PCF formula is defined by the polynomials that generate it. This work focuses on polynomials up to 2nd degree: an A2n² + Ayn + Ag. b, Byn² + B + Bo, with integer coefficients in the domain -5 5 < A; S 5, 5 < B; < 5. We removed the 0 and b = 0 cases, they break the PCF structure, leaving us with 1,768,900 formulas. Some of these PCFs do not converge to single limit, rendering their measured metrics meaningless (see Appendix B for the classification method we developed to predict PCF convergence). We filtered out all formulas that do not converge, providing the final filtered dataset of 1,543,926 formulas.
The conventional classification of the PCFs is by the coefficients their polynomials ag. bn (A,,A1 Ao, B2, B1, Bo), and by their numerical limit L, which we evaluate here at depth = 2000.
Going beyond these conventional classification, our methodology relies dynamics-based metrics calculated for each formula:
The irrationality measure: for each PCF, we calculate ppediittdd (Eq 5) and compute directly using the Blind-5 algorithm (presented in section 3.4 at depth n 1000. Fig 2h presents example ó evaluations.

---

The convergence rate dynamics, comprised of three parameters: for each PCF, we fit the approximation error c(n) = Ps L, which scales as c(n) ~ n! e n for large n. We store the fitted 17 (factorial coefficient), 7 (exponential coefficient), and / (polynomial coefficient). The process is detailed in Appendix A
The growth rate of qn. comprised of three parameters: For each PCF, we fit the denominator to qn n!" ennnn n for large n. We store the fitted parameters n/ 7'p"".
Based on this set of metries, we applied unsupervised clustering for unlabeled data (using the hierarchical density-based HDBSCAN algorithm [McInnes et al., 20177) The clustering is the key component in the algorithm we developed (Fig M. leading to the discovery of a variety of formulas and data patterns that exposed formula families (see sections
and 4.4 for selected results).
[a] --- Asymptate
Numerical
Predicted Evalustions
(c)
1.4
1.0
degree algebraic exporsent convergence
Exponential coefficient of i.
0.5
Rational Numbers
1.2
0.0
1.0-
0.5
100
2000
100
-1
3
conC1)
25.0 -
0.6409261
X
2n
(b)
Irattionlity
Proving
-3.0
Exponential coefficient
Limi
convergencee rate
(d)
064202268
0662299789
+5
24%
dn' 80
Figure 2: Dynamics- based metries for formulas of mathematical constants. Analysing the convergence of polynomial continued fraction (PCF) formulas provide dynamical metrics that prove useful for their automated clustering and identification. (a) Irrationality measure ō vs. PCF depth, for two example formulas of the constants cot(1) and the Silver Ratio. The of these constants known to be (green dashed lines). The blue dots show the numerical convergence of 8 (Eq to the correct value. The red dots show the evaluated å-Predictor formula 5E following the numerical 5 very closely in the Silver Ratio formula, while taking a completely different (and much slower) trajectory in the (1) formula; yet both converge to the correct value = 1. For the purposes evaluated at 10 providing an accurate estimation for 8. (b) 8n (n = 1000) vs. the limit value for PCFs in our dataset. While 8 values seem to follow a pattern, the limit value distribution does not contain relevant information (the higher density of PCFs near the axis arises from the small coefficients of the polynomials in our dataset). Our dataset contains 913, 056 maaiionaiity-rroinn formulas, most of which are not yet linked any known constant. (c) Exponential growth coefficients of On and of c(n) for PCFs with deg( B) = 2deg(A). Note the surprising "band-structures" that this view reveals. A few of the clusters have been identified, but the reason for the appearance of these "bands" and the properties of most clusters remain as open questions. (d) Example PCFs in the dataset that converge (#1000) and yet are not related to it, showcasing the challenge of mathematical formula discovery. For visual clarity, error bars not shown. See Appendix A for discussion regarding measurement errors.
of clustering, Spredicted
value close to the constant cot(1)
3.4 The Blind-ō Algorithm
The irrationality measure 8 of PCF is of mathematical interest, and (as will see in section is powerful dynamical metric. Unfortunately, evaluating 8 using Eq reqquires knowing the series limit L, making its estimation for a large set of unlabeled PCFs impractical.
The Blind-é algorithm was created in order to circumvent this limitation. Instead of inspecting the convergence behavior of Pa + L, we inspect the convergence behavior of Pa + Pm for specific
go
gn. gm
5

---

m > n. Given rational approximation Pu. + L, we approximate the error rate <(n) with
qu
Pa
Pm
= =((n)
<(mm)
4n
Qm
(())
If0 <r0<<<>
<(m)
< S is bounded away from zero and infinity for all large enough, then the
<())
approximation of the Blind-ó algorithm has the same convergence rate as Eq bypassing the need to
evaluate L. Intuitively, this condition holds whenever (c(n)] 0 fast enough, which is true for the
vast majority of PCFs (see Appendix F for details).
Note that m has to grow with In practice, our implementation of the algorithm in this work uses m 2n, so order study 8 up to n 1000, we use m 2000.
3.5 Choice of Metries for Clustering
As part of the automated formula discovery flow we choose the best metric (for each step), in terms of representation power, which is measured by applying the Davies-Bouldin Index (Davies and Bouldin 1979] on clustering using each metric individually. Table[] results for randomly chosen sample of 25K converging PCFs. Note the extremely poor performance of the PCF limit L. in agreement with p.d This dimensionality reduction is important for efficiency during the clustering step and for better explainability. The former is because the dataset size grows exponentially with the PCF degree and with the magnitude of the polynomial coefficients.
Table I: Comparison of the representation power of the main dynamical metrics (lower is better). A, 3' and (A2, A1, Ap, B2, B1, Bo) provide little value for the initial clustering and are not shown.
Metric
Davies-Bouldin Index
Limit
67.23
Irrationality measure å
1.11
Reduced denominator Jn growth factors in ~ n!! e>'n. nn
Exponential coefficient y
0.51
Factorial coefficient r/'
0.13
Error rate (e(n)) growth factors [e(n)] n! e n
Exponential coefficient 7
14.83
Factorial coefficient
0.77
Other metrics were tested. Some have been shown to have little to no representation power (e.g. Pn and yn. as defined Eq[2] modulo various primes, their sign, their GCD etc.) while others show potential and are left for future study (e.g. the leading Fourier coefficients of the "noise" around the fit of 9n)) ). A relatively small number metrics were measured and used, which helped keep the results mathematically explainable. Nevertheless, the clustering using the metrics in Table showcases the strength of our dynamical metrics approach.
Results
4.1
Discovered Formulas for Mathematical Constants
The first step in validating the dynamical metries approach is using basic heuristics on the metric space to find PCFs related to mathematical constants. There are some PCFs in the dataset that have known irrational limit (like the examples in and the PCF family
B
2B
B
A+ A 4B
A+
A+
for constant A and B), expected to find some of them. Through this test, also found previously unknown PCF formulas related to mathematical constants. Note that known mathematical formulas both the anchors for labeling and test set in our method.
natural heuristic inspecting PCFs with 8 1, marking their limits as irrational. Another heuristic we used is focusing PCFs with n¹ x 0, as it was very strong indicator for mathematical constant

---

Clustering using dynamic metrics
Characterization within clusters
(a) Cluster by sequence dynamics
(b) Clustering reveals connected formulas
(c)
Keeping
q Factorial Coefficient
@ Exponential Coefficient
data points
with B2 1
-2
2.0
-4
(d)
1.0
Rational
Golden
cluster
Ratio cluster
10g
Ao
lnn
Constant
2deg(A) < deg(B) 2 deg(A) (B) deg(A) > deg(8)
* Rational
Lemniscate Constant
Figure 3: Discovery of mathematical structures via analysis of dynamical metrics of formulas. (a) Projecting the data the 5 vs. n/' (ân, factorial coefficient) plane, it is easy to the emerging subsets. We focus on PCFs with v/ x 0, as a previous work [Elimelech et al. 2023) indicated this important property. (b) Clustering in the 5 7 (%n exponential coefficient) plane shows examples of common properties within a cluster, like rationality or convergence to a specific constant (up to linear fractional transformation). Focusing further the deg( (B) > 2deg( A) cluster (as it is a clear anomaly in the r/ RJ subset), we used a PSLQ algorithm to identify links between these formulas and mathematical constants. This identification was feasible since a preliminary step identified promising subset ~ 5, 000 times smaller than the initial dataset. (c) The result of this clustering and identification procedure a structured arrangement of formulas that reveal range of novel formulas related to constants such as π,, In(2), V2, Gauss' constant, and Lemniscate's 's constant. (d) Keeping only PCFs with B2 = 1 we are left with a highly symmetrical "checkerboard pattern" of formulas for π and (2),,, which was generalized into infinite formula families hypotheses (see section 4.3). Error bars not shown for visual clarity, see Appendix A for discussion regarding measurement errors.
formulas in a previous work [Elimelech et al. 2023 top left) that contains PCFs such as:
Combining the two gives a subset (see Fig
-2
5+
5n - 5m = 2 6
- -3
V13
(6)
5n + +
-3
Removing the limitation on T growth rate, one can find the c1) formula shown in Fig
cot(1)
2n 1+
On the other hand, relaxing the limitation 8, focusing only on n/ 0. rich structure emerges (FigBb). Diving deeper into the B-dominated subset, we find formulas (Fig for the Gauss constant
GGa Finch 2003
6
2GGA
4
4GGA
4+
4n
+ 2n
4GGA
-
4n
+ 2n 2
3G - 2
(8)
4 4+
+ '
Lemniscate constant L Lemosecate
Finch 2003
7

---

2
-6
4 +
4n
2n
Lemmiscaaee
(9)
4
As well as for second order roots, and (2) (see section 4.3 Note that unlike the formulas in Eq
and [ which are analytically proven, the formulas in Eq [ and Eq 2 are (to the best of the authors knowledge) novel. Their limits were numerically validated to large precision, yet formal proofs for these formula hypotheses remain an open challenge.
It should be noted that usually in number theory, bigger ò is considered "good", whereas smaller (often negative) 5 is considered "bad". We use å as a metric, without "judgment". These novel formulas (Eq
g Eq Bqgaand and the infinite family of formulas shown in section 4.3),, which have the "bad" 5 Rz 1, are demonstration that our "non-judgmental" approach successful.
4.2
Clustering in Dynamics- Based Metric Latent Space
This section shows that clusters in the latent space of dynamics-based metrics are successful in grouping together different formulas in a way that exposes their shared properties, such as the mathematical constant to which they relate.
Looking at the top left cluster in Fig B (defined by 9. exponential coefficient 0.6 and 5 > 0.9), recognize the canonical form of the golden ratio PCF (shown in [. This cluster also contains 21 additional PCFs, with different generating polynomials, some of higher degree. As it turns out, all of them linear fractional transformations of V5 (see Appendix C). which were labeled automatically by the formula discovery algorithm (Fig D. Another example of property conservation within clusters the rational cluster marked in green on Fp. The limits of the PCFs in this subset are varied, and its spread is real (i.e., not only due to numerical imperfections). Yet, all the PCFs in this cluster converge to rationals which not directly measured by any of the latent space dimensions.
Data Points
Anchor Points
0.31 0.44''' 0.38*1 0.42%Y + 0.62*M
17
v2
Muldi unchor clurter
e&ee
GeF
0.05 0.42*y' 0.10*n' + 0.88ty 0.20*n
Figure 4: Automated Formula Discovery Results: Showcasing the automated clustering and labeling of PCFs using set of 126 known anchor formulas, connected to constants such π., e?, CCF (the continued fraction constant), the golden ratio o, V2, v3, and V17. The clustering is visualized here via the 2 leading PCA components, revealing 441 novel, automatically discovered, mathematical formula hypotheses. For visual clarity, error bars are not shown. See Appendix A for discussion regarding measurement errors. See Appendix for additional visualizations.
Fig showcases a collection of clusters with shared properties. Using set of 126 (mathematically unique) known anchor formulas, 441 novel mathematical formula hypotheses were automatically
8

---

discovered. The constants which are related to the most new conjectures are: c² (28 anchor formulas gave 178 new conjectures), (39 anchor formulas gave 116 new conjectures), e (44 anchor formulas gave 80 new conjectures), and /17 (1 anchor formula gave 55 new conjectures). Some of the novel formulas are equivalent to known PCFs (see Appendix C for a discussion about equivalence) while other formulas were analytically proven (see Appendix A.3 and Appendix G
Note the multi-anchor clusters of and e², as well as the algebraic roots: specific constant, yet relate constants of similar nature suggesting meaningful clustering nevertheless. For the sake of visualization the algorithm stopped after the second iteration. In a standard run these multi-anchor clusters would have been separated via additional metrics.
these clusters failed to single
4.3 Detecting Patterns and Underlying Structure
As mentioned in section 4.1 the deg(B) > 2deg(A) (A). '7' = O cluster, contains many formulas of interest (see Fi and d). They were discovered via PSLQ algorithm, identifying linear fractional relations between the limit values of PCFs in the subset and notable mathematical constants (such as e). This is computationally heavy operation, and it would be challenging to run it on all 1.5M formulas in the data set. Yet by first identifying the promising clusters, we reduce the search space 5, 000 times, allowing for deeper inspection of each PCF.
Once the "checkerboard" pattern Fig was discovered, the hypothesis was expanded into infinite families of PCFs with sub-exponential convergence relating
and In(2):
an i+ + b, n² + + k)m, with integers i,j 2 0, and E (0. This is expected to be related to if k 1. and to ln(2) if k: = 0 (in fact, this pattern can be generalized even further, into a novel 3-dimensional Conservative Matrix Field, provided in Appendix C. See [Elimelech et al. 2023 for the definition of Conservative Matrix Fields).
Another formula family was discovered via clustering in the vs. 7 space. The algebraic roots subset (marked by green circle in F was generalized into:
an = 2n + - 1, b,, = -n² + jn + for integer j. k such that 6n has real roots that are not positive integers. This is expected to converge to root of bn
These are novel experimental results and mathematical hypotheses - awaiting proof.
4.4 Higher Degree PCF Identification
q Exponential Coefficient
Novel formulas
(1)
Training formulas
3.0
New equivalent formulas
2n 1II2EE
& In cluster
Truly novel formulas
Novel Golden Ratio formula
(1,2)
2.0
(3)
Novel square formulas
1.0
(5)
Golden Ratio cluster
(4)
20 -20004
(3)
=
61
1.0
1.0
Figure 5: Higher-degree PCF formulas for mathematical constants, overlaid on Fig.3b. Formulas that are equivalent to an existing element in the original dataset are marked with "o", while "x" marks truly novel formulas. The higher- degree PCFs fit, using the same dynamical metrics, into clusters trained on lower-degree PCFs, showing the ability identify novel formulas despite being of mathematical forms not seen during training.

---

The final validation for the dynamical metrics approach is to show its potential even on test sets of a different mathematical structure than the training set. For that purpose, the clusters created based on the original dataset were treated as a classifier, and test set of higher-degree PCFs (up to 3rd degree A, and 4th degree Bn, coefficient range [-2,2]) was created, measured and classified.
Naturally, not all high-degree PCFs fit neatly into the existing clusters (as they represent constants that were not present in the original dataset), but some were correctly identified and labeled, discovering novel formulas in the process (see Fig 5 for a sample of the results).
5 Discussion and Outlook
This work marks an important step toward the vision of automated on-demand formula creation in mathematics. Going beyond all previous algorithms in this field, we connect the challenge of formula creation to robust ML methods. This methodology provides a wide variety of automatically generated formulas, including both previously known and previously unknown ones, exposing their underlying mathematical structure and enabling new proofs.
The next research step directly building on our methodology could help to finally reveal the complete intricate mathematical structure of PCFs. For example, starting with the "band-structure" found in Fig or with clusters of formulas with various structures representing the same mathematical con- stant. Further exploration of our conjectures from section 4cculld have more impact on mathematics, perhaps achieving further generalizations and prescriptive formula generation.
The technique presented here can be applied a larger scope of continued fractions and for completely different types of formulas. For more general continued fractions, dynamical metrics such as the numerical trajectories and the corresponding sequences of (in addition to its asymptotic value) hold valuable information even in continued fractions that do not converge at all. We expect these dynamical metrics to provide a "fingerprint" for wider families of formulas and perhaps even for the mathematical constants themselves. This approach was directly applied in this work to higher polynomial degrees, larger polynomial coefficients, and can be expanded to continued fractions not based on polynomials. Looking beyond continued fractions, metrics that are derived from the dynamies of numerical calculation of certain formulas are an especially good fit for automated computer-assisted investigations. Such metrics can be measured for variety of mathematical structures, including ones whose evaluation is iterative or recursive, that are defined via an infinite sum, or any other process which produces rational approximants. Any such mathematical structure can be measured, clustered, and identified using the proposed method- treating the generating functions black box. We believe that such dynamical metrics can unveil patterns and underlying structures in broad fields of mathematics and in other areas of science.
To exemplify this universal concept, we looked into higher depth recursion relations, which are promising research direction because little known about their global structure, yet they are involved in several important conjectures. For example, the best rational approximation formula known for Euler's gamma constant constructed via such a recursion relation [Aptekarev] 2009]. This family of formulas is broader than continued fractions, yet they can be described by the same metrics as PCFs. Another type of mathematical structure successfully analyzed using the same method is hypergeometric functions, showing the applicability of our measurement-clustene -egteriingngnnntttionn approach to a bigger family of mathematical functions. This generalization can be useful in a wide variety of contexts, such investigations of integral formulas (c.g., Beukers-type integrals [Beukers] 1979 Dougherty- Bliss et al. 2022 Brown and Zudilin 2022
Our work was based on a limited-size dataset and on a small set of metrics. It would be intriguing to test the extracted conjectures on larger datasets, which can help reveal additional, more intricate, phenomena. Considering the success we had using relatively small set of metrics, we would like to use an -fagntue- larger set of metrics and find what new predictions can be recovered. In fact, the creation and evaluation of the metrics themselves can be automated.
Taking a broader perspective. the methodology presented in this work can be seen as a general prescription for tackling scientific discovery challenges in mathematics and theoretical physics that rely on numerical evaluations and generalizations. Such an advance is especially exciting for such challenges that were considered in the past to require intuitive leaps of creativity.
10

---

Acknowledgments
This research received support through Schmidt Sciences, L.C.C
References
References
David P Anderson. Boinc: A system for public-resource computing and storage. In Fifth IEEE/ACM international workshop on grid computing, pages 10. IEEE, 2004.
A. I. Aptekarev. On linear forms containing the culer constant, 2009. URL httth::ttaaxiiiii org/
abs abs/0902. 1768
David Bailey, Jonathan Borwein, Neil Calkin, Russell Luke, Roland Girgensohn, and Victor Moll. Experimental mathematics in action. CRC press, 2007.
Lennart Berggren, Jonathan Borwein, Peter Borwein, and M Lambert. Mémoire sur quelques propriétés remarquables des quantités transcendentes circulaires logarithmiques. Pi: Source Book, pages 129-140, 2004.
Frits Beukers. A note the irrationality of ((2) and ((3). Bulletin of the London Mathematical Society, 11(3):268-22 1979.
Douglas Bowman and James McLaughlin. Polynomial continued fractions. Acta Arith., 103:329-342. 2002.
Francis Brown and Wadim Zudilin. On cellular rational approximations to ((5). arXiv preprint arXiv:2210.03391, 2022.
Bruno Buchberger, Adrian Crăciun, Tudor Jebelean, Laura Kovács, Temur Kutsia, Koji Nakagawa, Florina Piroi, Nikolaj Popov, Judit Robu, Markus Rosenkranz, et al. Theorema: Towards computer- aided mathematical theory exploration. Journal of applied logic, 4:470-504, 2006.
Benjamin Church. Diophantine approximation and transcendence theory. 2019.
Annie Cuyt, Vigdis Petersen, Brigitte Verdonk, Haakon Waadeland, and William B. Jones. Handbook of continued fractions for special functions. Springer Science & Business Media, 2008
Nadav Ben David, Guy Nimri, Uri Mendlovic, Yahel Manor, and Ido Kaminer. On the connection be- tween irrationality measures and polynomial continued fractions. arXiv preprint arXiv:2111.044000 2021.
Alex Davies, Petar Veličković, Lars Buesing, Sam Blackwell, Daniel Zheng, Nenad Tomašev, Richard Tanburn, Peter Battaglia, Charles Blundell, András Juhász, et al. Advancing mathematics by guiding human intuition with AI. Nature, 600:70-74, 2021.
David L. Davies and Donald W. Bouldin. A cluster separation measure. IEEE Transactions on Pattern Analysis and Machine Intelligence, PAMI-1(2):24-2277, 1979. doi: 10.1109/TPAMI.1999. 4766909.
R. Davis and D.B. Lenat. Knowledge-basee Systems in Artificial Intelligence. A McGraw-Hii advertising classic, McGraw- Hill International Book Company, 1982. ISBN 9780070155572. URL https://books. google. co. 11/boo2ss1dd-M///AAAAAA
Robert Dougherty-Bliss Christoph Koutschan, and Doron Zeilberger. Tweaking the beukers integrals in search of more miraculous irrationality proofs a la apéry. The Ramanujan Journal, 58(3):
973-994, 2022.
Rotem Elimelech, Ofir David, Carlos De la Cruz Mengual, Rotem Kalisch, Wolfgang Berndt, Michael Shalyt, Mark Silberstein, Yaron Hadad, and Ido Kaminer. Algorith discovery of an intrinsic order among mathematical constants. arXiv preprint arXiv:2308.11829 2023.
11

---

Leonhard Euler. Introductio in analysin infinitorum, volume 1,2. Apud Marcum Michaallem Bousquet & Socios, 1748.
Siemion Fajtlowicz. On conjectures of graffiti. J. Akiyama. Y. Egawa, and H. Enomoto, editors, Graph Theory and Applications, volume 38 of Annals of Discrete Mathematicx, pages 113- 118. Elsevier, 1988. doi: http:///oooggoooooooooooo6660000 sciencedirect
URLhttts:://
Alhussein Fawzi, Matej Balog, Aja Huang. Thomas Hubert, Bernardino Nomera-Paredess Moham- madamin Barckatain, Alexander Novikov, Francisco J R Ruiz, Julian Schrittwieser, Grzegorz Swirszez, et al. Discovering faster matrix multiplication algorithms with reinforcement learning. Nature, 610:47-53, 2022.
Steven R. Finch. Mathematical Constants, Cambridge University Press, 2003.
Godfrey Harold Hardy, Edward Maitland Wright, et al. An introduction to the theory of numbers. Oxford university press. 1979.
Charles R. Harris, K. Jarrod Millman, Stéfan J. van der Walt, Ralf Gommers, Pauli Virtanen, David Cournapeau, Eric Wieser, Julian Taylor, Sebastian Berg, Nathaniel J. Smith, Robert Kern, Matti Picus, Stephan Hoyer, Marten H. van Kerkwijk, Matthew Brett, Allan Haldane, Jaime Fernández del Río, Mark Wiebe, Pearu Peterson, Pierre Gérard-Marchant, Kevin Sheppard, Tyler Reddy, Warren Weckesser, Hameer Abbasi, Christoph Gohlke, and Travis E. Oliphant. Array programming doi: 15-11338/41156--22--6666---
with NumPy, Nature, 585(7825) September 2020.
URL https: org 00000000/00011000022220-29999990 220-2649-2
Cezary Kaliszyk and Josef Urban. Learning- autoomam automated reasoning with flyspeck. CoRR, abs/1211.7012, 2012.
Jeffrey C. Lagarias. Fuler's s constant: Euler's work and modern developments. Bulletin of the Ameri- can Mathematical Society, 50, 2013. ISSN 02730979. doi: 10.1099/50233-09999 00007773-99992211--111 2013--023--------
Johann Heinrich Lambert. Mémoires sur quelques propriétés remarquables des quantités transcen- dantes, circulaires et logarit hmiques. Mémoires de l'Académie royale des sciences de Berlin, pages 265-322, 1768.
James Mc Laughlin and Nancy J Wyshinski. Real numbers with polynomial continued fraction expansions. arXiv preprint math/0402462, 2004.
Douglas B. Lenat. The nature of heuristics. Artificial Intellignne, 19(2):1892244 1982. ISSN 0004- 3702. doi: http:///aauurr////.ooo0000000100
BtLphttps://wwt URl.ttpp:///ww..iimmiiiiioo sciencedirect
Douglas B Lenat and John Seely Brown. Why am and eurisko appear to work. Artificial intelligence, 23(3):26--294, 1984.
Leland McInnes, John Healy, and Steve Astels. hdbscan: Hierarchical density based clustering. Journal of Open Source Software, 2(11):205, 2017. doi: 10.21 105/joss.00205. URL https
21105/joss 00205
Marko Petkovsek, Herbert S. Wilf, and Doron Zeilberger. A=B. AK Peters Ltd, 1996.
Gal Raayoni, Shahar Gottlieb, Yahel Manor, George Pisha, Yoav Harris, Uri Mendlovic, Doron Haviv, Yaron Hadad, and Ido Kaminer. Generating conjectures on fundamental constants with the ramanujan machine. Nature, 590, 2021. ISSN 14764687. doi: 10. 10000/6411566001-0322999
Tal Ridnik, Dedy Kredo, and Itamar Friedman. Code generation with alphacodium: From prompt engineering to flow engineering, arXiv preprint arXiv:2401.08500, 2024.
Trieu H Trinh, Yuhuai Wu, Quoe V Le, He He, and Thang Luong. Solving olympiad geometry without human demonstrations. Nature, 62767995):476-4882 2024.
Josef Urban. Malarea: metasystem for automated reasoning in large theories. ESARLT, 257, 2007.
12

---

Pauli Virtanen, Ralf Gommers, Travis E. Oliphant, Matt Haberland. Tyler Reddy, David Coumapeau,, Evgeni Burovski, Pearu Peterson, Warren Weckesser, Jonathan Bright, Stéfan J. van der Walt, Matthew Brett, Joshua Wilson, K. Jarrod Millman, Nikolay Mayorov, Andrew R. J. Nelson, Eric Jones, Robert Kern, Eric Larson, C J Carey, Ilhan Polat, Yu Feng. Eric W. Moore, Jake VanderPlas, Denis Laxalde, Josef Perktold, Robert Cimrman, Ian Henriksen, E. A. Quintero, Charles R. Harris, Anne M. Archibald, Antônio H. Ribeiro, Fabian Pedregosa, Paul van Mulbregt, and SciPy 1.0 Contributors, SciPy 1.0: Fundamental Algorithms for Scientific Computing in Python. Nature Methods, 17:261-272, 2020, doi: 10.1038/s4 1592-019-0686-..
H. Wang. Toward mechanical mathematics. IBM Journal of Research and Development, 4(1):2-22, 1960. doi: 10. 41.02.
Stephen Wolfram et al. A new kind of science, volume 5. Wolfram media Champaign, 2002.
13

---

A Numerical Measurements, Curve Fitting and Validation of Formulas
A.1 Numerical Measurements
When characterizing PCFs, we use several metrics extracted from the dynamic behavior of the formula:
The growth coefficients 7, Y, (of the form n!" e n of the convergence rate e(n)..
In (as defined in Eq growth coefficients: , (of the form n!! e²"n).
The (as defined in Eq calculated using the Blind-ò Algorithm described in section 3.4
To measure the growth coefficients of Gn and E(1), the values of log (e(n)) (see section 3.4 and of log (Āx) were evaluated up to depth 1000.
The most resource-intensiig values that are generated are Pn, 9n and ged(pn. qn) all other values calculated from them (and require less precision). For the worst case PCF this requires 36MB of memory (without optimizations) and ~ 1.9 seconds of run time on a single core of a basic workstation, which translates to upper cap of ~ 900 hours for the whole data set. In practice power cluster with 64 cores, which ran each iteration of the measurements in 8.5 hours.
used high
Once these values are calculated, using scipy [Virtanen et al. 2020) and numpy (Harris et al. 2020 fit of the form log (n1"... e n was calculated for 4n. and <(n), producing the dynamical metrics.
A.2 Curve Fitting
A curve fit using 1000 points is fairly heavy operation, unsuited for large scale investigations. Instead, we used an extreme waw-lag. Specifically, only 5 points were used for the fit.
One may justifiably wonder if 5 data points are sufficient to fit accurately enough the desired metrics.
A test comparing between 5 data point fit and 1000 data point fit done. As the test set, 50 PCFs were randomly chosen out of each of 9 categories (450 total test cases). The categories were all combinations of deg(a)
0, 1, and deg() (b) = 0, Focusing the dominant coefficients (y and 1)), for each case, a full (1000 point) fit was performed (producing Tf,1f) 11y), and compared to the down sampled fit of 5 points (producing Tp: 7pp) We tested 2 methods of choosing the points,
even (i = 6, 206, 406, 606, 806) and logarithmic (i = 6, 125, 250, 500, 1000). The relative error was then calculated ( (1op~ and "y--"s for c(n) and q... The relative errors were then averaged the test (results summarised in table 2) showing the 5-point fit be almost as good as the full 1000-point fit. In our measurements we use the logarithmic point distribution as it gives better results for most metrics.
lysl
Table 2: Comparison between 1000 point fit results and point fits results (even spread and logarithmic spread).
Behavior
4n
Convergence Rate
Coefficient
Y'
Y
Relative Error Average (even spread)
0.0124
0.0007
0.0078 0.0024
0.0005
Relative Error Average (logarithmic spread)
0.0175
0.0004
0.00012
Licences and versions of Python packages (used for curve fitting, clustering and large number mathematics):
Scipy (Version: 1.11.3) BSD License (Copyright (c) 2001-2002 Enthought, Inc. 2003-2024, SciPy Developers. All rights reserved.)
gmpy2 (Version: 2.1.5) GNU Lesser General Public License v3 later
Numpy (Version 1. 1.26.1)- BSD License (Copyright (c) 2005-2023, NumPy Developers. All rights reserved.)
14

---

A.3 Validation of Automatically Generated Formulas
After a PCF has been identified as potentially connected known constant C, undergoes verification process in order to establish whether it indeed converges to a known mathematical expression involving C. This process consists of several steps:
Expression Identification Through PSI. Analysis
The PCF limit is computed and targeted PSLQ search is used to identify an expression that matches the resulting digits and incorporates the constant C.
2. Expression Verification
After candidate expression has been identified, the PCF is computed greater depth. The resulting values are compared against the proposed expression to assess the accuracy of the match and verify convergence.
Systematic Proof
Once the expression has been verified, an automated analytical proof is attempted via the following steps.
Given two polynomials a(n) and b(n) defining a PCF, Euler's formula Euler 1748 states that
when there exist polynomials hy (n), (n),,, fn) such that b(n) = hy (2) n1) J (n)a(()
f(n - (n) + fn + 1)h2(n + 1). the limit of the PCF equals:
((-1)h1(0)
f(1)h2(1) f(0)
f(0)((1) (k))((++1)
h1(1)
II
(10)
f(0)
h (1+++1)
The proof is then completed using known identities of infinite sums.
the PCF's polynomials do not satisfy Euler's formula's 'as conditions, the PCF is flagged for further examination.
For example, one may take the PCF defined by the polynomials en, 2 and b, 4n2 + 4n + which was labeled related to The PCF numerically identified and validated as converging to 3 Next, the polynomials h1, fi2 and f were identified as 2n 1, -(2n + 1) and respectively,
--
which corresponds to the sum
3
This infinite sum is known and was indeed
Er Ih 2043
proven converge to 1 3-
For the entire of proven formulas, see table
5 in appendix G
B Classification of Continued Fractions
Not all PCFs converge. Clearly, if does not have well defined limit, then some of our numerically measured metrics lose their meaning. Though had algorithmic safeguards detect such cases and remove them from the analyzed set, it was valuable to identify a pattern and formulate a rule-set that predicts the convergence of PCF.
4n
For that purpose we turned to the matrix representation of continued fraction to depth n (see Appendix F.1 for details):
N-1
pN-
pn
0
bn On
(N-1
UN
n=l
N-
0
f do
0 aN
On-1
0
n=1
Assuming a, + 0 for n 2 0.
Analyzing the eigenvalues of the matrices within the matrix product - allows for examining the asymptotic behavior of the continued fraction. Their characteristic polynomial is X²
by
R2 Cn
15

---

and we propose observing the discriminant of this polynomial more specifically its dominant power of
4b,
n
Cgn + O(n²²))
(11)
0pan-1
Here assume C3 * and is some integer. Based the data, compiled table as summary of the conjectured behavior of any polynomial continued fraction based on 8 and Cs.
Table 3: Summary of PCF behavior characterized by and C2 defined in Eq 11
Convergence
C.00 0
Cg
8 823 3
82 0
8 $52
8 -1
We can further claborate the converging cases by discussing the conjectured rate of convergence. Usually, PCF is expected to converge at a sub-exponential rate, but the case of 8 = 0, C. > 0 expected to converge faster:
If Co + then the PCF will converge at an exponential rate, and the exact rate of convergence increases monotonically as Co 1, with vertical asymptote Co- The convergence rate is identical for Co and
C
If Co 1 then the PCF will converge second most dominant power A, = Co + C. + 0 प for some C, ş 0 and integer t >> 0
factorial rate. More specifically, if we find the nt
then the precision will grow at rate of Onn!).
We used these rules (in conjunction with the measurements mentioned in section 3.3 to validate that all PCFs we analyze and cluster do converge and their measured metries are well defined.
C Discovering equivalence of continued fractions
Polynomial continued fractions use two polynomials an a(n) and b, = b(n) to generate sequence of rationals Pa/qu. However, same sequences with identical behaviour can be generated using more then one set of polynomials. By identifying transformations under which the dynamics of p//4g remains invariant, we can formally prove equivalence between data points, validating the clustering power of the chosen metrics.
By rearranging the continued fraction definition, we can see how equivalent a, and bn series can arise:
a
b
bz
00
1+
4g01
b2
agCo
C0
1 1+
Cg
b2 02
a1
+
by
1 +
1+
02
+
on
1
dg
1
+
16
an '.
1 +
14
(12)
Indeed, by defining pair of polynomials d' Cn 1/ bn we get equivalent continued fraction which converges to cops. Clearly, since the resulting sequence P' is identical to the original one, it exhibits the same dynamics. We call this process "Inflation by c,
%
The metrics we are interested in are mostly not affected by a finite number of elements in the sequence. For example, both the convergence rate and 8 discuss an overall trend as grows. Consequently, we can initiate sequence at different values of n e 0 without changing the latent parameters. When expressing these transformations modification to the continued fraction definition, we see that the limit of the continued fraction might change due this shift in sequence initiation, but only by rational fractional transform.
16

---

b1
pn
b2
b1
00
b2
qn
01
b3
Pn.
dg
01
+
02 bn
On
b3
02
D
Og
+
On
For example, consider the cluster of formulas related the golden ratio shown in figure 33p A large portion of these PCFs stem from transforming the known formula for the golden ratio shown in Eq via the methods aforementioned. The exact transformations are depicted in Table 4
Table 4: Continued fractions converging to linear fractional transformations of the golden ratio é, found using the top left cluster of Figure3p. Numerous data points in this cluster exhibit identical sequence dynamics and are equivalent under the inflation and index indentation transformations. The equivalent data points create families of continued fractions in the cluster. Discrepancies between the calculated irrationality measure within the same family is ascribed numerical inaccuracies, typically on the order 0.001. However, when comparing families, discrepancies in the irrationality measure rise to magnitude of 0.04, suggesting potential deeper distinctions among these PCFs.
Transtornation
Irationnlity measure
Family's a canonical form
100168
Inflation by On
100 168
20
Initation by
00023
-266
Inflation by Cn
1.00023
nin 1)
Inilation o.,
1.00168
(n
nin 1)
Inllation
-(1
00168
+ 2
(n + 1 +
20
Inilation by
12
1 000223
(n + 2)
(n + 1]
-26
Inflation by on -11 + +2)
000223
2n
(2n (](2n
Inflation by cn (2n
1.00 168
(2m 1)
(2n [] +
Inilation by = 12n
1.00168
2 + 1)
4m +
2g
Inflation by on 2(n +
1.00023
2(n 1)
Imin + 1)
Inclation by Cn. -2im
00023
642
Family s canonical
1.00
[0+ +
Inflation by Cn
168
5(m 1)
- (m +
db 2
Inflation by
1.00 168
5(n + 1)
-5a(y + +)+0
- [6b + 2)
Inilation Cn - (n
1.00168
+ 2
nin 3)
(30p + 6)/19
Family's i camonical form
0.96967
-(n + 2)
(1 + 3)
(300 + 2)/19
Intlation by Cn = -1
0.96967
() 4)
(300 + 23/1T
Indent
0.97245
(n 3)
(n ( 1](n +
[300 + 21/11
Indent
and inflation by
0.97245
n(n 5)
(750p + 2401/361
Family's 's canonical form
0.95243
(n + 3)
n(n 5)
(7500 + 240)/361
Inflation by On -1
0.95243
D Scalability to Larger Datasets
The methodology can be scaled to larger data sets in multiple ways. For example, by extending the range of PCF coefficients from [-5, 5] to [-10, 10], the size of the data increases by approximately 50 times. Additionally, by considering polynomials of higher degrees such as advancing to third- degree 0n and fourth degree b, with coefficients in [-5, 5] the dataset size can be amplified by approximately 1,333 times.
To manage the substantial computational demands associated with these expansions, a three-fold solution is proposed.
First, dynamically chosen measurement depth is implemented during evaluation, aiming for fixed precision across all PCFs rather than maintaining a constant depth for all computations. This approach optimizes computational efficiency by adjusting the measurement depth based the specific convergence rate of each PCF.
Second, using known equivalences of PCFs such as inflation (see Appendix C it is possible to substantially decrease the effective size of the dataset that needs to be measured. particular, when On -1 observe that the sign of o, does not affect the dynamics of the sequence only flips the
17

---

sign of the limit to L. For every PCF its inflation by -1 is also contained in the current data set, and clearly will have the same dynamics-basee based metrics. This equivalence single handedly de-facto cuts the size of the current data set by half (to 771, 771,963 converging formulas).
Third, the inherently parallelizable nature of computing metrics for each formula is leveraged. The algorithm has been adapted and deployed within the Berkeley Open Infrastructure for Network Computing [Anderson] 2004), enabling parallel computation across thousands volunteer computers. Assuming a typical contribution of approximately 1,000 BOINC volunteer cores, processing the expanded data requires about one month of computational time.
The approaches described enable the methodology to handle larger datasets efficiently, facilitating the analysis of more complex polynomial continued fractions within practical computational limits.
E Clustering Visualizations
Multi-anchor cluster
3
17
CCF
& Multi anchor cluster
Anchor formula
Identified, related via
PSLQ to anchor
Unidentified
Figure 6: Automated clustering and labeling of PCFs via a 2D iSNE with perplexity 10. Clusters were verified via PSLQ relations between members of the cluster and known formulas. For visual clarity not all points are shown and error bars are not shown, see Appendix A for discussion regarding measurement errors.
Fig Fig show alternate clustering and 2D visualization approaches (in addition to Fig4)) The same set of anchors was used for all versions. The tSNE algorithm (Fig. (6) provides a visualization that separates well between clusters, but loses out explainability. In Fig the opposite approach was taken using only 2 dynamical metrics for clustering allows each cluster to be defined very clearly, but information from other dynamical metrics (which can be used for better cluster separation) is lost along the way.
Despite their differences, all 3 techniques point to similar results and conclusions, providing additional validation for our conclusions.
F Analysis of the convergence rate
The growth rate for simple continued fraction or equivalently for constant linear recurrences is well understood, and usually boils down to the matrix defining the recurrence, and its eigenvalues. In our case, the coefficient the recurrence also depend on n, so their study is more involved, however the ideas are similar, which now describe
18

---

Andlee Prinds
2.0
1.5
Cop
7f 10-
e qjar
0.5
Multi auchor cluster
v3
Ø
v2
-Q
-0.75
-0.50
Figure 7: Automated clustering and labeling of PCFs via HDBSCAN. Only dynamical metrics were used, d and n/ (as in F), and yet the clustering is already informative. For visual clarity eиror bars are not shown, see Appendix A a discussion regarding measurement errors.
F1 Approximating the error rate
To find whether or not the sequence Pa converges and so what is its convergence rate, we note the continued fraction formula
b (1)
Pn 4T
b 62)
(1) +
b (3)
a (2)
M(n 1)
aln 1)40
can be rewritten in matrix form as
Pn-1
Pn
72
II
(1
b 6(k) a (k)
9n-1 9n
In particular this implies that both Pr, and On satisfy the same linear recurrence:
Un+1 = a (n) 26, + b (n) Un-11
with initial conditions
Po go 91
Pi = 1 9)
Trying to determine if there is convergence, use the Cauchy condition. For any > we have that
1 det
Pk
Pk++1 [k+1
m-1
m
i= b(i)
Pin
Pn
Σ
Pk+1
Pk:
Pk+ 4K
qk+1Pk
qm
gb++1
9k
q gk4k++1
72
19

---

Ma) Corollary 1. The sequence Kig a[m] converges if and only if 21 same limit L. More over, the convergence rate is
11;-, 80)
converges, and to the
gE02+1
1
c (n)
Pn
L
On
4x-94+1
This suggests that we should understand the growth rate of both and II b().
Remark 1. Note that the convergence and its rate might depend on the sign of (-1)" 1 (()
(-1)* 11 60)
1. Suppose that
64 If the signs do not alternate, then
01
Σ (-1)*
IT: 6()
En
1 kd-
This diverge if d = and has order of magnitude
for d >> 1. However, with alternating signs we get the smaller bound
80
00
(2k (2k+1)- (2k) -22 (2k) (2k 1)ad
d d(kk)d 2k
Σ
Σ
(2k)
(2k (2k 1)
nd
Thus, it always converges and with better rates.
2. However, for faster converging sequences we do not expect alternating sign to affect the convergence rate. For example, if [IT"=(-b(<E) xm for some 0 < À < 1, then with only
2m
positive signs the limit will be A
1+X while for alternating signs it will be (-A)"
14X so in any
case the convergence rate is exponential.
F.2 Growth rate of I 16 (k)]
Claim 1. Let b (I:) be polynomial of degree d, with leading coefficient of absolute value B. Then there exists constant C' 0 such that for any integer N have
N
b (k)
(Ne)-C
II
S (
Bk4
Proof 1. We may assume that the leading coefficient of b positive. Writing b (xx Σó bعot with by B & O, want approximate the product (of the absolute value) of
b (kk) = Σ kaj
Hence, can find integer constant Co 2 1 such that for all le 2 1 we have
(1-1)
Co (R)) <
Co
For all large enough, all the expression above are positive, so we get
Co k
Co
In
1
< In b (1))
In
k
With the goal of summing up these expressions from to infinity, we claim that there is some constant M >> 0 such that for any C" IR, and 2 |C'| < n N we have that
N Σ n
C'
N )
In
1
C' In
$ M.
(13)
k
20

---

Given this claim we conclude that
N
(k:
- (C (N) + [M - Co In (2Co)))
In
S Coln + [M-
- Co In (2Co)
Bkd
k=2Co+1
For another C' large enough (independent of N), start the summation from k get
N
(I) +1 &
Σ In [Kk)
S CIn (N) + 1).
Finally, exponenting it back we get the result we wanted:
(Ne)
b (k)) s (Ne)
We are left to prove Equation 13
Using the Taylor expansion of In (1 + x) for [x|| S 1.4 we know that there is some large enough
0 < Mo such that
Il (1 + x| Mox²
follows that for |C' S N we have
In
+
S M
Σ 12
S
MOO (2).
k
k
In addition, we have that
Σ
1 dx
< 1, and
N
dx
In
n-1
n-1
Therefore
N
C³
N
Σ
In
1
1
C²
$ +|0*| (2
k
n2
is uniformly bounded.
F33
Growth rate of qn
The sequence qu satisfies the linear recurrence
go+1 aa (1) gn + b (n) qn-1,
or in matrix form
(()
(n))
(ga, qn+ -4m- (4
1
If both a (z). 6 (z) are constant, and therefore M = M (n) is a constant matrix, then this problem reduces to simply (gn, 9n+1) = (go. q1) Ar". Its a standard exereise to approximate 4.. using the eigenvectors decomposition of M. However, in general not only M (n) is non-constant, its entries have different orders of magnitude.
Thus, we would like to move to an "equivalent" system where at the very least M (n) converges to some matrix Mac, and then hope to show that the behavior qu can be read from the system with M This equivalent system will be built in two steps: first we "balance" the matrix, so its coordinates growth rate are the same, and then taking outside as a scalar, the remaining sequence of matrices will converge.
21

---

F33.1 Matrix balancing
This balancing is split into two cases according to the degrees of d, = deg (a (x)) d. = deg (b (x)).
Let max { dd. } and denote by A, B the coefficients of xt, شي of a (ar) b (x) respectively. Note that both A, B are either the corresponding leading coefficients zero, depending
whether d = d, respectively d 2d. If 2d < d, and d, is odd, then d < d = de and we still consider A
2.
to be zero. Regardless the choice of d. we see that at least one of A or B is not zero (and both 2d,- ds. which we call "balanced" PCF).
With this choice, taking dn-
9a (xl)a
we obtain the linear recurrence
(n)
(n)
an+1
(n + 1) In +
fn-1-
(n (n + 1))
Letting ā (n) recurrence converge, and not both to zero:
a(n) fn+ and b (n)
(n(*+1))² by our choice of d we see that the coefficient or the
b(z)
lim (n) A
lim (n) = B. -2000
Here too we can also write it in matrix form, namely
b 6()) (ān, To+ 1) = (1)=1;90) () ā (n)
We now have limit matrix, and the dynamics of such matrix well known. If both eigenvalues are real which are distinct in absolute value, then we expect exponential convergence. If both are non real, and therefore complex conjugate we expect behave like a rotation, and therefore will not converge. In both of these cases, since the eigenvalues are distinct in the limit, this holds for almost all n, so this behavior should hold in general.
In the discriminant zero, the situation is much more delicate, since we can converge to zero in many ways. For example, the discriminant along the way can be negative, positive or zero. In this notes we will restrict the study only to the two real eigenvalues with different absolute values.
F.3.2 Asymptotics of the continued fraction recurrence
The main goal of this section to approximate the growth rate of solution un the recurrence
un++1 Un + b, Un-1.
where both an bn converge (and not both to zero) or in matrix form
Un 1o41 ) Un-1 Un Mn
Mn (1 am
bn
where Mn + M = 0 )
The first step the standard conjugation simpler matrix. Indeed, if D PMP simpler, e.g. diagonal, then D, := PM,,P- + D, and II; M, = P-1 II" D,P, so we more or less need to understand Il D,.
A1
0
In the constant diagonal case Dn
with A > /22/. we expect that for almost every 0 %
initial condition [((1,, B1) D*| ~ A/. This is true as long as the initial vector is not in R 62, and we have similar behaviour for other type of matrices. When the D,, are not constant, we need to take little bit more care. The image you should have mind is the following:
Instead of the two eigenvectors being the X and Y axes, they only converge to it, we only know that they are somewhere inside the red and blue regions. Thus, to understand this system we first need separation condition saying that these regions are disjoint. Assuming the X-axis is the
22

---

P
Figure 8: Convergence with variable coefficients
pulling axis (larger eigenvalue), we will need at least one point outside the error region around the Y axis, which we call the initial condition. Once both these conditions hold, standard investigation of diagonalizable product will show that the point's orbit converge towards the eigenvector in the X X-region. As this region shrinks X in the limit, we that the limit of the orbit is there well.
Lemma 1. Suppose that Dn D where D
("o A+ 0 x_) with 0 <
A=
and let Kn.
A
[x+] maxk2a DD DI
Fix some initial 21 and let 2 (21) k-1
II1-
Dn. Assuming that for some we have
Separation condition:
A d
+ 4hin 1 and
Initial condition;
|2nz
Mn
(un - 2N Uten + 2kn)
fh =1-1
2K
2NH
Then lim [x 0. k-000
Remark 2. Note that t (H., (M, + +ren)
1-|11 + 00 as f6p + 0, so this initial condition
2k.
becomes easier to satisfy as n -> D0.
Proof. First, proving the claim for x Dn instead of Dn, may assume that the limit is D
where
Next, note that whenever = [ 2x > 2k 0, we have that v(u 2n. (a - 2x.
22 4x² < μ. Setting
Px + v (Mx
2x) (Nxx + 2x)
(R)
2x
get that v_ (k) < 1 (k) are real numbers, and the condition in the assumption is |2,l D4 (Na)). Our main goal is to prove our process satisfies:
1. [ < 1 (Na) for all k 2 n and,
2. We have lim sup, Izu| < v_ (Kn).
Assuming these two steps are true, the full proof is not too far behind. Indeed, since D,, D. the sequence Kn = sup DD, - D|| converges to zero, and note that as Kn, - 0 we get that 1/4 (k,, / oo
k2
and v_ (Kn) y 0. Assuming step (1), for 2 m 2 we have [zx] < 24 (Kw) < U4+ (Km). and by
step (2) get that lim limsup; |z p_ (km) 0.
For the remaining of the proof, without loss of generality we may assume that n 1 and just write K, instead of Kns Pn-
23

---

To prove these two steps, consider the change from 24 to 2++1- Writing Dj since Ek+1 (2x) Da and
[1tei 61,2 F2.1 A+62,2
D 1o S Ks get that
F122 + 2 + E2.2)
K |2 (11) + x) 1 n - 16 12
Ek+1|
(1 + 61,1) + 2x,2,1
Note that the final denominator is positive, so that the last inequality is valid. Indeed, using the conditions of the claim we get
11+ H v(p 2x) (1r + 2n²)
1 1 K (1+ + |xx)) 21 1I-
1
> 1 - (K + P) ||] + x>
2x
Thus, we can rewrite the inequality as
Mg = (1 + K
|zk+1| S Me (lzx)),
K 1 1-k)
(14)
The goal now is to show that if |=x| is "large", then | is much smaller, and if |&x is small, then |zk+1| cannot increase too much.
simple computations shows that the eigenvalues of this matrix are
7 Al + 1 v (p - 2x) (μ 2x)
2
and since V(u+ (k)) (1 - 2x) S μ 5 - |*/. we get that
?+ 7- 0.
Finally, the corresponding (right) eigenvectors
0=
(4t
V4 V-
)) to obtain
To simplify the notations, let us conjugate by the matrix T
1
7
T-T
7+1
Note that the Mobius map
T (3)
vー
V_ V
V_ V+
V_
-1
V+
2
1/
sends v_ + 0, // + Da and 0 + that our two steps from above equivalent
K= < 0. In particular, it is monotone increasing on v4,,, so
1. T (1zx ) |- 14 00).
2. lim sup, T (1zx) )
and the claim's original assumption is that 7 (|21)) E 1- 20). However, now this claim is simple, since in these notations we get that
(Me ()) (T- M.T) (T (1=xl)) 7- -- (12xl)
74
and 0 < I= < 1. Thus, if T-1 (|zul) E I- -h x)). then so is T (Me (1zx))) E [- 1- xo), so by
Equation 14] and the monotonicity of T, we obtain that
T ((z4+1l) < 2= F.T T (12x1).
7+
which implies the two steps.
24

---

Returning back to the recursion, we get the following.
Theorem Suppose that we have a solution to the recurrence un+1 = an
Un + b, where
an bn - b and suppose that r are the roots of x² = az + with |r_l < )4. Writing
Kin Pa max max {lax= al. |b bl} and (A1) 1-1a+ Assume that for some n we have
k2n
| + 4C (A+)
Separation condition:
R/n and
Initial condition:
A_
Un ox-1
2 ((+t)
K
A
1 x
10
Then
Un
+
X
Un-1
Proof 2. Set Mn = (1 bn) on
and M = (9%) as in the beginning of this section. With P=
(1
and p-1 -ー A -> we have that D = PMP-1 A 2 ). We would like to apply
Lemma to the matrices D,, = PM,PP
For the separation condition on the infinity norm, we have
PM, p-1 Dll = =|| M) 10
(1 (0 bn- ( - )
2 - 1
c(A+)
A__ 2+1 (122) (b-by an
1+ |
Tدl X_ || M
Thus, the separation condition of this theorem implies the separation condition of Lemma
+ 4m
S
+ 40 40(()))
max || - M 1 R2n
A+
Next, for the initial condition, setting
k-1 II Mn
1-1 (vo 11) p-1 II
(Uk-1 uk) = (vo v1)
Dm P
we have
k-1
(ax, 3&) = (vo u1) p-1
II Dn = -VK--2, 2/k P-1
Setting 2n
Pn a.
we get that
8n
\ + Un
A x_
x_
En
+ X Mi
1 +
On
X_ 1n 1 Un
Un
ox
Using the assumption that -_ en
C (A+) R/ 1- -
1 -_
4C(A2 1x%
Kn
A4 - x_
2
4x..
that the expression above is
A4
x+
4km
2pn
2Kn
pn + V(nn
2kn)
(Min + 2kn
(+)
< 14
Fin
2k
2k
This was the second condition needed for Lemma D. so we can now conclude that
d
x_
Bn
1 +
__
Un
=
+ 0
Ux
which implies that Un. 1n-1
+ At.
25

---

F.4 Conclusion
We return now the original problem with a Ki kpaa(a) b(n)
a(n) and assume that a (2), (x) have degrees
da dp As mentioned before, we split our study into two cases:
The balanced case
Assume that d, = 2d, 2d, and let A, B be the leading coefficients of a (x) (xx) respectively.
00 and we assume that the roots A of x²² Ax + B
In this case the limit matrix is Mo
A
satisfy A-1 X4. Using Theorem
Jonce the conditions hold, obtain
9n+1/ (n 1)!d
gn+1 (n 1)d
X4:
In
qn/
implying that on mn! exp (o (n)). As for the product of the (k)), using Claim 1 we have that
Il 6 (k) exp (N)) B Nisd
Putting them together in the error rate expression, get:
T (k)]
[B] 1B|" (m 1)!2d
4m 19m (m 1) (m) | exp (o (m)) (*)
Note that [B] = |det (Moc = 11_ so that the expression above
(+)
/- tm exp (m)) exp (m log /A/ + (m))
Thus, for given E
where
+ < 1, and for any m large enough we see that (+) s
m-1
We conclude that the error rate for all large enough is bounded from above by
(k)]
= -(2 x
A+ +
+
On
9m 19m
x+
+
m=n+1
m=
x+
follows that
Pn
S nn ((/211 ++)) 1
A_
++)) ~ n In (12m)
In
In
x+
The unbalanced case
b (n)
Suppose now that d, 2d, 2d, so that B lim
0. This time the two roots of 222 = Ar + 0 are x = 0, A. If needed, we can use a simple continued fraction inflation
no+00 (n (n + 1))
(1)) and assume that > 0. Using Theorem the two conditions hold, we obtain K7 (- T)(())
In nd r exp (n)).
Letting i be the leading coefficient of (x) in absolute value, Claim implies that
II k=1
1 (+)) exp (o (N)) BN Ni4.
Again, together we obtain that
T (k))
mm-1
(m 1)!d
TB
(m md! A2m-1 exp (o (m)) (m 1)!2d,
exp (o (m))
4m 9ml
A2
26

---

Similarly to the previous case, given > 0, and using the fact that 2d - do 2 1, for all n large enough we obtain
Pre
A=1
(k)
1
B
+
Gn
qm
19m
(m
1).2d
-o
A2
m=n+1
m=n+1
B
8c
n!
2d. -d
B
m
+
n2!2d de
A2
(n + mmm
B
8o
2d -d
B
+
+
71. 22. ds
A
A
Lm=0
The infinite sum in the last expression converges to finite limit C, conclude that
Pn
a S (db 2da) In (n!) + aln
+
+
Ö
(ds 2da) n [n|.
On
A
G
List of Proven Formulas
Table 5: This is a table showing the automatically generated conjectured formulas that were analyt- ically proven. For the method of proving, see appendix [A.3] Note that lines 1,4 and are proven infinite families of formulas, generalized from the cases found in the dataset.
An
B.
hu (n))
h2(12)
f(n
Known limit of resulting infinite
w
+
? + wn
72
n + w
(w + 1)/2fi 1, 2, 1)
n² + An
+ 4
131
n² +
-
+
12)
w
(wn + 1)²
-m
wm +
(w + 1)/2 Fi (1, w+1 2wit1
4n²
An +
-2
2n
n²² + 2n +
-
7 +
log (2)
log (2)
n² (w + 1)m +
1
+
(w + 1) (1, -1) -
n² + 4n +
-0
+
34 48 log (2)
2
n² + 3n +
f2
+ 2
n² +
n
+
n 3
17- Ing(2]
n²? + 4n +
-
+
T n
3 (2))
120 log, (2)-83
n²
+
1
n 3m +
n
+
n n
16 log (2) -1
4
4n² + An
2n
2n +
2log (2)
n²
n
n
+
1-log (2)
n² + 2m +
-n
+
n
log (2) -2
log (2)
n²
4n
2
4(3 log (2)-2)
(2)
n² + 4n +
71 + 2
-n 2
2log (2)-1 2
4n 1
-2n +
2n +
+
27

---

4n
+ 2n
-2
2n + 2
22-12v/2
4n
2n
-2n
2n
3 3+ 2/
An 2 2n
+
2n
+ v2
2nd + 2t
Tt
x²²
x²²
C(2)
2n
n
=1
n²
((2)
n² +
n + n² +
"(n² + n + 1) (v2 ()
(Φ - 1)( 1( 2
+ n 1)
2nd 2m 2
n² n² +1
(V2 1)( n² +
+ /2
2n² + 2n 2
2n 2n +
(v3 + 1)(n² + n + 1)
(V3 ()(-n² + n 1)
+ v
-5
n² + 2n
2
1 + 3
22 log(2)
-5
n² + 4n
-n 4
658 +80
-4
n² +
11
-
T n+
12- 16 (og
-4
n²² + 3n
11
- 3
1-32 (2)
-4
r 3n
2
T
(2))
n 5m
+
-
160 log
4n + 4r
2n
-2 (2)
33
n²
11
-Y 2
3-3 log
-3
n²² + 2n
1
2
+12 log (2)
3
v=²
2n
+
YY 1
3
20 (2)
-3
n² + 4n +
+
- 3
3 48 log (2)
-3
n² 4m +
+ 2
2
21
5 14
105 log (2)
22
n
π1
1
2: log
-2
n + 3n +
7 +
-n 2
1 9- 12 log (2)
2
An
2n
2n
log
n
n
n
log (2)
-1
n + 2m +
+ 1
-n 1
2 log (2)
n² 4n
2
73
2
b +3log (2)
-4
An²
2n
-2n
#
-2
4n
4n +
2n
-2n +
-2
4n2 + Am +
+
-2n
4n² + 2n
+
-2n
20 10V2
-5
4n 4 2n
2n +
-nn
32r +
561
)
(r(3
14 (())
-3
4n 2n
2n
2n
-2+2y2
28

---

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: Yes, the main results, motivations and aspirational goals are included in the abstract (section 1 and introduction (section 2
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 the paper and important assumptions and limitations. A No or NA answer to this question will not be 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 is clear that these goals attained by the paper.
2. Limitations
Question:
Does the paper discuss the limitations the work performed by the authors?
Answer: [Yes]
Justification: We discuss limitations and required assumptions of the Blind-ô Algorithm in section (..4] the dynamical metric estimation via down-sampled curve fitting Appendix and of PCF convergence and the delta-pu cetta-predictor formula (Eq 5) in Appendix
Guidelines:
The answer NA means that the paper has limitation while the answer No means that the paper has limitations, but those are not discussed in the paper.
The authors are encouraged to create separate "I "Limitations" section in their paper.
The paper should point out any strong assumptions and how robust the results are 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 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 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 are not acknowledged in the paper. The authors should use their best judgment and recognize that individual actions in favor of transparency play an important role developing norms that preserve the integrity of the community. Reviewers will be specifically instructed to not penalize honesty concerning limitations.
3. Theory Assumptions and Proofs
Question: For each theoretical result, does the paper provide the full of assumptions and complete (and correct) proof?
29

---

Answer: [Yes]
Justification: Most of the results in this work are mathematical hypotheses (numerically validated) presented without proof. The main theoretical result (Eq 5) is proven in Appendix
F
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 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 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 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 or not)?
Question:
Answer: [Yes]
Justification: The data set used was explicitly defined, the algorithmic steps presented and numerical techniques described.
Guidelines:
The answer NA means that the paper does not include experiments.
If the paper includes experiments, No answer 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.
If the contribution is dataset and/or model, the authors should describe the steps taken to make their results reproducible or verifiable.
Depending on the contribution, reproducibility can be accomplished in various ways. For example, if the contribution is a novel architecture, describing the architecture fully might suffice, or if the contribution is specific model and empirical evaluation, it may be necessary to either make it possible for others to replicate the model with the same dataset, or provide access the model. In general. releasing code and data is often one good way to accomplish this, but reproducibility can also be provided via detailed instructions for how replicate the results, access a hosted model (e.g.. in the case of large language model), releasing of a model checkpoint, or other means that are appropriate to the research performed.
While NeurIPS does not require releasing code, the conference does require submis- sions to provide some reasonable avenue for reproducibility, which may depend on the nature of the contribution. For example
(a) the contribution is primarily a new algorithm, the paper should make clear how reproduce that algorithm. (b) If the contribution is primarily
model architecture, the paper should describe the architecture clearly and fully.
(c) the contribution is a new model (e.g., a large language model), then there should either be way to access this model for reproducing the results or way to reproduce the model (e.g., with an open-source dataset or instructions for how to construct the dataset).
(d) We recognize that reproducibility may be tricky in some cases, in which case authors are welcome describe the particular way they provide for reproducibility. In the case of closed-source models, it may be that access to the model is limited in some way (e.g., to registered users), but it should possible for other researchers to have some path to reproducing verifying the results.
30

---

5. Open access to data and code
Does the paper provide open access to the data and code, with sufficient instruc- tions faithfully reproduce the main experimental results, as described in supplemental material?
Question:
Answer: [Yes]
The git repository linked in this camera-ready version. The data set is mathematical formulas and constants so freely available to all.
Justification:
Guidelines:
The answer NA means that paper does not include experiments requiring code.
Please see the NeurIPS code and data submission guidelines (https://nips.cc.
for more details.
While we encourage the release of code and data, we understand that this might not be possible, so "No" an acceptable answer. Papers cannot be rejected simply for not including code, unless this is central to the contribution (e.g., for new open-source benchmark).
The instructions should contain the exact command and environment needed to run to reproduce the results. See the NeurIPS code and data submission guidelines (https for more details.
7 //nips. //nips.cc/publll wwtpc//pppp.g.udeeeeoeeepppeppee httpi///guides//dceeuubbiiieee prrooeSubmissionool11..
The authors should provide instructions on data access and preparation, including how to access the raw data, preprocessed data, intermediate data, and generated data, etc.
The authors should provide scripts to reproduce all experimental results for the new proposed method and baselines. only subset of experiments are reproducible, they should state which ones are omitted from the script and why.
At submission time, to preserve anonymity, the authors should release anonymized versions (if applicable).
Providing as much information as possible in supplemental material (appended to the paper) recommended, but including URLs to data and code permitted.
6. Experimental Setting/Details
Question: Does the paper specify all the training and test details (e.g., data splits, hyper- parameters, how they were chosen, type of optimizer, etc.) necessary to understand the results?
Answer: [Yes]
In this work we are using unsupervised recurring clustering as the main machine learning method, and the results are mostly validated using the (test) subset of known PCF formulas for mathematical constants. This subset included in the full data set but was not given any special treatment during clustering (as mentioned in section 4.1).
Justification:
Guidelines:
The answer NA means that the paper does not include experiments.
The experimental setting should be presented in the core the paper to a level of detail that is necessary appreciate the results and make sense of them.
The full details can provided either with the code, appendix, or as supplemental material.
7. Experiment Statistical Significance
Question: Does the paper report error bars suitably and correctly defined or other appropriate information about the statistical significance of the experiments?
Answer: [Yss]
Justification: The main errors in this work originate in the dynamical metrics measurement stage. Error bars are not shown (for visual clarity), but these errors are discussed in Appendix A and mentioned in every figure that shows measured dynamical metrics.
Guidelines:
The answer NA means that the paper does not include experiments.
31

---

The authors should answer "Yes" if the results are accompanied by error bars, confi- dence intervals, or statistical significance tests, least for the experiments that support the main claims of the paper.
The factors of variability that the error bars capturing should be clearly stated (for example, train/test split, initialization, random drawing of some parameter, or overall with given experimental conditions).
The method for calculating the error bars should be explained (closed form formula, call to a library function, bootstrap, etc.)
The assumptions made should be given (e.g., Normally distributed errors).
should be clear whether the error bar is the standard deviation the standard error of the mean.
It is OK to report 1-sigma error bars, but should state it. The authors should preferably report a 2-sigma error bar than state that they have 96% CI, if the hypothesis of Normality of errors is not verified.
For asymmetric distributions, the authors should be careful not to show in tables or figures symmetric error bars that would yield results that are out of range (e.g. negative eπror rates).
If error bars are reported in tables or plots, The authors should explain in the text how they were calculated and reference the corresponding figures tables in the text.
8. Experiments Compute Resources
For each experiment, does the paper provide sufficient information on the puter resources (type of compute workers, memory, time of execution) needed to reproduce the experiments?
Question:
Answer: [Yes]
Justification: There special or extreme resource requirements, did not focus
this question, but a brief discussion of required memory and runtime is in Appendix A
Guidelines:
The answer NA means that the paper does not include experiments.
The paper should indicate the type of compute workers CPU or GPU, internal cluster, or cloud provider, including relevant memory and storage.
The paper should provide the amount compute required for each of the individual experimental rums well estimate the total compute.
The paper should disclose whether the full research project required more compute than the experiments reported in the paper (e.g., preliminary or failed experiments that did make it into the paper).
9. Code Of Ethics
Question: Does the research conducted in the paper conform, in every respect, with the NeurIPS Code of Ethics
Answer: [Yes]
Justification: This work does not involve human participants, special data sets, does not have implications for the broader public (only for scientists) and cannot be used do harm.
Guidelines:
The answer NA means that the authors have not reviewed the NeurIPS Code of Ethics.
If the authors answer No, they should explain the special circumstances that require deviation from the Code of Ethics.
The authors should make sure to preserve anonymity (e.g.. if there is special consid- eration due to laws or regulations their jurisdiction).
10. Broader Impacts
Question: Does the paper discuss both potential positive socictal impacts and negative socictal impacts of the work performed?
Answer: [NA]
32

---

Justification: Although our work is applied ML, its impact is on mathematical research, not society at large.
Guidelines:
The answer NA means that there no societal impact of the work performed.
If the authors answer NA or No. they should explain why their work has no societal impact why the paper does not address societal impact.
Examples of negative societal impacts include potential malicious unintended uses (e.g., disinformation, generating fake profiles, surveillance), fairness considerations (c.g.. deployment of technologies that could make decisions that unfairly impact specific groups), privacy considerations, and security considerations.
The conference expects that many papers will be foundational research and not tied to particular applications, let alone deployments. However, if there is a direct path to any negative applications, the authors should point it out. For example, it is legitimate to point out that an improvement in the quality of generative models could be used to generate deepfakes for disinformation. On the other hand, it is not needed to point that a generic algorithm for optimizing neural networks could enable people to train models that generate Deepfakes faster.
The authors should consider possible harms that could arise when the technology is being used as intended and functioning correctly, harms that could arise when the technology is being used as intended but gives incorrect results, and harms following from (intentional unintentional) misuse of the technology.
If there are negative societal impacts, the authors could also discuss possible mitigation strategies (c.g., gated release of models, providing defenses in addition to attacks, mechanisms for monitoring misuse, mechanisms to monitor how system learns from feedback over time, improving the efficiency and accessibility of ML).
11. Safeguards
Does the paper describe safeguards that have been put in place for responsible release of data models that have high risk for misuse (e.g., pretrained language models, image generators, or scraped datasets)?
Question:
Answer: [NA]
Justification:
The methods in this paper do not pose such risks.
Guidelines:
The answer NA means that the paper poses no such risks.
Released models that have high risk for misuse or dual-us should be released with necessary safeguards to allow for controlled use of the model. for example by requiring that users adhere to usage guidelines or restrictions to access the model or implementing safety filters.
Datasets that have been scraped from the Internet could pose safety risks. The authors should describe how they avoided releasing unsafe images.
We recognize that providing effective safeguards challenging, and many papers do not require this, but we encourage authors to take this into account and make best faith effort.
12. Licenses for existing assets
Are the creators or original owners of assets (e.g., code, data, models), used in the paper, properly credited and are the license and terms of use explicitly mentioned and properly respected?
Question:
Answer: [Yes]
Justification: The only existing assets that were used are open source Python packages, that were properly credited in Appendix A
Guidelines:
The answer NA means that the paper does not use existing assets.
The authors should cite the original paper that produced the code package dataset.
33

---

The authors should state which version of the asset is used and, if possible, include URL.
The name of the license (e.g., CC-BY 4.0) should be included for each asset. For scraped data from a particular source (e.g.. website), the copyright and terms of service of that source should be provided.
If assets are released, the license, copyright information, and terms of use in the package should be provided. For popular datasets, paperssi thcode com/ datasets has curated licenses for some datasets. Their licensing guide can help determine the license of dataset.
For existing datasets that re-packaged, both the original license and the license of the derived asset (if it has changed) should provided.
If this information is not available online, the authors are encouraged to reach out to the asset's creators.
13. New Assets
Question: Are new assets introduced in the paper well documented and is the documentation provided alongside the assets?
Answer: [Yes]
Justification:
The code is released as open source with readme and documentation.
Guidelines:
The answer NA means that the paper does not release new assets.
Researchers should communicate the details the dataset/code/model as part of their submissions via structured templates. This includes details about training, license, limitations, etc.
The paper should discuss whether and how consent was obtained from people whose asset used.
At submission time, remember to anonymize your assets (if applicable). You can either create anonymized URL or include an anonymized zip file.
14. Crowdsourcing and Research with Human Subjects
Question: For crowdsourcing experiments and research with human subjects, does the paper include the full text of instructions given to participants and screenshots, if applicable, as well as details about compensation (if any)?
Answer: [NA]
Justification: This research does involve crowdsoureing nor research with human subjects.
Guidelines:
The answer NA means that the paper does not involve crowdsoureing nor research with human subjects.
Including this information in the supplemental material is fine, but if the main contribu- tion of the paper involves human subjects, then as much detail as possible should be included in the main paper.
According the NeurIPS Code of Ethics, workers involved in data collection, curation, or other labor should be paid at least the minimum wage in the country of the data collector.
15. Institutional Review Board (IRB) Approvals or Equivalent for Research with Human Subjects
Question: Does the paper describe potential risks incurred by study participants, whether such risks were disclosed to the subjects, and whether Institutional Review Board (IRB) approvals (or an equivalent approval/review based on the requirements of your country or institution) were obtained?
Answer: [NA]
Justification: This research does not involve crowdsourcing nor research with human subjects.
34

---

Guidelines:
The answer NA means that the paper does not involve crowdsoureing nor research with
human subjects.
Depending on the country in which research is conducted, IRB approval (or equivalent)
may be required for any human subjects research. If you obtained IRB approval, you should clearly state this in the paper.
We recognize that the procedures for this may vary significantly between institutions and locations, and expect authors to adhere to the NeurIPS Code of Ethics and the guidelines for their institution.
For initial submissions, do not include any information that would break anonymity (if applicable), such the institution conducting the review.
35


Reviews:
[{"review_id": "8Oyn4tcMya", "reviewer": "Reviewer_zxY9", "summary": "The paper addresses a long-standing challenge in number theory by proposing a new methodology for the categorization, characterization, and pattern identification of mathematical formulas, specifically Polynomial Continued Fraction (PCF) formulas. The authors introduce metrics based on the convergence dynamics of these formulas, enabling the first automated clustering of mathematical formulas. The methodology is demonstrated on a dataset of 1.7M PCF formulas, leading to the identification of both known and previously unknown formulas for significant mathematical constants such as π, ln(2), Gauss, and Lemniscate constants. The uncovered patterns allow for the generalization of individual formulas to infinite families, revealing rich mathematical structures. This work sets the stage for a generative model capable of creating continued fractions with specified mathematical properties, potentially accelerating the discovery of useful formulas.", "questions": "How do the authors plan to further verify the newly discovered formulas for significant constants such as π and ln(2)?\n\nCan the authors provide more details on how their methodology can be scaled to larger datasets and more complex formulas?\n\nHow can the approach be adapted or extended to other types of mathematical formulas beyond Polynomial Continued Fractions?", "limitations": "The paper could benefit from a more detailed discussion on the impact of the limited dataset size and the potential benefits of using larger and more diverse datasets.\n\nThe choice of metrics is crucial for the clustering and characterization of formulas. Providing a rationale for the selected metrics and discussing potential additional metrics would strengthen the paper.\n\nFurther elaboration on the methods for verifying the newly discovered formulas would enhance the credibility of the findings.", "rating": 5, "confidence": 3, "soundness": 2, "presentation": 2, "contribution": 3, "strengths": "The work introduces a novel methodology for the automated investigation of mathematical formulas, specifically focusing on Polynomial Continued Fractions. This approach is new in its use of convergence dynamics as metrics for clustering and categorization.\n\nThe methodology is rigorously tested on a large dataset, resulting in the discovery of both known and previously unknown formulas for important mathematical constants. This demonstrates the robustness and potential of the approach.\n\nThe paper clearly outlines the problem, the new methodology proposed, and the important findings. The inclusion of detailed explanations and relevant figures helps in understanding the approach and its implications.\n\nBy automating the discovery of mathematical formulas, this work has the potential to impact the field of number theory and mathematical discovery. The ability to generalize formulas into infinite families could lead to new insights and advancements in mathematics.", "weaknesses": "The paper has some weaknesses that should be addressed to improve its quality:\n\n\nWhile the addressed task and the methodology is unique, it may not be the best fit for a machine learning conference like NeurIPS. The focus on mathematical discovery might be better suited for a specialized conference or journal in mathematics or computational mathematics.\n\nThe paper’s writing can be further improved for clarity and conciseness. The abstract could be more concise, and the figures are currently blurry and not well-organized, detracting from the overall presentation.\n\nThe study is based on a limited-size dataset and a small set of metrics. Expanding the dataset and incorporating a broader range of metrics could enhance the robustness and applicability of the findings.\n\nThe newly identified formulas for significant constants need to be further verified. Ensuring their correctness and utility is crucial for the validity of the contributions.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "* **“While the addressed task and the methodology is unique, it may not be the best fit for a machine learning conference like NeurIPS. The focus on mathematical discovery might be better suited for a specialized conference or journal in mathematics”**\n\n*Please see the joint rebuttal about this important point.*\n\n* **“The paper’s writing can be further improved for clarity and conciseness. The abstract could be more concise, and the figures are currently blurry and not well-organized, detracting from the overall presentation.”**\n\nWe thank the referee for this constructive feedback.\nThe abstract has been shortened and made more concise, and the figures' quality and readability improved.\n\n* **“The _choice of metrics_ is crucial for the clustering and characterization of formulas. Providing a _rationale for the selected metrics_ and discussing _potential additional metrics_ would strengthen the paper.”**\n* **“...and a _small set of metrics_. Expanding the dataset and _incorporating a broader range of metrics_ could enhance the robustness and applicability of the findings.”**\n\nIndeed, the choice of metrics is crucial. We have tried several other metrics during this research (like measuring $p_n$ and $q_n$ modulo primes), which gave no apparent value.\nIn practice, our selection is automated (Fig. 1), selecting the metrics that provide the largest immediate improvement in clustering.\n\nIn the revised manuscript, we now discuss the broader set of possible metrics in Section 3.5, describing the relative value of including additional metrics and the rationale for the metric selection.\n\nWe thank the referee for this constructive and useful comment.\n\n* **“The paper could benefit from a more detailed discussion on the impact of the _limited dataset size_ and the potential benefits of _using larger_ and more diverse _datasets_.”**\n* **“Can the authors provide more details on how their methodology can be _scaled to larger datasets_…?”**\n* **“The study is based on a _limited-size dataset_…“**\n\nOur methodology can be scaled to larger datasets in multiple ways. For example, extend the range of PCF coefficients from [-5,5] to [-10,10], increasing the size of the dataset ~50X, or going up to 3rd degree $A_n$ and 4th degree $B_n$ with coefficients in [-5,5], increasing the dataset ~1333X.\n\nTo combat this rapid growth in required compute, we propose a 2-fold approach:\n* Make the measurement depth dynamically chosen (instead of constant for all) during the evaluation - aiming at a fixed precision for all PCFs instead of a fixed fraction depth.\n* The heaviest part of the computation - calculating the metrics for each formula - is “embarrassingly parallelizable”. We recently adapted our algorithm to the Berkeley Open Infrastructure for Network Computing (BOINC), enabling parallel experiments on thousands of volunteer computers. Assuming a typical contribution of 1000 BOINC volunteer cores, we expect the above dataset to require about ~1 month of compute.\n\nThese estimations and advances are now added to Appendix A.\n\n* **“Can the authors provide more details on how their methodology can be scaled to … _more complex formulas_?”**\n* **“How can the approach be adapted or _extended to other types of mathematical formulas_ beyond Polynomial Continued Fractions?”**\n\nFirst, please note the surprising generality of PCFs. Many useful sums, Taylor series and families of integrals, are all equivalent to continued fractions via Euler’s formula (see PDF). This way, studying PCFs covers relations to trigonometric, hyperbolic, Bessel and other important functions.\n\nThis fact is now stressed further in the revised manuscript.\n\nOur dynamical metrics clustering methodology can be scaled to more complex formulas as it does not depend on the specific structure of the underlying function - including ones whose evaluation is iterative / recursive / an infinite sum / any process producing rational approximants. Any such mathematical structure can be measured, clustered, and identified using the proposed method - treating the generating functions as a black box.\n\nTo exemplify this universal concept, we also specifically looked into higher depth recursion relations, which are a promising research direction because little is known about their global structure, yet they are involved in several important conjectures. For example, the best rational approximation formula known for Euler’s gamma constant is constructed via such a recursion relation [Aptekarov et al., Trans. Moscow Math. Soc. 70, 237 (2009)]. This family of formulas is broader than continued fractions, yet our analysis showed that it is **described by the same metrics** we originally discovered for PCFs.\n\nAnother type of mathematical formula that we analyzed in the revised manuscript are hypergeometric functions, which are given as infinite sums and can be analyzed by the exact same metrics that we originally developed for PCFs. Hypergeometric functions show the applicability of our approach to an even bigger family of functions for measurement, clustering, and conjecture generation. This can be useful in a wide variety of contexts, e.g., in investigations of integral formulas (e.g., Beukers-type integrals [Beukers, B. Lond. Math. Soc. 11, 268 (1979); Dougherty-Bliss et al., The Ramanujan J. 58, 973 (2022); Brown et al., arXiv:2210.03391 (2022)]).\n\nThese prospects and additional potential generalizations are now expanded on in Section 5, with the above references included therein.\n\n* **“The newly identified formulas for significant constants need to be further verified. Ensuring their correctness and utility is crucial for the validity of the contributions.”**\n* **“How do the authors plan to further verify the newly discovered formulas for significant constants such as π and ln(2)?”**\n* **“Further elaboration on the methods for verifying the newly discovered formulas would enhance the credibility of the findings.”**\n\n*Please see the joint rebuttal about this important point.*", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "641aBAdVTf", "reviewer": "Reviewer_YVDn", "summary": "They generate continued fraction formulas and test if they evaluate to mathematical constants. \nThey introduce a distance metric to compare formulas.\nThey discover novel formulas for known constants.", "questions": "all formulas are continued fraction formula of quadratic polynomials? is that not rather limited?\n\nin (1) you give the example tan(x), but in the definition of a/b (p4 118), there is only n and no x.  So you cannot actually get a formula for tan(x), can you? \n\nis there a list of all 1,543,926 formulas? how many are equal to pi or e? is there pi^pi or e^e among them? or ln(pi)?\ndo you know how many are irrational? or even how many are transcendent?\n\nhow many are proven to be correct and how many remain hypotheses?\n\nhave you consulted mathematicians if they are going to use these formulas for anything?\n\n>(4)\n\nis it necessary to introduce q̃_n and not use q_n without tilde there?", "limitations": "yes", "rating": 4, "confidence": 3, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "Mathematical constants are always used, so it is important to have formulas to calculate them\n\nwell-written\n\nthroughout experiments", "weaknesses": "It generates formula hypotheses, so we do not always know if the formulas are actually correct\n\nRather limited structure of the formulas", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "* **“It generates formula hypotheses, so we do not always know if the formulas are actually correct”**\n* **“how many are proven to be correct and how many remain hypotheses?”**\n\nWe thank the referee for the constructive feedback.\n*Please see the joint rebuttal.*\n\n* **“Rather limited structure of the formulas”**\n* **“All formulas are continued fraction formulas of quadratic polynomials? is that not rather limited?”**\n\nFirst, please note the surprising generality of polynomial continued fractions (PCFs). Many useful sums, Taylor series of ubiquitous functions, and common families of integrals, are all equivalent to continued fractions via Euler’s continued fraction formula. This way, studying PCFs covers relations to trigonometric, hyperbolic, Bessel and other important functions.\nThis fact is now stressed further in the revised manuscript.\n\nFor more complex formulas, our dynamical metrics clustering approach can be directly extended, as it does not depend on the specific structure of the formula. Our work directly applies to a wide range of parametric families of functions: including ones whose evaluation is iterative / recursive / an infinite sum / any process producing rational approximants. Any mathematical structure of these types can be measured, clustered, and identified using the proposed method - the underlying generating functions can be thought of as a black box.\n\nTo exemplify this universal concept, we also specifically looked into higher depth recursion relations, which are a promising research direction because little is known about their global structure, yet they are involved in several important conjectures. For example, the best rational approximation formula known for Euler’s gamma constant is constructed via such a recursion relation [Aptekarov et al., Trans. Moscow Math. Soc. 70, 237 (2009)]. This family of formulas is broader than continued fractions, yet our analysis showed that it is **described by the same metrics** we originally discovered for PCFs.\n\nAnother type of mathematical formula that we analyzed in the revised manuscript are hypergeometric functions, which are given as infinite sums and can be analyzed by the exact same metrics that we originally developed for PCFs. Hypergeometric functions show the applicability of our approach to an even bigger family of functions for measurement, clustering, and conjecture generation. This can be useful in a wide variety of contexts, e.g., in investigations of integral formulas (e.g., Beukers-type integrals [Beukers, B. Lond. Math. Soc. 11, 268 (1979); Dougherty-Bliss et al., The Ramanujan J. 58, 973 (2022); Brown et al., arXiv:2210.03391 (2022)]).\n\nThese prospects and additional potential generalizations are now expanded on in Section 5, with the above references included therein.\n\n* **“in (1) you give the example tan(x), but in the definition of a/b (p4 118), there is only n and no x. So you cannot actually get a formula for tan(x), can you?”**\n\nYes, we can get formulas for functions of $x$ like $\\tan(x)$. For this to be possible, we need $A_n,B_n$ to be functions of $n$ and of $x$.\n\nIn eq. (1) the polynomials are:\n$A_n = 2n - 1$\n$B_n = -x^2$\nand the PCF converges to $x*\\tan(x) - 1$.\n\nIn general, given any rational $x$, there are polynomials $A_n,B_n$ with integer coefficients that produce a PCF that converges to $\\tan(x)$. But the parametric family applies to any value $x$. In fig.2a, we show an example of a PCF that we found and converges to $1/\\tan(1)$.\n\nGenerating an $x$-dependent conjecture thus requires an additional step, but it is just as automatable as the rest. For example, once a large-enough dataset is measured and automatically identified, the family of formulas converging to $\\tan(1),\\tan(2)$ etc. can be extrapolated and we get a formula for $\\tan(x)$.\n\n* **“Is there a list of all 1,543,926 formulas? how many are equal to pi or e? is there pi^pi or e^e among them? or ln(pi)? do you know how many are irrational? or even how many are transcendent?”**\n\nDue to its size the full dataset cannot be explicitly shown inside the article, but the code used to generate and measure it is attached to the original paper submission.\nSpecifically:\n\n$\\pi =$ 39 previously known + 116 new conjectures\n\n$e =$ 44 previously known + 80 new conjectures\n\n$e^2 =$ 28 previously known + 178 new conjectures\n\nPositive $\\delta$ (proving irrationality) = 913,056\n\nThese numbers are now stated explicitly in the manuscript. \nWe thank the referee for this constructive comment.\n\nThere are no $\\ln(\\pi)$ formulas in the dataset.\nThere were no $\\pi^2$ formulas in the original dataset, but we have now discovered and proved 2 new high degree PCFs for $\\pi^2$ (see the PDF).\n\nRegarding the question of transcendence, there were many conjectures found for known transcendental constants (like $\\pi$ and $e$) and known non-transcendental constants (like the Golden Ratio). We do not know whether the unidentified PCFs are transcendental or not.\n\n* **“have you consulted mathematicians if they are going to use these formulas for anything?”**\n\nYes, we are in contact with experts in several fields, like Doron Zeilberger (Rutgers), Jeffrey Lagarias (University of Michigan), Uri Bader (Weizmann), Dzmitry Badziahin (University of Sydney) and others.\nWe also have researchers with PhDs in mathematics as part of the team (and co-authors).\n\nOne common usage is for irrationality proofs, which require a series of rational approximations with $\\delta>0$. Out of the 1,543,926 converging formulas, 913,056 have $\\delta>0$ and are thus irrationality proving formulas. In the next stage, we will explore how many of these constants were not known to be irrational. Such discoveries can be of substantial importance as new irrationality proofs are scarce and far between.\n\n* **“(4) is it necessary to introduce q̃_n and not use q_n without tilde there?”**\n\nWe thank the referee for the suggestion.  $q̃_n$ is no longer introduced.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "shQkYpwvAV", "reviewer": "Reviewer_uW6k", "summary": "The paper presents a classification of 1.5 million polynomial continued fractions (PCF), continued fractions having as coefficients the integer values of two polynomials, $A(n)$ and $B(n)$, with $A$ $B$ of degree two, with integer coefficients in $[-5,5]$. PCF are classified according to the asymptotic properties of the sequence of approximation errors (difference between the convergents and the limit), the asymptotic properties of the denominator of the convergents (in simplest terms), and the measure of irrationality, which is the difference between the logs of the two previous metrics.\n\nAlong these metrics, the authors observe that groups of PCF with the same limit tend to cluster together, and that by \"anchoring\" these clusters on known formulas, one can discover new PCF decompositions of mathematical constants.", "questions": "* Figure 1 caption is very long, in the interest of clarity, it might be worth describing your methodology in a specific section (3.2?)\n* l.96: couldn't we assume that the convergents $p_n/q_n$ are always in simplest terms? this would avoid having to introduce of $\\tilde q_n$. Besides, I believe it is assumed in the usual definition of the measure of irrationality.\n* in section 3.1, you explain that the irrationality measure is either 0 or larger than 1. You then claim that the blind-$\\delta$ method provides a good estimator of $\\delta$, and figure 2.a seems to support this, yet in figure 2.b most estimates of $\\delta$ are below $1$, and some are even negative. What happens? Doesn't this compromise the use of blind-$\\delta$ as an estimator of the irrationality measure?\n* Section 3.5, can you elaborate on \"representation power\", what it is? why is it important?\n* Table 1: The exponential factor of the growth coefficient seems useless, why is it? Also, could the larger value of the Davies Boulding Index the irrationality measure be a sign of the problem with its estimation?\n* Figure 4: can you provide a description of the axes in the tSNE graph? (maybe switch to PCA for more explainability)\n* l.29: shouldn't Lambert's original paper be quoted, instead of a modern compilation?\n* l. 100 \"equal\" (typo)\n* l.135: the error rate $\\epsilon$ is used before it is defined (in section 3.4)\n* Figure 4, constant $C1$ is not defined anywhere, lemniscate?", "limitations": "yes", "rating": 7, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "The paper is well written, and interesting to read. It is an original work, and both theoretical and experimental results are adequately supported. The results are intriguing, and seem to point to scientific discovery.", "weaknesses": "The link with AI or machine learning is not completely clear to me. Whereas the clustering techniques used in the paper, and the t-SNE projections used to represent the results, are commonly used by ML practitionners, most of the analyses conducted in this paper amount to descriptive statistical analysis of a specific mathematical dataset. To demonstrate a possible link with Machine Learning, it would be useful that the authors discuss (and perhaps demonstrate) how their approach scales to large sets of PCF, by letting the coefficients and degrees of $A$ and $B$ grow larger. \n\nI lean towards acceptance because of the potential interest of such approaches in AI for Science.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "* **“To demonstrate a possible link with Machine Learning, it would be useful that the authors discuss (and perhaps demonstrate) how their approach scales to large sets of PCF, by letting the coefficients and degrees of A and B grow larger.”**\n\nWe thank the referee for this suggestion. \n\nWe successfully tested our hypothesis regarding the value of identified clusters for broader forms of formulas. Specifically, we showed how the clustering applies to 3rd and 4th degree polynomial continued fractions (PCFs), showing that they belong to the same clusters created by 2nd degree PCFs, and importantly, using the same dynamical metrics. The 2nd degree PCFs in our initial dataset served as training data for a classifier (based on the automatically identified clusters) that was then tested on higher degree PCFs.\nThese results are now part of a new appendix in the paper (please see the figure in the joint rebuttal PDF).\n\n* **“Figure 1 caption is very long, in the interest of clarity, it might be worth describing your methodology in a specific section (3.2?)”**\n\nWe thank the referee for this suggestion. \n\nThe content of Figure 1 caption is now split between the caption and the main text.\n\n* **“l.96: couldn't we assume that the convergents 𝑝𝑛/𝑞𝑛 are always in simplest terms? this would avoid having to introduce of  𝑞~𝑛.”**\n\nWe thank the referee for this comment.\n\n$q̃_n$ is no longer introduced.\n\n* **“In section 3.1, you explain that the irrationality measure is either 0 or larger than 1. You then claim that the blind-𝛿 method provides a good estimator of 𝛿, and figure 2.a seems to support this, yet in figure 2.b most estimates of 𝛿 are below 1, and some are even negative. What happens? Doesn't this compromise the use of blind-𝛿 as an estimator of the irrationality measure?”**\n\nThis is a delicate point in the concept of irrationality measure. Given a converging series of rational approximations $p_n / q_n$, $\\delta$ is defined as the irrationality measure of that _series_ - and it can be any number $\\geq$-1 (Eq.4). The irrationality measure of a _number_ is defined as the supremum of all possible $\\delta$’s (Eq.3).\n \nAny converging PCF produces a $\\delta$ measure - which is almost never exactly 0 or 1. But the _true_ irrationality measure of their limit will always be 0 or $\\geq$1. The challenge in proving the irrationality of a mathematical constant is mostly finding a series that produces a positive $\\delta$, as they are not known in advance and are notoriously hard to construct.\n\nWe thank the referee for bringing this to our attention. We now stress this point in section 3.1 and reverse the order in which these two concepts are introduced.\n\n* **“Section 3.5, can you elaborate on \"representation power\", what it is? why is it important?”**\n\n“Representation power” of a metric can be thought of as maximal conditional information (conditioned on the metrics already included). We don’t measure information - we aim for best clustering and identification - so the added value is measured by the Davies-Bouldin Index.\n\nThe core idea of step (e.2) in figure 1 is to gradually choose the metrics that give the most value for the resulting clustering of unidentified PCFs - choosing the one with the most representation power each time we need additional granularity / quality of clustering. \n\n* **“Table 1: The exponential factor of the growth coefficient seems useless, why is it?“**\n\nWe believe the reason is that for the majority of PCFs the dominant convergence factor is factorial. Only when the error factorial coefficient ($\\eta$) is $\\approx0$, then the exponential coefficient has true meaning.\n\nThe dataset contains only 72,610 PCFs with $|\\eta|<0.1$. When measuring the DB Index for this subset we get:\n\nExponential factor of the growth coefficient = 0.479275\n\nFactorial factor of the growth coefficient = 2.87425\n\nThe factorial rate is now worse, as expected, but the exponential growth rate metric becomes valuable for clustering, supporting the hypothesis.\n\n* **“Also, could the larger value of the Davies Boulding Index the irrationality measure be a sign of the problem with its estimation?”**\n\nYes, a situation where the underlying structure has good clusters but the measurement of some of the metrics is noisy can indeed produce high Davies Bouldin Index values. \n\nWe believe that is not the case here:\n1. The \"Blind-$\\delta$\" algorithm was validated on multiple examples of PCFs with known, analytically proven, $\\delta$'s.\n2. The $\\delta$ DB Index remains consistent between different random samplings of the dataset (as described in section 3.5). If the $\\delta$ estimation noise was substantial enough to mask the underlying clustering, it would also create substantial variance in the clustering quality assessment. \n\n\n* **“Figure 4: can you provide a description of the axes in the tSNE graph? (maybe switch to PCA for more explainability)”**\n\nIn this case there is a tradeoff between visualization and explainability. We opted to go for a better visualization. \n\nDue to the nature of tSNE, we cannot provide a simple description of the axes for the existing clustering graph, but following the referee’s comment we are now creating a new visualization based on non linear axis scales - to better combine explainability and graphical fidelity.\n\n* **“l.29: shouldn't Lambert's original paper be quoted, instead of a modern compilation?”**\n\nWe thank the referee for the remark. Lambert’s work is now cited in addition to Berggren’s.\n\n* **“l. 100 \"equal\" (typo)**\n* **l.135: the error rate 𝜖 is used before it is defined (in section 3.4)**\n* **Figure 4, constant 𝐶1 is not defined anywhere, lemniscate?”**\n\nWe thank the referee for the careful review and for spotting these issues.\n\nL. 100 is now corrected.\n\n$\\epsilon$ is now defined in section 3.3.\n\n$C1$ in this context is the Continued Fraction Constant. It is now renamed to $C_{\\mathrm{cf}}$ and referenced in the caption.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "ynOG6J0dDz", "reviewer": "Reviewer_Dgb8", "summary": "The authors propose an algorithm to filter, cluster and label polynomial continued fractions. It is based on several features linked to the asymptotic behaviour of their rational approximations. The authors detail how these are computed. They apply this algorithm to a large set of formulae ; they discuss the properties of the obtained clusters and highlight some novel PCFs linked to known mathematical constants.", "questions": "- Part 3.4 was a bit confusing. The Blind-δ Algorithm is presented as a proxy for delta eq. 4. Is it not rather to estimate the approximation error epsilon ?\n\n- There is an inconsistency : part 3.2 it is n^beta vs table 1 P(n) and no beta ; is it because beta is hard to estimate ? is it usefull for clustering ?\n\n- Could the authors comment on the fact that the predicted delta eq. 5 can be quite far from the actual delta (fig. 2a, fig. 2b) ? Yet it seems it provides relevant information for clustering ; what does it capture ?\n\n- More generally, did the authors try to extract more features, other metrics, not necessary having a mathematical interpretation, in an automated way ?\n\nl. 100 \"equval\"\n\nl. 152 \"but how is it related to the actual series delta?\" is not clear\n\nl. 197 \"We’ll start\" informal", "limitations": "-", "rating": 5, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "This work seems to be an interesting application of unsupervised data analysis to fundamental science. It proposes a way to build relevant metrics from sequences of numbers, and to numerically estimate them.\n\nMore generally, the structures revealed on fig. 3 and 4 are intriguing and the article is well written.", "weaknesses": "The claim that \"we connect the challenge of formula creation to modern approaches in AI for Science\" seems a bit bold. To my understanding this work does not involve modern AI in the sense that it requires manual and careful feature extraction and that nothing is learnt.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "* **“The claim that \"we connect the challenge of formula creation to modern approaches in AI for Science\" seems a bit bold. To my understanding this work does not involve modern AI in the sense that it requires manual and careful feature extraction and that nothing is learnt.”**\n\nIn the revised manuscript, we softened this sentence in l. 234 to express a more specific claim.\n \nFor context, there is no previous work that succeeded applying modern AI to conjecture generation in number theory. Our work is the first in this field to succeed with clustering algorithms, and this resulted in finding novel formulas on a large scale.\n\n*For further discussion regarding the connection of our work to modern AI for Science, please see the joint rebuttal section.*\n\nRegarding the feature extraction itself, the full set of metrics was chosen manually, because the mathematics of rational approximant series limits the possibilities of additional metrics which give independent information about the formula. \nDuring the conjecture generation process (Fig. 1), the subset of metrics used was selected automatically.\n\nIn the revised manuscript, we now discuss the broader set of possible metrics in section 3.5, describing the relative value of including additional metrics and the rationale for the metric selection.\n\nRegarding the use of the term learning, please see Fig.5 in the attached PDF. There we show that the clusters trained on our original dataset successfully classify (based on the same dynamical metrics) new formulas outside the original dataset. The new formulas are based on higher-order polynomials than the ones used for building the clusters - showing the trained classifier can work even on higher complexity formula structures.\n\n* **“Part 3.4 was a bit confusing. The Blind-δ Algorithm is presented as a proxy for delta eq. 4. Is it not rather to estimate the approximation error epsilon ?”**\n\nThis is a good point. “Blind-$\\delta$” is the name we gave the entire algorithm, although this name arises from eq. 4, where indeed it is the approximation error $\\epsilon$ that is being calculated without prior knowledge about the limit. It is just a terminology issue, but we should have explained it better. The ability to estimate $\\delta$ without prior knowledge provides us with a unique and powerful dynamical metric, which is why we consider it a key to the algorithm.\n\nFollowing this comment, we clarify the use of terminology. We thank the referee for raising this point and helping to improve our work.\n\n* **“There is an inconsistency : part 3.2 it is n^beta vs table 1 P(n) and no beta ; is it because beta is hard to estimate ? is it useful for clustering ?”**\n\nWe thank the referee for raising this point.\n\nThe vast majority of our dataset consists of PCFs with an exponential or factorial error decay rate - in which cases the polynomial coefficient is noisy and less useful for characterizing the PCF. So yes, $\\beta$ is hard to measure accurately for most PCFs.\nIn fact, only 534 PCFs were measured to have both $\\left| \\gamma \\right| < 0.1$ and $\\left| \\eta \\right| < 0.1$ (as defined in Table 1). In those cases we believe the polynomial coefficient to be potentially useful, but a larger subset is required to test the automated process. \n\nThe definitions in section 3.3 now follow the same form as Table 1, not mentioning $\\beta$ in the manuscript any more.\n\n* **“Could the authors comment on the fact that the predicted delta eq. 5 can be quite far from the actual delta (fig. 2a, fig. 2b) ? Yet it seems it provides relevant information for clustering ; what does it capture ?”**\n\nWe thank the referee for this remark.\n\nFigures 2a and 2b intentionally show the possible discrepancy between the direct calculation of $\\delta_n$ and the value of $\\delta_{\\mathrm{predicted}}$ (eq. 5) for $n < 1000$. The prediction formula is accurate in the limit as $n \\rightarrow \\infty$. In practice, we set the numerical limit at $n = 10^7$, making $\\delta_{\\mathrm{predicted}}$ much closer to the actual $\\delta$ value. This is now explicitly stated in the caption of fig. 2.\n\n$\\delta_{\\mathrm{predicted}}$ provides significant validation for the numerical $\\delta$ in clustering. We used several sanity checks to identify anomalies or execution issues. One of the flags was a large discrepancy between the numerical and the predicted $\\delta$.\n\nIn the future, we plan to use the prediction formula in a gradient-descent-based approach to search for high-$\\delta$ formulas, which we will explore in further research.\n\nThe $\\delta$ prediction formula holds additional importance in our research. The extension of the basic formula in Elimelech et al. (2023) to the general case was motivated and conjectured based on the results of the numerical $\\delta$ measurements. In a sense, it is the first conjecture that arose from the dataset and was later proven analytically.\n\n* **“More generally, did the authors try to extract more features, other metrics, not necessarily having a mathematical interpretation, in an automated way ?”**\n\nThe dynamical features are extracted automatically for the 1,543,926 formulas, but as the referee suggests, the choice of which metrics is predetermined.\n\nWe have tried several other dynamic metrics during this research (like measuring $p_n$ and $q_n$ modulo primes, tracking their sign etc.) - which gave no apparent value. \n\nThe idea to automate the choice of dynamic features is intriguing. Such an idea was never attempted in number theory. We thank the referee for this suggestion. It is now mentioned in Section 5 (“Discussion and Outlook”), and we leave it for future research.\n\n* **“l. 100 \"equval\"**\n* **l. 152 \"but how is it related to the actual series delta?\" is not clear**\n* **l. 197 \"We’ll start\" informal”**\n\nWe thank the referee for the careful review and for spotting these issues.\n\nL. 100 is now corrected.\n\nL.152 is now removed.\n\nIn l. 197 the first sentence is now removed.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:
We would like to summarize and address the most important comments brought by more than one referee.

Regarding the link of our work to ML
---

* **“While the addressed task and the methodology is unique, it may not be the best fit for a machine learning conference like NeurIPS.”**
* **“The link with AI or machine learning is not completely clear to me.”**

Until now, leading methods in ML were not successful in problems of conjecture generation in number theory. This field has been harder to penetrate compared to other areas, such as theorem proving. Our work is the first to successfully apply a basic learning method to conjecture generation in number theory. This is also the first example of successful clustering for conjecture generation in any area of mathematics. Our clustering methods are well-known in ML, but the underlying dynamical metrics we found and used are completely new and they open a path for many other ML applications in this field.

For context, our work offers advances in automated conjecture generation (ACG), a subfield of AI for Science. Notably, current generation LLMs do not succeed in generating relevant new conjectures in number theory. The reason is attributed to the lack of metrics, needed to provide a measure of being “closer to correct”. In number theory, such metrics have not existed until now - our work is the first to suggest successful metrics, and to use them.

In the time that passed since our initial submission, we improved our manuscript especially in these aspects. We successfully tested our clustering method on broader forms of formulas and found clusters applicable to formulas in ranges of parameters outside those used for training: 3rd and 4th degree polynomial continued fractions (PCFs) are captured by the same clusters trained on (constructed by) 2nd degree PCFs, and importantly, using the same dynamical metrics. These results are now part of the new appendix D (see the revised Fig. 2b in the attached PDF).

Our systematic approach for formula generation, with these new validations, provided the first example of automated learning for conjecture generation in number theory, hopefully inspiring other efforts for ML applications in this field.

Regarding the correctness of the automatically generated formulas
---

* **“The newly identified formulas for significant constants need to be further verified. Ensuring their correctness and utility is crucial for the validity of the contributions”.**
* **“How do the authors plan to further verify the newly discovered formulas for significant constants such as π and ln(2)?”**
* **“Further elaboration on the methods for verifying the newly discovered formulas would enhance the credibility of the findings.”**
* **“It generates formula hypotheses, so we do not always know if the formulas are actually correct”**
* **“how many are proven to be correct and how many remain hypotheses?”**

Following these remarks, we have taken two important verification steps:

1. The novel formulas are now evaluated to a higher depth: 4 million steps or more, producing between 13 digits to thousand of digits of precision, depending on the convergence rate. These additional digits reduce the chance of an incorrect accidental identification by a measure independent of our clustering (elaborated in the revised Appendix A). Even with this additional verification, none of the previously identified constants was found erroneous, further supporting the robustness of our approach.

2. We worked to prove selected formulas in cases of especially slow convergence (see the attached PDF). 47 of the automatically generated conjectures are now analytically proven. These proofs help validate our approach and show that the results can be relied on in future research efforts. We are working in parallel with mathematicians on general mathematical approaches for proofs that can be applied in scale, and automatically, which is necessary to cope with the large number of newly discovered formulas.

In addition, we emphasize that the value of unproven formulas (or generally conjectures) cannot be understated, as it is usually such a conjecture that acts as the first step that eventually leads to a discovery of a new theory. For example, consider Srinivasa Ramanujan’s contributions to mathematics, many of which were initially unproven formulas, yet his impact on the mathematical world is undeniable. In a similar way, our formula generation algorithm provides new leads for mathematical research that can have long-term impact.

Meta-review: This paper presents an original contribution to automatized discovery of mathematical formulas. While the machine learning methods it uses are rather basic, it does an extensive job of evaluating a large number of continued fractions. It discovers previously unknown (at least to the authors and the reviewers) formulas. This is a valuable contribution to the field and will be of interest to the NeurIPS community. 

The reviews contain many points that improve the presentation of the paper. The authors are strongly encouraged to carefully take them into account in order to increase the impact of their interesting work.
Decision: Accept (poster)