also at ]Michigan State University, East Lansing, MI 48823, USA

# Cheat sites and artificial intelligence usage in online introductory physics
courses:
what is the extent and what effect does it have on assessments?

Gerd Kortemeyer<EMAIL_ADDRESS>Rectorate and AI Center, ETH Zurich, 8092
Zurich, Switzerland  [  Wolfgang Bauer<EMAIL_ADDRESS>Department of Physics
and Astronomy, Michigan State University, East Lansing, MI 48824, USA

###### Abstract

As a result of the pandemic, many physics courses moved online. Alongside, the
popularity of internet-based problem-solving sites and forums rose. With the
emergence of Large Language Models, another shift occurred. One year into the
public availability of these models, how has online help-seeking behavior
among introductory physics students changed, and what is the effect of
different patterns of online-resource usage? In a mixed-method approach, we
investigate student choices and their impact on assessment components of an
online introductory physics course for scientists and engineers. We find that
students still mostly rely on traditional internet resources, and that their
usage strongly influences the outcome of low-stake unsupervised quizzes.
However, we also find that the impact of different help-seeking patterns on
the supervised assessment components of the course is non-significant.

##  I Introduction

The general assumption in teaching introductory physics courses is that we
need to teach a few essential concepts, for example conservation laws, and,
based on those, to derive basic equations that govern the motion of all
objects in our universe, from atomic nuclei to galaxies. Almost all of us
teaching professionals subscribe to the notion that it aids the students’
learning processes to flesh out these basic concepts with exercises, in class
and as homework. Mastering our subject requires time spent on wrestling with
conceptual questions, and familiarity with basic equations is acquired by
rearranging and combining them and solving for the unknown quantities.

“I think, however, that there isn’t any solution to this problem of education
other than to realize that the best teaching can be done only when there is a
direct individual relationship between a student and a good teacher–a
situation in which the student discusses the ideas, thinks about the things,
and talks about the things. It’s impossible to learn very much by simply
sitting in a lecture, or even by simply doing problems that are assigned. But
in our modern times we have so many students to teach that we have to try to
find some substitute for the ideal.” These sentences were written by Richard
Feynman. In 1963 [1]!

Some of us, the authors included [2, 3], have spent enormous amounts of time
to construct learning management systems, audience feedback systems,
simulation software, and other apps to help with giving the students an
opportunity to spend meaningful time on task. The ultimate goal may have been
to construct a software system that acts as a personal tutor for each
individual student. The rapid rise of Large Language Models (LLMs) and
artificial intelligence (AI) systems gives hope that the substitute for the
ideal that Feynman described might become reality. Alternatively, students
might just use their new AI tools to sidestep their learning process and just
let the AI complete the assignments for them.

For the last 25 year, Michigan State University has been running online
sections of the introductory physics courses [4]. For decades, even preceding
online courses, the prevalent notion regarding course delivery modes had been
“no significant difference” [5, 6, 7]. More recently, we confirmed this for
the courses under investigation in this study when we found no significant
difference between attendance choices with regards to learning success [8] and
preparation for subsequent courses [9].

Since the courses first came online, the amount of additional resources
available online has greatly increased, particularly during COVID-19. The
solution to virtually any introductory-physics problem is available shortly
after it is published [10], including on commercial sites like Chegg [11, 12];
a typical example can be seen in Fig. 1. Students find these resources useful
for homework or unsupervised online exam problems that have been “recycled”
from earlier in the semester (e.g., homework problems making an encore
appearance on exams).

Figure 1: A problem written and copyrighted by one of the authors (GK) in LON-
CAPA [13] (top panel) and a typical solution found on a problem-solving site
(bottom panel). The problem numbers are randomized, so the student would need
to identify their own numerical values and insert them. Depicted is one of 50
answers found on the site.

Instructors have been fighting these internet sites and forums by editing the
problem content, removing any references to problem numbers, randomizing the
solutions, selecting problems from older editions of textbooks, or writing new
problems every time [14]. All of these counter measures rely on there being
some time delay between publishing a problem and it appearing online (even
contracted problem solvers need a little time) and on the sites having one
static version of the problem (e.g., not being able to adapt to
randomizations).

Figure 2: The problem from Fig. 1 in its original form (left panel) and after
on-the-fly modification (right panel), correctly solved by GPT-4 on the first
attempt.

These same counter measures against cheat sites will not work against
artificial intelligence. Tools like ChatGPT [15] and Bard [16] deliver
solutions ad-hoc and on-demand, they are immune against changing wording or
numbers, and they solve problems independent of them having been published
days ago or appearing for the first time on an online exam. This is
illustrated in Fig. 2, where the original friction problem has been modified
by introducing additional distractors and different randomized numbers. Not
only does the availability of these solutions depend on some “expert” having
solved the problem before, but as opposed to the forum answer in Fig. 1, the
response includes the actual numbers encountered by the student and the
physics explanation is arguably better and more helpful.

Chegg can serve as a proxy to the popularity of online problem-solving sites:
while the share price of Chegg (NYSE: CHGG) greatly increased when courses
went online due to the onset of the pandemic in 2020, with the appearance of
ChatGPT in late 2022, they dropped below pre-pandemic levels [17, 18], see
Fig. 3.

Figure 3: Historic share prices in USD of Chegg (NYSE CHGG) as a proxy for the
popularity of online problem-solving sites. Prices greatly increased when
courses went online as a result of the pandemic, but fell again dramatically
coinciding with the emergence of Large Language Models.

Physics may still have been spared from this, since Large Language Models as
“calculators for words” are still notoriously bad at math, and other online
resources may still be more reliable. However, it has been shown that even
older versions of popular AI chatbots can (barely) pass the assessment
components of introductory physics courses [19], and as Fig. 2 shows, newer
versions make less calculation errors.

In our study, we investigate if one year into the public availability of
powerful Large Language Models, online help-seeking behavior of students in an
introductory physics course has shifted from traditional resources to AI. We
also investigate the possible impact of different help-seeking patterns on the
assessment components of such a course.

##  II Setting

Michigan State University is a public, large-enrollment ($>50,000$ students)
R-1 university. Almost 78% of the undergraduate population are from Michigan.
The online courses in this study were taught asynchronously using a variety of
multimedia components [4].

The study takes place in a calculus-based introductory physics course sequence
for scientists and engineers, where both a first-semester mechanics and a
second-semester E&M course were offered during Fall semester 2023. Each course
offered several asynchronous video lessons every week and online homework
using LON-CAPA [13]. The courses had 11 low-stakes weekly exams (“quizzes”)
[20], of which nine were conducted online and two of which had to be taken on-
campus under supervision. Faculty sanctioned the use of the textbook and the
LON-CAPA materials during these exams, no other resources were allowed..

The course also included a high-stakes on-campus final exam. The final exams
included five questions which were randomized duplicates of problems had been
assigned earlier in the semester. As a resource for the students, the course
also offered an on-campus and online help room staffed by course faculty and
staff.

At the end of the semester, a survey was given asking students to report how
frequently they consulted artificial intelligence tools and other online
resources during homework and online quizzes, and how often they conversed
with fellow students and course faculty and staff while working on homework.

##  III Methodology

###  III.1 Survey administration

The survey contained the following items, which for the numerical answers had
sliders ranging from 0–100%:

  * •

Homework: Estimate the percentage of your homework and lecture problems, for
which the following is true:

    * –

You used AI tools like ChatGPT, Khanmigo, …to solve them (HwkAI).

    * –

You used Internet resources like help sites or forums to solve them (HwkInt).

    * –

You consulted other student to solve them (HwkPeer).

    * –

You consulted the TAs/prof to solve them (HwkFac).

  * •

Online Exams: Estimate the percentage of your online exam problems, for which
the following is true:

    * –

You used AI tool to solve them (OnlAI).

    * –

You used Internet resources like help sites or forums to solve them (OnlInt).

  * •

Your Opinion: Please tell us what you think about using AI in online classes;
what should ideally be done; what should not be done?

The survey was administered online during the last week of the semester, but
results were not viewed or analyzed until the grades for the course had been
turned in. A nominal participation credit was given for submitting the survey,
regardless of whether or not the students agreed to be part of the study. The
students were aware of this protocol as part of the informed consent, and data
was only analyzed for students who agreed to participate. The study was
approved under MSU-IRB-STUDY00009987.

###  III.2 Considered variables

We compiled a range of variables that capture various aspects of student
performance and behavior, shown in Table 1. Key performance metrics include
Hwk (homework score), OnlExams (score from online exams), CamExams (score from
on-campus exams with supervision), and Final (score from the final exam, also
conducted on-campus with supervision). These scores are presented as
percentages, reflecting the students’ achievement in each respective
assessment. Additionally, Sem5 represents the scores for five problems
initially available online, Final5 for the same problems when included in the
final exam, and Diff5 indicating the score difference between these two
settings. Thus, Diff5 can be used as a proxy for retention of concepts between
the semester and the final exam. Finally, DiffExams quantifies the score
difference between online and on-campus quizzes, offering insight into
performance variations across different assessment environments.

The dataset also encompasses variables related to the use of digital resources
and student interactions. HwkAI and OnlAI denote the self-reported percentage
of problems for which artificial intelligence (AI) tools were used during
homework and online quizzes, respectively. Similarly, HwkInt and OnlInt
represent the usage of other internet resources in these settings. Finally,
HwkPeer and HwkFac quantify the self-reported extent of peer discussions and
interactions with faculty during homework.

Table 1: Summary of variables in the dataset Variable | Description | Min | Max | Mean | Std. Deviation
---|---|---|---|---|---
CamExams | On-Campus Exam Score (%) | 0.00 | 100.00 | 53.80 | 22.73
Diff5 | Difference in Online/Final Duplicate Problem Scores | -100.00 | 40.00 | -39.00 | 33.80
DiffExams | Difference in Online/On-Campus Exam Scores | -81.55 | 23.57 | -29.30 | 20.04
Final | Final Exam Score (%) | 0.00 | 100.00 | 49.19 | 27.64
Final5 | Final Exam Duplicate Problem Score (%) | 0.00 | 100.00 | 51.22 | 33.37
Hwk | Homework Score (%) | 25.00 | 100.00 | 86.60 | 15.68
HwkAI | AI Usage During Homework (%) | 0.00 | 100.00 | 17.08 | 23.94
HwkFac | Discussions with Faculty During Homework (%) | 0.00 | 100.00 | 16.32 | 24.29
HwkInt | Internet Usage During Homework (%) | 0.00 | 100.00 | 50.00 | 30.18
HwkPeer | Peer Discussions During Homework (%) | 0.00 | 100.00 | 32.79 | 31.44
OnlAI | AI Usage During Online Exams (%) | 0.00 | 88.00 | 8.07 | 17.54
OnlExams | Online Exam Score (%) | 43.69 | 98.06 | 83.10 | 11.77
OnlInt | Internet Usage During Online Exams (%) | 0.00 | 100.00 | 23.38 | 29.39
Sem5 | Online Problem Score (%) | 20.00 | 100.00 | 90.23 | 17.33

###  III.3 Statistical methods

Data were downloaded from the course management system, and survey results
were merged using Python scripts. Calculations for this project were carried
out using ChatGPT-4 Advanced Data Analysis [15] and R [21] (in particular
qgraph [22] and CTT [23]).

##  IV Results

###  IV.1 Response rate

The first and second semester courses were completed by 156 and 183 students,
respectively. Of these, 90 and 131 students agreed to participate in the
study, bringing the total to 221 participants.

###  IV.2 Online versus on-campus exams

Figure 4 shows the score distributions for the nine exams that were conducted
online and the two exams that were conducted on-campus under supervision. In a
$t$-test, these distributions are significantly different ($p\approx 4.7\cdot
10^{-47}$).

0-1010-2020-3030-4040-5050-6060-7070-8080-9090-100$0$$20$$40$$60$$80$$100$$5$$7$$20$$24$$27$$36$$37$$29$$23$$13$$0$$0$$0$$0$$3$$10$$20$$32$$75$$81$Score
RangeNumber of StudentsCamExamsOnlExams Figure 4: Comparison between score
distributions for the nine low-stakes exams that were conducted online (red)
and the two low-stakes exams that were conducted on-campus under supervision
(blue).

The conditions under which these exams were conducted led to vastly different
outcomes, and an immediate assumption would be that this is related to the use
of external resources during unsupervised assessments. As an example, for nine
of the ten questions on the last online exam, solutions could be found on
Chegg within about one minute each.

###  IV.3 Usage of resources during unsupervised assessments

Figure 5 illustrates the average self-reported usage of AI and other internet
resources, as well as self-reported consultation with peers and course
personnel.

OnlAIOnlIntHwkAIHwkIntHwkPeerHwkFac$0$$20$$40$$60$$80$$8.07$$23.38$$17.08$$50$$32.79$$16.32$VariablesMean
Value Figure 5: Self-reported usage of resources while working on homework and
online exams (see Table 1 for the explanation of the variables). Figure 6:
Distribution of the survey responses.

Overall, students report less usage of resources during exams than during
homework, but not significantly. The only one-sigma significant differences
are between usage of AI and talking to faculty on the one hand, and using
other internet resources while working on homework; students have not yet
adopted AI and stick with “traditional” problem-solving sites. On the average,
students use other internet resources for half of the homework problems. The
distributions of these types of resources usages, however, are very different,
see Fig. 6, which suggests that there a different classes of resource usage.

Using $k$-means clustering and elbow method, we identified four different
classes as indicated in Table 2. Cluster 1, the smallest group, is comprised
of students who appear to prefer human interaction to any online resources,
and these students mostly adhere to rules for the online exams. Students in
Cluster 2, the largest cluster, state that they make little use of resources
overall, and that they most closely adhere to rules for the online exams.
Students in Cluster 3 make heavy use of internet resources other than AI in
both homework and exams, thus not following rules. Finally, students in
Cluster 4 use all available resources and disregard exam rules.

Table 2: Number of members and mean values of variables in identified clusters of resource usage. Cluster | # members | HwkAI | HwkInt | HwkPeer | HwkFac | OnlAI | OnlInt | Interpretation
---|---|---|---|---|---|---|---|---
1 | 27 | 9.6 | 39.9 | 53.1 | 66.4 | 3.6 | 14.3 | Using mostly peer discussions and helprooms
2 | 96 | 8.3 | 29.3 | 20.6 | 5.7 | 1.2 | 3.1 | Using resources on homework but not exams
3 | 56 | 5.1 | 80.1 | 35.8 | 7.4 | 1.9 | 51.3 | Heavily using internet on homework and exams
4 | 42 | 58.0 | 63.7 | 43.6 | 20.3 | 34.8 | 38.2 | Using all resources including AI everywhere

###  IV.4 Correlations of attributes

Figure 7 shows a Fruchterman-Reingold [24, 22] representation of the
correlation matrix between the variables. Indicated in light blue are the
online, unsupervised assessments, in green the on-campus, supervised
assessments, and in gray the differences in scores between selected subsets of
assessments. The percentages of AI-usage are indicated in beige, usage of
internet resources in yellow, and discussions with humans in orange. Green
edges denote positive correlations, red edges negative correlations, and the
thickness their absolute strength. Due to the force-directed nature of
Fruchterman-Reingold graphs, closely correlated vertices tend to cluster,
while unrelated vertices tend to be further apart from each other.

Figure 7: Fruchterman-Reingold [24, 22] representation of all correlations
between the variables in Table 1 for all students.

It is apparent that the scores achieved online and those achieved under
supervision each cluster together, but both are disconnected from the survey
answers. Taking Final as a proxy for learning success in the course, after
discarding the derived variable DiffExams and all variables related to the
five duplicate problems, in a linear regression of all remaining assessment
and survey variables, only CamExam emerges as a statistically significant
predictor of the final exam score ($p\approx 7.3\cdot 10^{-16}$); OnlExams
comes close to statistical significance with $p=0.05$. Table 3 shows the
outcome of this linear regression, which has a week correlation of
$R^{2}=0.36$.

Table 3: Linear regression for Final. | Estimate | Std. Dev. | $t$ | $p$
---|---|---|---|---
Hwk | $-0.08$ | $0.13$ | $-0.6$ | $0.55$
OnlExams | $0.33$ | $0.17$ | $1.96$ | $0.05$
CamExams | $0.69$ | $0.08$ | $8.74$ | $7.31\cdot 10^{-16}$
HwkAI | $-0.06$ | $0.09$ | $-0.68$ | $0.50$
HwkInt | $-0.05$ | $0.07$ | $-0.70$ | $0.49$
HwkPeer | $-0.04$ | $0.05$ | $-0.78$ | $0.44$
HwkFac | $0.01$ | $0.07$ | $0.20$ | $0.84$
OnlAI | $0.06$ | $0.12$ | $0.46$ | $0.65$
OnlInt | $0.05$ | $0.07$ | $0.74$ | $0.46$

Also within the usage classes (Table 2), there are very few significant
correlations between resource usage and assessment performance. Figure 8 shows
the statistically significant correlations between the variables ($p<0.05$).
For the heavy internet users (Cluster 3), the usage of internet resources
other than AI during online exams (OnlInt) is significantly negatively
correlated with the scores on the exams (OnlExams) ($r=-0.28$; $p=0.04$),
which may indicate that relying on the internet, the students were not able to
quickly enough find what they needed to correctly solve the problems,
including replacement of the numbers by their values. For the users who made
use of all resources everywhere (Cluster 4), the use of AI during online exams
(OnlAI) is significantly positively related to Diff5 ($r=0.31$; $p=0.04$);
this means that AI-usage during online exams is positively correlated with
doing better on the final exam instance of duplicate problems than on their
first occurrence in the course.

Notably, within all usage classes, significant correlations between the
supervised final exam and the unsupervised assessment components of the course
remain. This means that in spite of even the heaviest use of external
resources, if not as a significant predictor, unsupervised assessment still
retains formative relevance.

Figure 8: Fruchterman-Reingold [24, 22] representation of the statistically
significant correlations ($p<0.05$) between the variables in Table 1 for the
clusters in Table 2. Note the rotation and handedness of these representations
are random.

Using Final as a proxy for learning success, we also investigated if higher or
lower performing quartiles of the students may have benefitted or been harmed
by the use of online resources, but we found no difference in that regard
between these populations.

###  IV.5 Item analysis

The use of external resources is detrimental to the validity of assessments.
Table 4 shows the average item parameters for assessment items on homework and
unsupervised online exams. The mean reflects the average percentage of
correctly solved items, and the point biserial (“pBis”) the discrimination of
these items (ranging from $-1$ to $1$ where negative values usually denote
invalid assessment items), that is, how well the item distinguishes between
students who generally have a good grasp of the concepts and those who do not.

Table 4: Average problem solving mean and average point biserial correlations for homework and online exams by clusters. | Homework | Online Exams
---|---|---
Cluster | Mean | pBis | Mean | pBis
1 | 0.87 | 0.46 | 0.82 | 0.33
2 | 0.86 | 0.45 | 0.82 | 0.30
3 | 0.84 | 0.40 | 0.81 | 0.28
4 | 0.85 | 0.49 | 0.83 | 0.26

The values only insignificantly differ between different classes of online
resource usage patterns. Overall, they are consistent with the low predictive
power of these unsupervised assessments, but they still indicate that items
can provide feedback to students and instructors.

###  IV.6 Student comments about AI

Based on the replies to the open-ended question on the survey, many students
recognize AI as a valuable tool for assisting in learning, particularly for
understanding complex topics and guiding problem-solving. They particularly
value its ability to quickly provide information without having to flip
through textbook materials or scroll through video. They appreciate AI’s
ability to provide alternate explanations and solutions, which can be
especially helpful when traditional teaching methods fall short. However,
there’s a consensus that AI should not replace genuine learning and effort.
Students suggest that AI’s role should be that of an assistant rather than a
solution provider, and its usage should be context-dependent. For instance, in
major-related courses, students advocate for minimal AI use to ensure a solid
understanding of essential concepts. Conversely, in subjects that are less
critical for them, they see AI as a more acceptable aid.

Concerns about academic integrity and the potential for AI to promote laziness
and dependency are prominent. Students worry that reliance on AI for problem-
solving or essay writing could lead to a superficial understanding of course
material and hinder the development of critical thinking skills. They propose
a balanced approach, where AI is used judiciously to enhance learning without
becoming a crutch. This balance involves using AI for initial guidance or
concept clarification while avoiding its use for directly solving assignments
or exams. Some students commented on having some of the exams during the
semester being in-person as beneficial. Ironically, but of course also very
temptingly, based on style considerations, it appears that several students
filled out the free-response question using ChatGPT.

Finally, the practicality of regulating AI use in online education is a
significant concern. Some students acknowledge that while AI tools like
ChatGPT may not be sophisticated enough currently to solve complex academic
problems accurately, they could still be misused. Several students find
ChatGPT is not yet trustworthy, but expect this to improve in the future.
There’s an acknowledgment that AI is a part of the evolving educational
landscape, and rather than outright banning it, educators should find ways to
integrate it responsibly into the curriculum. This integration could involve
designing assessments that still require a deep understanding of the material,
even with AI assistance, and teaching students how to use AI ethically and
effectively as part of their learning toolkit.

##  V Discussion

Students are making extensive use of external resources when working on
unsupervised assessments. On year after becoming available, students have not
made the jump from “traditional” problem-solving sites to AI. The reasons
might be manifold: the free version of GPT (at the time of writing version
3.5) is much less powerful than version 4, which is only available to
subscribers. A GPT-subscription costs $20/month, while for example Chegg costs
$14.95/month with promotion sales for half that price. Students may also be
used to the “traditional” resources from high school and carry over their
habits to college. For questions that contain figures or graphs, on forums, it
is sufficient to submit only the text in order to locate the question, while
with AI tools, these illustrations need to be described in words [19] (this
will change as the multimodal capabilities of these systems develop further).
Finally, with “traditional” sites, students would find the exact problem with
the expected answer, while all Large Language Models still hallucinate.

A surprising result of this study is the lack of correlations between self-
reported resource usage and assessment outcomes. While the discrepancy between
the score distributions of online and on-campus assessments (Fig. 4) could be
explained by the usage of AI and other online resources (Fig. 5), one would
have expected a correlation [25, 26]: the more resources are used, the higher
the discrepancy; this, however, is not the case. This null result may be due
to students being worried about punitive measures in spite of the strict
research protocol or students underestimating their reliance on external
resources; underreporting of academic dishonesty by approximately
$1/3^{\mbox{\scriptsize rd}}$ has been reported before [26]. It is also
surprising that in spite of all of the external resource usage, a significant
correlation remains between unsupervised and supervised assessments; while the
best and only statistically significant predictor of the score on the final
exam are the scores on the supervised in-semester exams, the other assessments
have not lost their formative relevance.

From the results, it is clear that high-stake exams like the final exam cannot
be conducted in unsupervised settings. At the moment, the time it takes to
look up solutions on the internet may still be a hinderance to overly relying
on those resources (as was also found by the negative correlation between the
extent of using the internet and performance on online exams among students
who heavily rely on external resources), on the long run, AI will likely be
reliable enough that answers to any introductory physics problem, including
newly created ones, can be obtained instantaneously.

Simple usage of lockdown browsers such as Respondus [27] or Safe Exam Browser
[28] are no remedy in an online setting, as students can simply use another
machine or their phone to access sites such as ChatGPT [15], Gemini [29] or
Chegg [11]. Instead, these lockdown browsers, which limit access to local
disks and particular internet sites, are useful in supervised on-campus
settings where students use their own devices (on-campus BYOD exams). If for
logistical reasons, high-stake exams have to be conducted online, there is no
alternative to intrusive proctoring systems that use cameras and microphones.

Students are well-aware of the possible pitfalls associated with AI usage.
While they argue that it will be part of their professional lives, they
support a balanced approach to its use, in particular over-dependence and
over-reliance. They are also are aware of reliability and trustworthiness
issues, which agrees with earlier findings regarding students’ ability to
judge the quality of answers [30, 31]. Overall, though, some statements about
using these tools for learning purposes may have to be taken with the same
grain of salt as the statement about the expert solution in Fig. 1 being
“designed to help students […] learn core concepts;” students may believe
these statements, but still not act accordingly [32]. Remarks about courses
that are “not critical for them” suggest that non-sanctioned external resource
usage may be particularly strong when the goal is simply to pass the course
[33].

##  VI Limitations

Students self-selected into this study, and students who are self-aware that
they are using external resources in non-constructive ways may have refrained
from participating. As the survey and consent form were administered at the
end of the semester, students who dropped out of the course earlier were not
considered.

##  VII Conclusion

Our findings reveal that despite the emergence of LLMs, students predominantly
rely on traditional internet resources for unsupervised quizzes. However, this
reliance does not significantly affect supervised assessments.

Our data indicates that the unsupervised online exams and supervised on-campus
exams yield markedly different outcomes, presumably due to the use of external
resources in unsupervised settings. Interestingly, there’s no strong
correlation between self-reported resource usage and exam performance,
suggesting other factors at play or potential underreporting of resource
usage. Despite heavy resource use, significant correlations between supervised
and unsupervised assessments persist, indicating that unsupervised
assessments, while having no significant predictive properties, retain
formative value.

While students advocate for a balanced approach to AI use, emphasizing its
role as an assistant rather than a solution provider, the study underscores
the necessity of carrying out high-stakes exams in supervised settings to
ensure academic integrity.

Only one quarter of our survey respondents claims to have utilized at least
some minimal form of AI tools in the solution of their homework problems. AI
systems, though, are becoming exponentially more competent. They will rapidly
penetrate all education settings. Our present results do not show significant
causal effects on student learning success from using AI tools. In this sense
our study was conducted perhaps a bit early. But it is clear, nevertheless,
that all of us need to rethink our course offerings, and particularly our
assessment tools, due to the rise of AI tools. Now is the time to help shape
AI environments into the perfect one-on-one tutor for students, similar to
what Feynman envisioned, instead of a means to avoid learning physics.

###### Acknowledgements.

We would like to thank the students who participated in this study. We would
also like to thank Christine Kortemeyer for her help with the manuscript.

## References

  * Feynman  _et al._ [1965] R. Feynman, R. Leighton, and M. Sands,  _The Feynman Lectures on Physics_ (Addison Wesley, Reading, MA, 1965).
  * Kortemeyer and Bauer [1999] G. Kortemeyer and W. Bauer, Multimedia collaborative content creation (mc3): The msu lecture online system, Journal of Engineering Education 88, 421 (1999).
  * Bauer and Westfall [2023] W. Bauer and G. D. Westfall,  _University physics with modern physics (ebook)_ ((McGraw-Hill Higher Education), New York, NY, 2023).
  * Kortemeyer [2014] G. Kortemeyer, Over two decades of blended and online physics courses at Michigan State University, eleed 10 (2014).
  * Russell [1997] T. L. Russell,  _The “No Significant Difference” phenomenon as reported in 248 research reports, summaries, and papers_ (North Carolina State University, 1997).
  * Cavanaugh and Jacquemin [2015] J. Cavanaugh and S. J. Jacquemin, A large sample comparison of grade based student learning outcomes in online vs. face-to-face courses, Online learning 19 (2015).
  * Bergeler and Read [2021] E. Bergeler and M. F. Read, Comparing learning outcomes and satisfaction of an online algebra-based physics course with a face-to-face course, Journal of Science Education and Technology 30, 97 (2021).
  * Kortemeyer  _et al._ [2022] G. Kortemeyer, W. Bauer, and W. Fisher, Hybrid teaching: A tale of two populations, Physical Review Physics Education Research 18, 020130 (2022).
  * Kortemeyer  _et al._ [2023] G. Kortemeyer, C. Kortemeyer, and W. Bauer, Taking introductory physics in studio, lecture, or online format: What difference does it make in subsequent courses, and for whom?, Phys. Rev. Phys. Educ. Res. 19, 020148 (2023).
  * Ruggieri [2020] C. Ruggieri, Students’ use and perception of textbooks and online resources in introductory physics, Physical Review Physics Education Research 16, 020123 (2020).
  * Chegg [2023] Chegg, Chegg, https://www.chegg.com (accessed December 2023).
  * Lancaster and Cotarlan [2021] T. Lancaster and C. Cotarlan, Contract cheating by stem students through a file sharing website: a covid-19 pandemic perspective, International Journal for Educational Integrity 17, 1 (2021).
  * Kortemeyer  _et al._ [2008] G. Kortemeyer, E. Kashy, W. Benenson, and W. Bauer, Experiences using the open-source learning content management and assessment system LON-CAPA in introductory physics courses, Am. J. Phys 76, 438 (2008).
  * Busch [2017] H. Busch, One method for inhibiting the copying of online homework, The Physics Teacher 55, 422 (2017).
  * OpenAI [2023] OpenAI, ChatGPT, https://chat.openai.com/ (accessed December 2023).
  * Google [2023a] Google, Google Bard, https://bard.google.com/ (accessed December 2023a).
  * Prakash [2023] P. Prakash, Chegg’s shares tumbled nearly 50% after the edtech company said its customers are using ChatGPT instead of paying for its study tools, Forbes May (2023).
  * Paresh [2023] D. Paresh, Chegg embraced ai. ChatGPT ate its lunch anyway, Wired June (2023).
  * Kortemeyer [2023] G. Kortemeyer, Could an artificial-intelligence agent pass an introductory physics course?, Phys. Rev. Phys. Educ. Res. 19, 010132 (2023).
  * Laverty  _et al._ [2012] J. T. Laverty, W. Bauer, G. Kortemeyer, and G. Westfall, Want to reduce guessing and cheating while making students happier? give more exams!, Phys. Teach. 50, 540 (2012).
  * R Core Team [2021] R Core Team,  _R: A Language and Environment for Statistical Computing_, R Foundation for Statistical Computing, Vienna, Austria (2021).
  * Epskamp  _et al._ [2012] S. Epskamp, A. O. J. Cramer, L. J. Waldorp, V. D. Schmittmann, and D. Borsboom, qgraph: Network visualizations of relationships in psychometric data, Journal of Statistical Software 48, 1 (2012).
  * Willse [2014] J. T. Willse, ct.: An r package for classical test theory functions (2014).
  * Fruchterman and Reingold [1991] T. M. Fruchterman and E. M. Reingold, Graph drawing by force-directed placement, Software: Practice and experience 21, 1129 (1991).
  * Kashy  _et al._ [2003] D. A. Kashy, G. Albertelli, W. Bauer, E. Kashy, and M. Thoennessen, Influence of nonmoderated and moderated discussion sites on student success, Journal of Asynchronous Learning Networks 7, 31 (2003).
  * Palazzo  _et al._ [2010] D. J. Palazzo, Y.-J. Lee, R. Warnakulasooriya, and D. E. Pritchard, Patterns, correlates, and reduction of homework copying, Physical Review Special Topics – Physics Education Research 6, 010104 (2010).
  * Respondus [2023] Respondus, Respondus Browser, https://web.respondus.com/he/lockdownbrowser/ (accessed December 2023).
  * SEB Alliance, ETH Zurich [2023] SEB Alliance, ETH Zurich, Save Exam Browser, https://www.safeexambrowser.org/ (accessed December 2023).
  * Google [2023b] Google, Gemini, https://deepmind.google/technologies/gemini/ (accessed December 2023b).
  * Shoufan [2023] A. Shoufan, Exploring students’perceptions of CHATGPT: Thematic analysis and follow-up survey, IEEE Access (2023).
  * Dahlkemper  _et al._ [2023] M. N. Dahlkemper, S. Z. Lahme, and P. Klein, How do physics students evaluate artificial intelligence responses on comprehension questions? a study on the perceived scientific accuracy and linguistic quality of ChatGPT, Phys. Rev. Phys. Educ. Res. 19, 010142 (2023).
  * Gray  _et al._ [2008] K. E. Gray, W. K. Adams, C. E. Wieman, and K. K. Perkins, Students know what physicists believe, but they don’t agree: A study using the class survey, Phys. Rev. ST Phys. Educ. Res. 4, 020106 (2008).
  * Lin [1982] H. Lin, Learning physics vs. passing courses, Phys. Teach. 20, 151 (1982).


\min & \dsum_{r\in \mathcal{R}} z_r\label{c:0}\\
& v_r \leq z_r, \forall r \in \mathcal{R}, \label{c:1}\\
& \dsum_{r \in \mathcal{R}} z_r \geq 2, \label{c:11}\\
& y_i + q_i \leq 1, \forall i \in \mathcal{S}, \label{c:2}\\
& y_i + q_i\geq z_r, \forall r \in \mathcal{R}, i \in \mathcal{S} \text{ with } \mathbb{S}^r_i>0 \neq 0,\label{c:3}\\
& \dsum_{i \in \mathcal{S}:\atop \mathbb{S}^r_i>0} y_i \geq z_r, \forall r \in \mathcal{R}, \label{c:5}\\
& \dsum_{i \in \mathcal{S}:\atop \mathbb{S}^r_i<0} y_i \geq z_r, \forall r \in \mathcal{R}, \label{c:6}\\
& \dsum_{r \in \mathcal{R}:\atop |\mathbb{S}^r_i|>0} z_r \geq q_i, \forall i \in \mathcal{S}, \label{c:7}\\
& \dsum_{r \in \mathcal{R}:\atop \mathbb{S}^r_i>0} z_r \geq y_i, \forall i \in \mathcal{S}, \label{c:8}\\
& \dsum_{r \in \mathcal{R}:\atop \mathbb{S}^r_i<0} z_r \geq y_i, \forall i \in \mathcal{S}, \label{c:9}\\
& \dsum_{r \in \mathcal{R}} \mathbb{S}_i^r v_r \geq \varepsilon - \Delta_i (1-y_i), \forall i \in \mathcal{S}, \label{c:12}\\
& y_i, q_i\in \{0,1\}, \forall i \in \mathcal{S}, \label{c:13}\\
& v_r \in [0,1], z_r \in \{0,1\}\forall r \in \mathcal{R}.
\end{align}

There, the objective function accounts for minimizing the number of reactions in the MAS. This objective assures, that each of the generated autocatalytic subnetworks is support-inclusion minimal. Constraints (<ref>) allow the correct representation  of variable $z_r$ (if $v_r>0$, then $z_r$ is forced to take value one, if $v_r=0$, by the minimization criterion, $z_r$ will also take value $0$). Constraint (<ref>) avoids solutions with a single reaction. Constraints (<ref>) assure that species are either core species or present non-core species but not both (it is also possible that they do not participate in the MAS). In case $y_i+q_i=1$, species $i$ is involved in the MASs either as food, member, core or waste species. Otherwise, species $i$ is not a part of the MAS. Constraints (<ref>) assure that in case a reaction $r$ is selected to be part of the MAS ($z_j=1$), the reactants and product species of reaction $r$ must be present (either as core species or present non-core species). Since $q_i$ and $y_i$ are binary variables, the sum of them being greater than one (but because of the previous constraint the equality holds) implies that either $i$ is a core species ($y_i=1$), or non-core present species ($q_i=1$). Otherwise, the constraint is redundant.  Constraints (<ref>) and (<ref>) ensure that in case a reaction is in the MAS, there must be a core species as a reactant and another as a product in the restricted stoichiometric matrix. Observe that in case $z_r=1$, the sum of the binary variables $y_i$ with $i$ being reactant/product of $r$ must be at least one, implying that at least one of the reactant/product species must be activated as core species. In case $z_r=0$, the constraint is redundant.  Similarly, Constraints (<ref>) enforces that in case a species is selected to be a present non-core species ($q_i=1$), then there must be a reaction where it is present.
(<ref>) is the productivity condition (see Remark <ref>). There, in case $y_i=1$ ($i$ is a core species in the motif), the right-hand side in the constraint becomes $1$, implying that $\dsum_{r \in \mathcal{R}} \mathbb{S}^r_i v_r \geq \varepsilon >0$ (here $\varepsilon$ is a given positive but small enough constant to assure that the production is positive).

Constants $\Delta_i$, for $i\in \mathcal{S}$, are assumed to be big enough assuring that in case $i$ is a non-core (present or not) species ($y_i=0$), the production of species $i$ with flow vector $\mathbf{v}$ must always be greater than $\varepsilon - \Delta_i$ (making the constraint redundant). Specifically, for those non-core species whose production is negative (as food species), the absolute value production is upper bounded by $\Delta_i$. We take $\Delta_i = -\sum_{r\in \mathcal{R}:\atop \mathbb{S}^r_i<0} \mathbb{S}^r_i$, for all $i\in \mathcal{S}$.

The above model belongs to the class of discrete optimization problems, which are known to be computationally challenging [42], but at the same time tremendously useful in determining solutions of complex decision problems in various fields [43]. Although the problem of detecting a MAS is known to be a NP-complete [44], there are several available softwares with implemented routines to obtain solutions of integer programming problems (such as Gurobi, CPLEX, ...) in reasonable CPU time for medium size instances.

In order to speed up the solution approach, the above master problem can be strengthened by adding valid inequalities that reduce the search of solutions. For instance, as shown in Theorem <ref>, since the number of reactions in an MAS coincides with the number of core species that are part of it, one can add this (redundant) constraint to the model:

\dsum_{i \in \mathcal{S}} y_i = \dsum_{i \in \mathcal{R}} z_j

Furthermore, in a reversible CRN, denoting by

\begin{align*}
\mathrm{Rev} = \{&(r,r') \in \mathcal{R}\times \mathcal{R}: \\
&\mbox{reaction $r$ is the reverse of reaction $r'$}\},
\end{align*}

one must also impose the following constraints that avoid using a reaction and its reverse in the same subnetwork:

z_r + z_{r'} \leq 1, \forall (r, r')\in \mathrm{Rev}.

That is, if $r$ and $r'$ are reverse reaction of each other, the $z$-variables of those reactions cannot be both one.

The solution of the problem above is a MAS with minimum number of reactions. Once the problem is solved, with optimal values in the variables $(\bar y, \bar q, \bar z, \overline{\mathbf{v}})$, the set of core species is obtained as:

\[{\rm Core} =\{i\in \mathcal{S}: \bar y_i=1\},\]

and the set of reactions involved in the MAS is:

\mathcal{R}'= \{r \in \mathcal{R}: \bar z_r=1\}.

Among the set of present non-core species, ${\rm Q}=\{i \in N: \bar q_i=1\}$, one can classify them based on the sign of the entries of the restricted stoichiometric matrix:

\begin{align*}
{\rm Food} &=\{i\in {\rm Q}: \mathbb{S}^r_i\leq 0, \forall r \in \mathcal{R}'\}, \\
{\rm Waste} &=\{i\in {\rm Q}: \mathbb{S}^r_i\geq 0, \forall r \in \mathcal{R}'\},
\\
{\rm Member} &=({\rm Q} \cup {\rm Core})\backslash ({\rm Food} \cup {\rm Waste})
\end{align*}

The algorithm to generate the whole set of MASs is described in Algorithm <ref>. There, at each iteration, $\ell$, the above Master problem is solved. Once a solution is obtained, say $(y^{\ell},   q^{\ell}, z^{\ell}, \textbf{v}^{\ell})$, the tuple of reactions that are part of the MAS is considered to restrict the next MASs. In particular, it is disallowed for the next MAS to use the same reactions tuples, which assures the minimality of the generated MASs. This restriction is incorporated in the model by means of Constraint (<ref>). The process is repeated until no more autocatalytic subnetwork can be found respecting the pool of incompatibilities that have been added during the procedure. With this algorithm the MASs are generated while being sorted by number of reactions involved in the subnetwork since each time the problem is solved the number of reactions is minimized.

$\mathcal{S}$, $\mathcal{R}$, $\mathbb{S}$, $\ell=0$

Solve Master: $\textsc{Mas}^{\ell} := (y^{\ell},   q^{\ell}, z^{\ell}, \textbf{v}^{\ell})$.

Master is feasible
Save the solution and add to Master the constraint:

\begin{equation}\label{incomp}
\dsum_{r \in \mathcal{R}:\atop z_r=1} z^{\ell}_j \leq |\{r\in \mathcal{R}: z_r^{\ell} =1\}|-1
\end{equation}

$\ell +1 \leftarrow \ell$.

Set of MASs of a CRN: $\textsc{Mas}^{1}, \ldots, \textsc{Mas}^{\ell}$.
Optimization-based approach to enumerate MASs.

One of the advantages of using mathematical optimization-based approaches to determine special sets from an underlying set (as MASs from a CRN) is its flexibility to be adapted to different features. For instance, one can easily filter the obtained MASs to have at most a number of reactions, MaxReactions, involved by adding the following constraint to the model:

\dsum_{r\in \mathcal{R}} z_r \leq \text{\sc MaxReactions}.

Analogously, one can also generate MASs where just a given set of species takes part in the core species set, $\mathcal{S}_0 \subseteq \mathcal{S}$, by adding:

\dsum_{i\in \mathcal{S}_0} y_i =0.

As already mentioned, computing the entire list of MASs in a CRN is computationally challenging. Each iteration in the proposed procedure requires solving a mixed integer linear optimization problem with the constraints in the initial Master problem, but also those in the shape of (<ref>) that are added each time a MASs is found. Thus, the difficulty of solving this optimization problem increases with the number of MASs for a given CRN. In Figure (<ref>) we show the performance of the cumulative CPU time required to generate the MASs for 1-constituent CCRNS with $L=4, 5,$ and $6$. As can be observed from the plots, the consumed CPU time increases linearly in the number of MASs, with a slope strictly greater than one.

CPU time vs. # MASs for 1-constituent CCRN.

Reaction id     Reaction
R1/R2     $2{\bar 1}$ <=> $ {\bar 2}$

R3/R4     ${\bar 1} +{\bar 2}$ <=> $ {\bar 3}$

R5/R6     ${\bar 1} +{\bar 3}$ <=> $ {\bar 4}$

R7/R8     ${\bar 1} +{\bar 4}$ <=> $ {\bar 2} + {\bar 3}$

R9/R10     ${\bar 1} +{\bar 3}$ <=> $ 2{\bar 2}$

R11/R12     $2{\bar 2}$ <=> $ {\bar 4}$

R13/R14     ${\bar 2} +{\bar 4}$ <=> $ 2{\bar 3}$

List of reactions for a CCRN with 1 constinuent and $L=4$.

§.§ Checking intersection of cones

Once the whole set of MASs has been generated, $\mathcal{M} = \{\textsc{Mas}_1, \ldots, \textsc{Mas}_K\}$, we have also developed an optimization-based methodology to check the pairwise intersection of the different polyhedral flow-productive cones and partition-productive cones induced from the set $\mathcal{M}$. Consider two of these cones in $\mathbb{R}^d_+$, say $C_1(v_1, \ldots, v_s)$ and $C_2(w_1, \ldots, w_r)$, described by means of its generating vectors (see Section <ref>). We first check if they the interior of the cones intersect or not by finding a non negative vector that belongs to the intersection of the closed cones maximizing the sum of the entries of that vector with respect to each of the generators of the cones, i.e.:

ρ^*(C_1,C_2) :=     max      ∑_l=1^s λ_l + ∑_l=1^r μ_l

x = _l=1^s λ_l v_l,

x = _l=1^r μ_l w_l,

x ∈ℝ^d_+,

λ_l ∈[0,1], ∀l=1, …, s,

μ_l ∈[0,1], ∀l=1, …, r.

If the optimal solution of the above linear programming problem is $\rho^*(C_1, C_2)=0$, then, the cones do not intersect (otherwise there should be another vector with greatest coefficients in the representation). If $\rho^*(C_1,C_2)>0$, then, the cones intersect.

In case the cones intersect, different situations are possible: the cones are identical, one of the cones is strictly contained in the other, or the cones partially intersect. To detect each of the above possibilities, we compute the distances between each of the  normalized generating vectors of one of the cones to the generating vectors in the other cones:

\phi^1_l = D(v_l, \{w_1, \ldots, w_r\}) = \min_{j=1, \ldots, r} \{\|v_l-w_j\|\}

for $l=1, \ldots, s,$

\phi^2_l = D(w_l, \{v_1, \ldots, v_s\}) = \min_{j=1, \ldots, s} \{\|w_l-v_j\|\}

for $l=1, \ldots, r$.

With these values, in case the cones intersect:

* If $\sum_{l=1}^s \phi^1_l = \sum_{l=1}^r \phi^2_l =0$, then for each of the generating vectors in both cones, there is a normalized generating vector of the other cone at distance zero from it. Thus, both cones are identical.

* If $\sum_{l=1}^s \phi^1_l = 0$, but $ \sum_{l=1}^r \phi^2_l >0$, then, $C_1 \subset C_2$ since for all the generating vectors in $C_1$, there exist a generating vector in $C_2$ at distance zero from it.

* If $\sum_{l=1}^s \phi^1_l >0$, but $ \sum_{l=1}^r \phi^2_l =0$, then, $C_2 \subset C_1$ by the same reasoning as above.

* If $\sum_{l=1}^s \phi^1_l >0$ and $ \sum_{l=1}^r \phi^2_l >0$, then, the cones intersect but only partially.

§ MINIMAL AUTOCATALYTIC SUBNETWORKS FOR CCRN WITH $L=4$ AND $L=5$

Tables <ref> and <ref> show the set of feasible reactions for 1-constituent CCRNs with $L=4$ and $L=5$, respectively. For those CCRNs we report in Tables <ref>, <ref>, and <ref> the list of MASs obtained with our approach. Each row in those tables represent a MAS. In the first column of those tables (id) indicates the identifier of the MAS. The second column (Reactions (Flow)) provides the list of reactions taking part of the MAS. In parenthesis we provide a feasible flow for autocatalysis. In the third column (FWMC Partition), we report the FWMC partition for such a MAS. Finally, in the fourth column (Production) we show the production vector for the feasible flow in the second column, for each of the core species in the MAS.

Reaction id     Reaction
R1/R2     $2 {\bar 1}$ <=> $ {\bar 2}$

R3/R4     ${\bar 1} +{\bar 2}$ <=> $ {\bar 3}$

R5/R6     ${\bar 1} +{\bar 3}$ <=> $ {\bar 4}$

R7/R8     ${\bar 1} +{\bar 4}$ <=> $ {\bar 5}$

R9/R10     ${\bar 1} +{\bar 5}$ <=> $ 2{\bar 3}$

R11/R12     ${\bar 1} +{\bar 5}$ <=> $ {\bar 2} + {\bar 4}$

R13/R14     ${\bar 1} +{\bar 4}$ <=> $ {\bar 2} + {\bar 3}$

R15/R16     ${\bar 1} +{\bar 3}$ <=> $ 2{\bar 2}$

R17/R18     $2 {\bar 2}$ <=> $ {\bar 4}$

R19/R20     ${\bar 2} +{\bar 3}$ <=> $ {\bar 5}$

R21/R22     ${\bar 2} +{\bar 5}$ <=> $ {\bar 3} + {\bar 4}$

R23/R24     ${\bar 2} +{\bar 4}$ <=> $ 2{\bar 3}$

R25/R26     ${\bar 3} +{\bar 5}$ <=> $ 2{\bar 4}$

List of reactions for a CCRN with 1 constinuent and $L=5$.

MAS id     Reactions (Flow)     FWMC Partition      Production
1     $\{ {\rm R2} (1),  {\rm R9} (1)\}$       $\{\{\bar 3\}, \{\},\{\{\bar 1,\bar 2\}\}$      $(1,  1)$

2     $\{ {\rm R9} (1),  {\rm R13} (1)\}$       $\{\{\bar 1,\bar 4\}, \{\},\{\{\bar 2,\bar 3\}\}$      $(1,  1)$

3     $\{ {\rm R4} (3),  {\rm R13} (2)\}$       $\{\{\bar 4\}, \{\bar 1\}, \{\{\bar 2,\bar 3\}\}$      $(1,  1)$

4     $\{ {\rm R3} (3),  {\rm R9} (2)\}$       $\{\{\bar 1\}, \{\},\{\{\bar 2,\bar 3\}\}$      $(1,  1)$

5     $\{ {\rm R8} (3),  {\rm R13} (2)\}$       $\{\{\bar 2\}, \{\bar 1\}, \{\{\bar 3,\bar 4\}\}$      $(1,  1)$

6     $\{ {\rm R5} (3),  {\rm R13} (2)\}$       $\{\{\bar 1,\bar 2\}, \{\},\{\{\bar 3,\bar 4\}\}$      $(1,  1)$

7     $\{ {\rm R8} (3),  {\rm R9} (2)\}$       $\{\{\bar 3\}, \{\bar 4\}, \{\{\bar 1,\bar 2\}\}$      $(1,  1)$

8     $\{ {\rm R2} (2),  {\rm R7} (3)\}$       $\{\{\bar 4\}, \{\bar 3\}, \{\{\bar 1,\bar 2\}\}$      $(1,  1)$

9     $\{ {\rm R8} (3),  {\rm R12} (2)\}$       $\{\{\bar 3\}, \{\bar 1\}, \{\{\bar 2,\bar 4\}\}$      $(1,  1)$

10     $\{ {\rm R4} (2),  {\rm R7} (2),  {\rm R10} (1)\}$       $\{\{\bar 4\}, \{\},\{\{\bar 1,\bar 2, \bar 3\}\}$      $(1,  2,  1)$

11     $\{ {\rm R3} (2),  {\rm R6} (1),  {\rm R8} (2)\}$       $\{\{\bar 2\}, \{\},\{\{\bar 1,\bar 3, \bar 4\}\}$      $(1,  1,  1)$

12     $\{ {\rm R2} (2),  {\rm R5} (3),  {\rm R12} (2)\}$       $\{\{\bar 3\}, \{\},\{\{\bar 1,\bar 2, \bar 4\}\}$      $(1,  2,  1)$

13     $\{ {\rm R3} (2),  {\rm R5} (4),  {\rm R7} (3)\}$       $\{\{\bar 1\}, \{\},\{\{\bar 2,\bar 3, \bar 4\}\}$      $(1,  1,  1)$

14     $\{ {\rm R5} (4),  {\rm R7} (5),  {\rm R11} (2)\}$       $\{\{\bar 1\}, \{\},\{\{\bar 2,\bar 3, \bar 4\}\}$      $(1,  1,  1)$

15     $\{ {\rm R3} (5),  {\rm R5} (4),  {\rm R12} (3)\}$       $\{\{\bar 1\}, \{\},\{\{\bar 2,\bar 3, \bar 4\}\}$      $(1,  1,  1)$

16     $\{ {\rm R3} (6),  {\rm R7} (3),  {\rm R14} (4)\}$       $\{\{\bar 1\}, \{\},\{\{\bar 2,\bar 3, \bar 4\}\}$      $(1,  1,  1)$

17     $\{ {\rm R5} (6),  {\rm R7} (5),  {\rm R10} (2)\}$       $\{\{\}, \{\},\{\bar 1, \{\bar 2,\bar 3, \bar 4\}\}$      $(1,  1,  1)$

18     $\{ {\rm R7} (5),  {\rm R9} (4),  {\rm R11} (6)\}$       $\{\{\bar 1\}, \{\},\{\{\bar 2,\bar 3, \bar 4\}\}$      $(1,  1,  1)$

19     $\{ {\rm R3} (9),  {\rm R12} (3),  {\rm R14} (4)\}$       $\{\{\bar 1\}, \{\},\{\{\bar 2,\bar 3, \bar 4\}\}$      $(1,  1,  1)$

20     $\{ {\rm R7} (9),  {\rm R11} (6),  {\rm R14} (4)\}$       $\{\{\bar 1\}, \{\},\{\{\bar 2,\bar 3, \bar 4\}\}$      $(1,  1,  1)$

List of MASs for 1-constituent CCRN $L=4$.

MAS id     Reactions (Flow)     FWMC Partition      Production
1     $\{ {\rm R2} (1),  {\rm R15} (1)\}$       $\{\{\bar 3\}, \{\},\{\{\bar 1,\bar 2\}\}$      $(1,  1)$

2     $\{ {\rm R23} (1),  {\rm R25} (1)\}$       $\{\{\bar 2,\bar 5\}, \{\},\{\{\bar 3,\bar 4\}\}$      $(1,  1)$

3     $\{ {\rm R15} (1),  {\rm R23} (1)\}$       $\{\{\bar 1,\bar 4\}, \{\},\{\{\bar 2,\bar 3\}\}$      $(1,  1)$

4     $\{ {\rm R13} (3),  {\rm R25} (2)\}$       $\{\{\bar 1,\bar 5\}, \{\bar 2\}, \{\{\bar 3,\bar 4\}\}$      $(1,  1)$

5     $\{ {\rm R22} (3),  {\rm R23} (2)\}$       $\{\{\bar 4\}, \{\bar 5\}, \{\{\bar 2,\bar 3\}\}$      $(1,  1)$

6     $\{ {\rm R2} (2),  {\rm R13} (3)\}$       $\{\{\bar 4\}, \{\bar 3\}, \{\{\bar 1,\bar 2\}\}$      $(1,  1)$

7     $\{ {\rm R12} (3),  {\rm R25} (2)\}$       $\{\{\bar 2,\bar 3\}, \{\bar 1\}, \{\{\bar 4,\bar 5\}\}$      $(1,  1)$

8     $\{ {\rm R7} (3),  {\rm R25} (2)\}$       $\{\{\bar 1,\bar 3\}, \{\},\{\{\bar 4,\bar 5\}\}$      $(1,  1)$

9     $\{ {\rm R9} (2),  {\rm R22} (3)\}$       $\{\{\bar 1,\bar 4\}, \{\bar 2\}, \{\{\bar 3,\bar 5\}\}$      $(1,  1)$

10     $\{ {\rm R6} (3),  {\rm R25} (2)\}$       $\{\{\bar 5\}, \{\bar 1\}, \{\{\bar 3,\bar 4\}\}$      $(1,  1)$

11     $\{ {\rm R14} (3),  {\rm R15} (2)\}$       $\{\{\bar 3\}, \{\bar 4\}, \{\{\bar 1,\bar 2\}\}$      $(1,  1)$

12     $\{ {\rm R5} (3),  {\rm R23} (2)\}$       $\{\{\bar 1,\bar 2\}, \{\},\{\{\bar 3,\bar 4\}\}$      $(1,  1)$

13     $\{ {\rm R14} (3),  {\rm R23} (2)\}$       $\{\{\bar 2\}, \{\bar 1\}, \{\{\bar 3,\bar 4\}\}$      $(1,  1)$

14     $\{ {\rm R15} (2),  {\rm R21} (3)\}$       $\{\{\bar 1,\bar 5\}, \{\bar 4\}, \{\{\bar 2,\bar 3\}\}$      $(1,  1)$

15     $\{ {\rm R22} (3),  {\rm R25} (2)\}$       $\{\{\bar 3\}, \{\bar 2\}, \{\{\bar 4,\bar 5\}\}$      $(1,  1)$

16     $\{ {\rm R12} (3),  {\rm R15} (2)\}$       $\{\{\bar 3,\bar 4\}, \{\bar 5\}, \{\{\bar 1,\bar 2\}\}$      $(1,  1)$

17     $\{ {\rm R14} (3),  {\rm R18} (2)\}$       $\{\{\bar 3\}, \{\bar 1\}, \{\{\bar 2,\bar 4\}\}$      $(1,  1)$

18     $\{ {\rm R2} (2),  {\rm R11} (3)\}$       $\{\{\bar 5\}, \{\bar 4\}, \{\{\bar 1,\bar 2\}\}$      $(1,  1)$

19     $\{ {\rm R4} (3),  {\rm R9} (2)\}$       $\{\{\bar 5\}, \{\bar 2\}, \{\{\bar 1,\bar 3\}\}$      $(1,  1)$

20     $\{ {\rm R9} (2),  {\rm R14} (3)\}$       $\{\{\bar 2,\bar 5\}, \{\bar 4\}, \{\{\bar 1,\bar 3\}\}$      $(1,  1)$

21     $\{ {\rm R18} (2),  {\rm R21} (3)\}$       $\{\{\bar 5\}, \{\bar 3\}, \{\{\bar 2,\bar 4\}\}$      $(1,  1)$

22     $\{ {\rm R3} (3),  {\rm R15} (2)\}$       $\{\{\bar 1\}, \{\},\{\{\bar 2,\bar 3\}\}$      $(1,  1)$

23     $\{ {\rm R4} (3),  {\rm R23} (2)\}$       $\{\{\bar 4\}, \{\bar 1\}, \{\{\bar 2,\bar 3\}\}$      $(1,  1)$

24     $\{ {\rm R9} (2),  {\rm R19} (3)\}$       $\{\{\bar 1,\bar 2\}, \{\},\{\{\bar 3,\bar 5\}\}$      $(1,  1)$

25     $\{ {\rm R11} (1),  {\rm R14} (2),  {\rm R22} (2)\}$       $\{\{\bar 3\}, \{\},\{\bar 1, \{\bar 2,\bar 4, \bar 5\}\}$      $(1,  1,  1)$

26     $\{ {\rm R13} (2),  {\rm R21} (2),  {\rm R24} (1)\}$       $\{\{\bar 1,\bar 5\}, \{\},\{\{\bar 2,\bar 3, \bar 4\}\}$      $(1,  2,  1)$

27     $\{ {\rm R11} (2),  {\rm R19} (2),  {\rm R22} (1)\}$       $\{\{\bar 1,\bar 3\}, \{\},\{\{\bar 2,\bar 4, \bar 5\}\}$      $(1,  1,  1)$

28     $\{ {\rm R3} (2),  {\rm R6} (1),  {\rm R14} (2)\}$       $\{\{\bar 2\}, \{\},\{\{\bar 1,\bar 3, \bar 4\}\}$      $(1,  1,  1)$

29     $\{ {\rm R12} (2),  {\rm R20} (2),  {\rm R22} (1)\}$       $\{\{\bar 4\}, \{\bar 1\}, \{\{\bar 2,\bar 3, \bar 5\}\}$      $(1,  1,  1)$

30     $\{ {\rm R4} (2),  {\rm R13} (1),  {\rm R21} (2)\}$       $\{\{\bar 5\}, \{\},\{\bar 1, \{\bar 2,\bar 3, \bar 4\}\}$      $(1,  1,  1)$

31     $\{ {\rm R6} (2),  {\rm R11} (2),  {\rm R14} (1)\}$       $\{\{\bar 5\}, \{\},\{\bar 2, \{\bar 1,\bar 3, \bar 4\}\}$      $(1,  1,  1)$

32     $\{ {\rm R3} (2),  {\rm R20} (1),  {\rm R22} (2)\}$       $\{\{\bar 1,\bar 4\}, \{\},\{\{\bar 2,\bar 3, \bar 5\}\}$      $(1,  1,  1)$

33     $\{ {\rm R4} (1),  {\rm R12} (2),  {\rm R13} (2)\}$       $\{\{\bar 4\}, \{\bar 5\}, \{\{\bar 1,\bar 2, \bar 3\}\}$      $(1,  1,  1)$

34     $\{ {\rm R4} (2),  {\rm R13} (2),  {\rm R16} (1)\}$       $\{\{\bar 4\}, \{\},\{\{\bar 1,\bar 2, \bar 3\}\}$      $(1,  2,  1)$

35     $\{ {\rm R5} (2),  {\rm R8} (1),  {\rm R12} (2)\}$       $\{\{\bar 2,\bar 3\}, \{\},\{\{\bar 1,\bar 4, \bar 5\}\}$      $(1,  1,  1)$

36     $\{ {\rm R5} (2),  {\rm R13} (2),  {\rm R21} (1)\}$       $\{\{\bar 1,\bar 5\}, \{\},\{\{\bar 2,\bar 3, \bar 4\}\}$      $(1,  1,  1)$

37     $\{ {\rm R11} (2),  {\rm R18} (1),  {\rm R19} (3)\}$       $\{\{\bar 1,\bar 3\}, \{\},\{\{\bar 2,\bar 4, \bar 5\}\}$      $(1,  1,  1)$

38     $\{ {\rm R19} (3),  {\rm R21} (2),  {\rm R23} (1)\}$       $\{\{\bar 2\}, \{\},\{\{\bar 3,\bar 4, \bar 5\}\}$      $(1,  1,  1)$

39     $\{ {\rm R3} (3),  {\rm R18} (2),  {\rm R25} (2)\}$       $\{\{\bar 1,\bar 5\}, \{\},\{\{\bar 2,\bar 3, \bar 4\}\}$      $(1,  1,  2)$

40     $\{ {\rm R7} (2),  {\rm R19} (2),  {\rm R21} (3)\}$       $\{\{\bar 1,\bar 2\}, \{\},\{\{\bar 3,\bar 4, \bar 5\}\}$      $(1,  1,  1)$

41     $\{ {\rm R8} (2),  {\rm R14} (2),  {\rm R22} (3)\}$       $\{\{\bar 3\}, \{\bar 1\}, \{\{\bar 2,\bar 4, \bar 5\}\}$      $(1,  1,  1)$

42     $\{ {\rm R12} (2),  {\rm R19} (2),  {\rm R21} (3)\}$       $\{\{\bar 2\}, \{\bar 1\}, \{\{\bar 3,\bar 4, \bar 5\}\}$      $(1,  1,  1)$

43     $\{ {\rm R11} (2),  {\rm R17} (2),  {\rm R22} (3)\}$       $\{\{\bar 1,\bar 3\}, \{\},\{\{\bar 2,\bar 4, \bar 5\}\}$      $(1,  1,  1)$

44     $\{ {\rm R5} (2),  {\rm R12} (3),  {\rm R21} (2)\}$       $\{\{\bar 2\}, \{\},\{\bar 3, \{\bar 1,\bar 4, \bar 5\}\}$      $(1,  1,  1)$

45     $\{ {\rm R18} (2),  {\rm R19} (3),  {\rm R25} (2)\}$       $\{\{\bar 3\}, \{\},\{\{\bar 2,\bar 4, \bar 5\}\}$      $(1,  2,  1)$

46     $\{ {\rm R12} (3),  {\rm R13} (2),  {\rm R20} (2)\}$       $\{\{\bar 4\}, \{\bar 3\}, \{\{\bar 1,\bar 2, \bar 5\}\}$      $(1,  1,  1)$

47     $\{ {\rm R19} (3),  {\rm R21} (3),  {\rm R26} (1)\}$       $\{\{\bar 2\}, \{\},\{\{\bar 3,\bar 4, \bar 5\}\}$      $(1,  1,  1)$

48     $\{ {\rm R7} (2),  {\rm R11} (3),  {\rm R19} (2)\}$       $\{\{\bar 1,\bar 3\}, \{\},\{\{\bar 2,\bar 4, \bar 5\}\}$      $(1,  1,  1)$

49     $\{ {\rm R2} (2),  {\rm R5} (3),  {\rm R18} (2)\}$       $\{\{\bar 3\}, \{\},\{\{\bar 1,\bar 2, \bar 4\}\}$      $(1,  2,  1)$

50     $\{ {\rm R16} (2),  {\rm R20} (2),  {\rm R22} (3)\}$       $\{\{\bar 4\}, \{\bar 1\}, \{\{\bar 2,\bar 3, \bar 5\}\}$      $(1,  1,  1)$

51     $\{ {\rm R4} (2),  {\rm R6} (3),  {\rm R11} (4)\}$       $\{\{\bar 5\}, \{\bar 2\}, \{\{\bar 1,\bar 3, \bar 4\}\}$      $(1,  1,  1)$

52     $\{ {\rm R14} (4),  {\rm R20} (2),  {\rm R22} (3)\}$       $\{\{\}, \{\bar 1\}, \{\bar 3, \{\bar 2,\bar 4, \bar 5\}\}$      $(1,  1,  1)$

53     $\{ {\rm R4} (2),  {\rm R12} (4),  {\rm R20} (3)\}$       $\{\{\bar 4\}, \{\bar 1\}, \{\{\bar 2,\bar 3, \bar 5\}\}$      $(1,  1,  1)$

54     $\{ {\rm R7} (4),  {\rm R11} (3),  {\rm R14} (2)\}$       $\{\{\bar 3\}, \{\},\{\bar 1, \{\bar 2,\bar 4, \bar 5\}\}$      $(1,  1,  1)$

55     $\{ {\rm R11} (3),  {\rm R13} (2),  {\rm R19} (4)\}$       $\{\{\bar 1\}, \{\},\{\bar 3, \{\bar 2,\bar 4, \bar 5\}\}$      $(1,  1,  1)$

56     $\{ {\rm R13} (2),  {\rm R19} (4),  {\rm R21} (3)\}$       $\{\{\bar 1\}, \{\},\{\bar 2, \{\bar 3,\bar 4, \bar 5\}\}$      $(1,  1,  1)$

57     $\{ {\rm R12} (4),  {\rm R14} (2),  {\rm R21} (3)\}$       $\{\{\bar 2\}, \{\bar 1\}, \{\{\bar 3,\bar 4, \bar 5\}\}$      $(1,  1,  1)$

58     $\{ {\rm R6} (4),  {\rm R11} (3),  {\rm R21} (2)\}$       $\{\{\bar 5\}, \{\bar 3\}, \{\{\bar 1,\bar 2, \bar 4\}\}$      $(1,  1,  1)$

59     $\{ {\rm R8} (4),  {\rm R19} (2),  {\rm R22} (3)\}$       $\{\{\bar 3\}, \{\bar 1\}, \{\{\bar 2,\bar 4, \bar 5\}\}$      $(1,  1,  1)$

60     $\{ {\rm R7} (4),  {\rm R14} (2),  {\rm R21} (3)\}$       $\{\{\bar 2\}, \{\},\{\bar 1, \{\bar 3,\bar 4, \bar 5\}\}$      $(1,  1,  1)$

61     $\{ {\rm R4} (4),  {\rm R6} (2),  {\rm R21} (3)\}$       $\{\{\bar 5\}, \{\bar 1\}, \{\{\bar 2,\bar 3, \bar 4\}\}$      $(1,  1,  1)$

62     $\{ {\rm R5} (2),  {\rm R7} (4),  {\rm R21} (3)\}$       $\{\{\bar 1,\bar 2\}, \{\},\{\{\bar 3,\bar 4, \bar 5\}\}$      $(1,  1,  1)$

63     $\{ {\rm R3} (4),  {\rm R11} (2),  {\rm R22} (3)\}$       $\{\{\bar 1\}, \{\},\{\bar 4, \{\bar 2,\bar 3, \bar 5\}\}$      $(1,  1,  1)$

64     $\{ {\rm R5} (2),  {\rm R14} (3),  {\rm R22} (4)\}$       $\{\{\bar 3\}, \{\bar 5\}, \{\{\bar 1,\bar 2, \bar 4\}\}$      $(1,  1,  1)$

65     $\{ {\rm R4} (3),  {\rm R11} (2),  {\rm R21} (4)\}$       $\{\{\bar 5\}, \{\bar 4\}, \{\{\bar 1,\bar 2, \bar 3\}\}$      $(1,  1,  1)$

66     $\{ {\rm R3} (2),  {\rm R5} (4),  {\rm R13} (3)\}$       $\{\{\bar 1\}, \{\},\{\{\bar 2,\bar 3, \bar 4\}\}$      $(1,  1,  1)$

67     $\{ {\rm R8} (2),  {\rm R12} (3),  {\rm R13} (4)\}$       $\{\{\}, \{\bar 3\}, \{\bar 4, \{\bar 1,\bar 2, \bar 5\}\}$      $(1,  1,  1)$

68     $\{ {\rm R6} (2),  {\rm R19} (4),  {\rm R21} (3)\}$       $\{\{\bar 2\}, \{\bar 1\}, \{\{\bar 3,\bar 4, \bar 5\}\}$      $(1,  1,  1)$

69     $\{ {\rm R3} (4),  {\rm R12} (2),  {\rm R14} (3)\}$       $\{\{\bar 2\}, \{\bar 5\}, \{\{\bar 1,\bar 3, \bar 4\}\}$      $(1,  1,  1)$

70     $\{ {\rm R5} (2),  {\rm R8} (3),  {\rm R22} (4)\}$       $\{\{\bar 3\}, \{\bar 2\}, \{\{\bar 1,\bar 4, \bar 5\}\}$      $(1,  1,  1)$

71     $\{ {\rm R7} (2),  {\rm R12} (3),  {\rm R20} (4)\}$       $\{\{\bar 4\}, \{\bar 3\}, \{\{\bar 1,\bar 2, \bar 5\}\}$      $(1,  1,  1)$

72     $\{ {\rm R12} (4),  {\rm R13} (3),  {\rm R22} (2)\}$       $\{\{\bar 4\}, \{\bar 5\}, \{\{\bar 1,\bar 2, \bar 3\}\}$      $(1,  1,  1)$

73     $\{ {\rm R3} (4),  {\rm R14} (5),  {\rm R26} (2)\}$       $\{\{\bar 2\}, \{\bar 5\}, \{\{\bar 1,\bar 3, \bar 4\}\}$      $(1,  1,  1)$

74     $\{ {\rm R11} (5),  {\rm R19} (4),  {\rm R26} (2)\}$       $\{\{\bar 1\}, \{\},\{\bar 3, \{\bar 2,\bar 4, \bar 5\}\}$      $(1,  1,  1)$

75     $\{ {\rm R8} (4),  {\rm R17} (2),  {\rm R22} (5)\}$       $\{\{\bar 3\}, \{\bar 1\}, \{\{\bar 2,\bar 4, \bar 5\}\}$      $(1,  1,  1)$

76     $\{ {\rm R10} (2),  {\rm R12} (4),  {\rm R13} (5)\}$       $\{\{\bar 4\}, \{\bar 5\}, \{\{\bar 1,\bar 2, \bar 3\}\}$      $(1,  1,  1)$

77     $\{ {\rm R10} (2),  {\rm R12} (4),  {\rm R20} (5)\}$       $\{\{\bar 4\}, \{\bar 1\}, \{\{\bar 2,\bar 3, \bar 5\}\}$      $(1,  1,  1)$

List of MASs for 1-constituent CCRN $L=5$.

MAS id     Reactions (Flow)     FWMC Partition      Production
78     $\{ {\rm R1} (2),  {\rm R12} (5),  {\rm R20} (4)\}$       $\{\{\bar 4\}, \{\bar 3\}, \{\{\bar 1,\bar 2, \bar 5\}\}$      $(1,  1,  1)$

79     $\{ {\rm R7} (4),  {\rm R10} (2),  {\rm R21} (5)\}$       $\{\{\bar 2\}, \{\},\{\bar 1, \{\bar 3,\bar 4, \bar 5\}\}$      $(1,  1,  1)$

80     $\{ {\rm R1} (2),  {\rm R14} (5),  {\rm R22} (4)\}$       $\{\{\bar 3\}, \{\bar 5\}, \{\{\bar 1,\bar 2, \bar 4\}\}$      $(1,  1,  1)$

81     $\{ {\rm R5} (4),  {\rm R13} (5),  {\rm R17} (2)\}$       $\{\{\bar 1\}, \{\},\{\{\bar 2,\bar 3, \bar 4\}\}$      $(1,  1,  1)$

82     $\{ {\rm R6} (4),  {\rm R11} (5),  {\rm R16} (2)\}$       $\{\{\bar 5\}, \{\bar 3\}, \{\{\bar 1,\bar 2, \bar 4\}\}$      $(1,  1,  1)$

83     $\{ {\rm R6} (5),  {\rm R11} (4),  {\rm R24} (2)\}$       $\{\{\bar 5\}, \{\bar 2\}, \{\{\bar 1,\bar 3, \bar 4\}\}$      $(1,  1,  1)$

84     $\{ {\rm R10} (2),  {\rm R12} (4),  {\rm R21} (5)\}$       $\{\{\bar 2\}, \{\bar 1\}, \{\{\bar 3,\bar 4, \bar 5\}\}$      $(1,  1,  1)$

85     $\{ {\rm R2} (3),  {\rm R9} (5),  {\rm R24} (4)\}$       $\{\{\bar 5\}, \{\bar 4\}, \{\{\bar 1,\bar 2, \bar 3\}\}$      $(1,  1,  2)$

86     $\{ {\rm R5} (5),  {\rm R9} (3),  {\rm R12} (4)\}$       $\{\{\bar 2\}, \{\},\{\bar 1, \{\bar 3,\bar 4, \bar 5\}\}$      $(1,  1,  1)$

87     $\{ {\rm R2} (3),  {\rm R5} (5),  {\rm R22} (4)\}$       $\{\{\bar 3\}, \{\bar 5\}, \{\{\bar 1,\bar 2, \bar 4\}\}$      $(1,  1,  1)$

88     $\{ {\rm R2} (3),  {\rm R7} (5),  {\rm R20} (4)\}$       $\{\{\bar 4\}, \{\bar 3\}, \{\{\bar 1,\bar 2, \bar 5\}\}$      $(1,  1,  1)$

89     $\{ {\rm R8} (4),  {\rm R15} (3),  {\rm R19} (5)\}$       $\{\{\bar 3\}, \{\bar 4\}, \{\{\bar 1,\bar 2, \bar 5\}\}$      $(1,  1,  1)$

90     $\{ {\rm R11} (4),  {\rm R19} (5),  {\rm R23} (3)\}$       $\{\{\bar 1\}, \{\},\{\bar 2, \{\bar 3,\bar 4, \bar 5\}\}$      $(1,  1,  1)$

91     $\{ {\rm R5} (5),  {\rm R7} (4),  {\rm R9} (3)\}$       $\{\{\bar 1\}, \{\},\{\{\bar 3,\bar 4, \bar 5\}\}$      $(1,  1,  1)$

92     $\{ {\rm R3} (5),  {\rm R5} (4),  {\rm R18} (3)\}$       $\{\{\bar 1\}, \{\},\{\{\bar 2,\bar 3, \bar 4\}\}$      $(1,  1,  1)$

93     $\{ {\rm R16} (4),  {\rm R18} (5),  {\rm R25} (3)\}$       $\{\{\bar 5\}, \{\bar 1\}, \{\{\bar 2,\bar 3, \bar 4\}\}$      $(2,  1,  1)$

94     $\{ {\rm R8} (4),  {\rm R19} (5),  {\rm R23} (3)\}$       $\{\{\bar 2\}, \{\bar 1\}, \{\{\bar 3,\bar 4, \bar 5\}\}$      $(1,  1,  1)$

95     $\{ {\rm R8} (4),  {\rm R18} (3),  {\rm R19} (5)\}$       $\{\{\bar 3\}, \{\bar 1\}, \{\{\bar 2,\bar 4, \bar 5\}\}$      $(1,  1,  1)$

96     $\{ {\rm R6} (6),  {\rm R11} (5),  {\rm R17} (2)\}$       $\{\{\bar 5\}, \{\bar 3\}, \{\{\bar 1,\bar 2, \bar 4\}\}$      $(1,  1,  1)$

97     $\{ {\rm R7} (6),  {\rm R11} (5),  {\rm R17} (2)\}$       $\{\{\bar 1\}, \{\},\{\{\bar 2,\bar 4, \bar 5\}\}$      $(1,  1,  1)$

98     $\{ {\rm R13} (3),  {\rm R16} (4),  {\rm R22} (6)\}$       $\{\{\bar 4\}, \{\bar 5\}, \{\{\bar 1,\bar 2, \bar 3\}\}$      $(1,  1,  1)$

99     $\{ {\rm R4} (6),  {\rm R21} (5),  {\rm R26} (2)\}$       $\{\{\}, \{\bar 1\}, \{\bar 5, \{\bar 2,\bar 3, \bar 4\}\}$      $(1,  1,  1)$

100     $\{ {\rm R1} (2),  {\rm R4} (5),  {\rm R21} (6)\}$       $\{\{\bar 5\}, \{\bar 4\}, \{\{\bar 1,\bar 2, \bar 3\}\}$      $(1,  1,  1)$

101     $\{ {\rm R3} (6),  {\rm R13} (3),  {\rm R24} (4)\}$       $\{\{\bar 1\}, \{\},\{\{\bar 2,\bar 3, \bar 4\}\}$      $(1,  1,  1)$

102     $\{ {\rm R5} (6),  {\rm R13} (5),  {\rm R16} (2)\}$       $\{\{\}, \{\},\{\bar 1, \{\bar 2,\bar 3, \bar 4\}\}$      $(1,  1,  1)$

103     $\{ {\rm R12} (6),  {\rm R21} (5),  {\rm R24} (2)\}$       $\{\{\}, \{\bar 1\}, \{\bar 2, \{\bar 3,\bar 4, \bar 5\}\}$      $(1,  1,  1)$

104     $\{ {\rm R6} (5),  {\rm R10} (2),  {\rm R11} (6)\}$       $\{\{\}, \{\bar 2\}, \{\bar 5, \{\bar 1,\bar 3, \bar 4\}\}$      $(1,  1,  1)$

105     $\{ {\rm R12} (6),  {\rm R20} (5),  {\rm R24} (2)\}$       $\{\{\}, \{\bar 1\}, \{\bar 4, \{\bar 2,\bar 3, \bar 5\}\}$      $(1,  1,  1)$

106     $\{ {\rm R5} (6),  {\rm R21} (3),  {\rm R26} (4)\}$       $\{\{\bar 1,\bar 2\}, \{\},\{\{\bar 3,\bar 4, \bar 5\}\}$      $(1,  1,  1)$

107     $\{ {\rm R12} (6),  {\rm R13} (5),  {\rm R24} (2)\}$       $\{\{\}, \{\bar 5\}, \{\bar 4, \{\bar 1,\bar 2, \bar 3\}\}$      $(1,  1,  1)$

108     $\{ {\rm R4} (3),  {\rm R11} (6),  {\rm R16} (4)\}$       $\{\{\bar 5\}, \{\bar 4\}, \{\{\bar 1,\bar 2, \bar 3\}\}$      $(1,  1,  1)$

109     $\{ {\rm R6} (6),  {\rm R21} (3),  {\rm R24} (4)\}$       $\{\{\bar 5\}, \{\bar 1\}, \{\{\bar 2,\bar 3, \bar 4\}\}$      $(1,  1,  1)$

110     $\{ {\rm R7} (6),  {\rm R21} (5),  {\rm R24} (2)\}$       $\{\{\bar 1\}, \{\},\{\bar 2, \{\bar 3,\bar 4, \bar 5\}\}$      $(1,  1,  1)$

111     $\{ {\rm R14} (6),  {\rm R21} (3),  {\rm R26} (4)\}$       $\{\{\bar 2\}, \{\bar 1\}, \{\{\bar 3,\bar 4, \bar 5\}\}$      $(1,  1,  1)$

112     $\{ {\rm R11} (5),  {\rm R16} (6),  {\rm R18} (4)\}$       $\{\{\bar 5\}, \{\bar 3\}, \{\{\bar 1,\bar 2, \bar 4\}\}$      $(1,  1,  1)$

113     $\{ {\rm R10} (6),  {\rm R21} (5),  {\rm R23} (4)\}$       $\{\{\bar 2\}, \{\bar 1\}, \{\{\bar 3,\bar 4, \bar 5\}\}$      $(1,  1,  1)$

114     $\{ {\rm R6} (5),  {\rm R9} (4),  {\rm R24} (6)\}$       $\{\{\bar 5\}, \{\bar 2\}, \{\{\bar 1,\bar 3, \bar 4\}\}$      $(1,  1,  1)$

115     $\{ {\rm R13} (5),  {\rm R15} (4),  {\rm R17} (6)\}$       $\{\{\bar 1\}, \{\},\{\{\bar 2,\bar 3, \bar 4\}\}$      $(1,  1,  1)$

116     $\{ {\rm R10} (6),  {\rm R13} (5),  {\rm R23} (4)\}$       $\{\{\bar 4\}, \{\bar 5\}, \{\{\bar 1,\bar 2, \bar 3\}\}$      $(1,  1,  1)$

117     $\{ {\rm R10} (6),  {\rm R20} (5),  {\rm R23} (4)\}$       $\{\{\bar 4\}, \{\bar 1\}, \{\{\bar 2,\bar 3, \bar 5\}\}$      $(1,  1,  1)$

118     $\{ {\rm R5} (9),  {\rm R9} (3),  {\rm R26} (4)\}$       $\{\{\bar 1\}, \{\},\{\{\bar 3,\bar 4, \bar 5\}\}$      $(1,  1,  1)$

119     $\{ {\rm R3} (9),  {\rm R18} (3),  {\rm R24} (4)\}$       $\{\{\bar 1\}, \{\},\{\{\bar 2,\bar 3, \bar 4\}\}$      $(1,  1,  1)$

120     $\{ {\rm R10} (6),  {\rm R13} (9),  {\rm R16} (4)\}$       $\{\{\bar 4\}, \{\bar 5\}, \{\{\bar 1,\bar 2, \bar 3\}\}$      $(1,  1,  1)$

121     $\{ {\rm R10} (6),  {\rm R21} (9),  {\rm R26} (4)\}$       $\{\{\bar 2\}, \{\bar 1\}, \{\{\bar 3,\bar 4, \bar 5\}\}$      $(1,  1,  1)$

122     $\{ {\rm R13} (9),  {\rm R17} (6),  {\rm R24} (4)\}$       $\{\{\bar 1\}, \{\},\{\{\bar 2,\bar 3, \bar 4\}\}$      $(1,  1,  1)$

123     $\{ {\rm R2} (1),  {\rm R9} (1),  {\rm R18} (1),  {\rm R25} (1)\}$       $\{\{\bar 5\}, \{\},\{\{\bar 1,\bar 2, \bar 3, \bar 4\}\}$      $(1,  1,  1,  1)$

124     $\{ {\rm R3} (3),  {\rm R8} (3),  {\rm R12} (2),  {\rm R19} (2)\}$       $\{\{\bar 2\}, \{\},\{\{\bar 1,\bar 3, \bar 4, \bar 5\}\}$      $(2,  1,  1,  1)$

125     $\{ {\rm R9} (2),  {\rm R16} (3),  {\rm R18} (2),  {\rm R24} (3)\}$       $\{\{\bar 5\}, \{\},\{\{\bar 1,\bar 2, \bar 3, \bar 4\}\}$      $(1,  1,  1,  1)$

126     $\{ {\rm R4} (3),  {\rm R7} (3),  {\rm R16} (2),  {\rm R20} (2)\}$       $\{\{\bar 4\}, \{\},\{\{\bar 1,\bar 2, \bar 3, \bar 5\}\}$      $(2,  1,  1,  1)$

127     $\{ {\rm R3} (5),  {\rm R8} (3),  {\rm R10} (2),  {\rm R12} (2)\}$       $\{\{\bar 2\}, \{\},\{\{\bar 1,\bar 3, \bar 4, \bar 5\}\}$      $(2,  1,  1,  1)$

128     $\{ {\rm R3} (3),  {\rm R6} (2),  {\rm R8} (3),  {\rm R19} (4)\}$       $\{\{\bar 2\}, \{\},\{\{\bar 1,\bar 3, \bar 4, \bar 5\}\}$      $(2,  1,  1,  1)$

129     $\{ {\rm R7} (3),  {\rm R10} (3),  {\rm R16} (2),  {\rm R20} (5)\}$       $\{\{\bar 4\}, \{\},\{\{\bar 1,\bar 2, \bar 3, \bar 5\}\}$      $(2,  1,  1,  1)$

130     $\{ {\rm R3} (4),  {\rm R6} (2),  {\rm R10} (4),  {\rm R21} (3)\}$       $\{\{\bar 2\}, \{\},\{\{\bar 1,\bar 3, \bar 4, \bar 5\}\}$      $(2,  1,  1,  1)$

131     $\{ {\rm R3} (5),  {\rm R8} (3),  {\rm R10} (4),  {\rm R23} (2)\}$       $\{\{\bar 2\}, \{\},\{\{\bar 1,\bar 3, \bar 4, \bar 5\}\}$      $(2,  1,  1,  1)$

132     $\{ {\rm R3} (3),  {\rm R5} (2),  {\rm R7} (5),  {\rm R11} (4)\}$       $\{\{\bar 1\}, \{\},\{\{\bar 2,\bar 3, \bar 4, \bar 5\}\}$      $(1,  1,  1,  1)$

133     $\{ {\rm R3} (3),  {\rm R8} (5),  {\rm R19} (4),  {\rm R26} (2)\}$       $\{\{\bar 2\}, \{\},\{\{\bar 1,\bar 3, \bar 4, \bar 5\}\}$      $(2,  1,  1,  1)$

134     $\{ {\rm R3} (5),  {\rm R8} (3),  {\rm R12} (4),  {\rm R24} (2)\}$       $\{\{\}, \{\},\{\bar 2, \{\bar 1,\bar 3, \bar 4, \bar 5\}\}$      $(2,  1,  1,  1)$

135     $\{ {\rm R3} (5),  {\rm R7} (5),  {\rm R11} (4),  {\rm R24} (2)\}$       $\{\{\bar 1\}, \{\},\{\{\bar 2,\bar 3, \bar 4, \bar 5\}\}$      $(1,  1,  1,  1)$

136     $\{ {\rm R3} (7),  {\rm R6} (2),  {\rm R8} (3),  {\rm R10} (4)\}$       $\{\{\bar 2\}, \{\},\{\{\bar 1,\bar 3, \bar 4, \bar 5\}\}$      $(2,  1,  1,  1)$

137     $\{ {\rm R3} (3),  {\rm R5} (6),  {\rm R7} (5),  {\rm R20} (4)\}$       $\{\{\bar 1\}, \{\},\{\{\bar 2,\bar 3, \bar 4, \bar 5\}\}$      $(1,  1,  1,  1)$

138     $\{ {\rm R3} (5),  {\rm R7} (5),  {\rm R10} (2),  {\rm R11} (6)\}$       $\{\{\}, \{\},\{\bar 1, \{\bar 2,\bar 3, \bar 4, \bar 5\}\}$      $(1,  1,  1,  1)$

139     $\{ {\rm R3} (7),  {\rm R8} (5),  {\rm R10} (4),  {\rm R26} (2)\}$       $\{\{\bar 2\}, \{\},\{\{\bar 1,\bar 3, \bar 4, \bar 5\}\}$      $(2,  1,  1,  1)$

140     $\{ {\rm R3} (3),  {\rm R5} (7),  {\rm R11} (4),  {\rm R26} (5)\}$       $\{\{\bar 1\}, \{\},\{\{\bar 2,\bar 3, \bar 4, \bar 5\}\}$      $(1,  1,  1,  1)$

141     $\{ {\rm R5} (2),  {\rm R7} (8),  {\rm R11} (7),  {\rm R16} (3)\}$       $\{\{\}, \{\},\{\bar 1, \{\bar 2,\bar 3, \bar 4, \bar 5\}\}$      $(1,  1,  1,  1)$

142     $\{ {\rm R3} (5),  {\rm R7} (5),  {\rm R9} (4),  {\rm R24} (6)\}$       $\{\{\bar 1\}, \{\},\{\{\bar 2,\bar 3, \bar 4, \bar 5\}\}$      $(1,  1,  1,  1)$

143     $\{ {\rm R7} (5),  {\rm R9} (4),  {\rm R15} (7),  {\rm R17} (6)\}$       $\{\{\bar 1\}, \{\},\{\{\bar 2,\bar 3, \bar 4, \bar 5\}\}$      $(2,  1,  1,  1)$

144     $\{ {\rm R3} (3),  {\rm R5} (11),  {\rm R20} (4),  {\rm R26} (5)\}$       $\{\{\bar 1\}, \{\},\{\{\bar 2,\bar 3, \bar 4, \bar 5\}\}$      $(1,  1,  1,  1)$

145     $\{ {\rm R5} (6),  {\rm R7} (8),  {\rm R17} (3),  {\rm R20} (7)\}$       $\{\{\bar 1\}, \{\},\{\{\bar 2,\bar 3, \bar 4, \bar 5\}\}$      $(1,  1,  1,  1)$

146     $\{ {\rm R3} (9),  {\rm R7} (5),  {\rm R20} (4),  {\rm R24} (6)\}$       $\{\{\bar 1\}, \{\},\{\{\bar 2,\bar 3, \bar 4, \bar 5\}\}$      $(1,  1,  1,  1)$

147     $\{ {\rm R5} (7),  {\rm R11} (7),  {\rm R17} (3),  {\rm R26} (8)\}$       $\{\{\bar 1\}, \{\},\{\{\bar 2,\bar 3, \bar 4, \bar 5\}\}$      $(1,  1,  1,  1)$

148     $\{ {\rm R3} (10),  {\rm R11} (4),  {\rm R24} (7),  {\rm R26} (5)\}$       $\{\{\bar 1\}, \{\},\{\{\bar 2,\bar 3, \bar 4, \bar 5\}\}$      $(1,  1,  1,  1)$

149     $\{ {\rm R7} (10),  {\rm R11} (9),  {\rm R16} (5),  {\rm R24} (2)\}$       $\{\{\}, \{\},\{\bar 1, \{\bar 2,\bar 3, \bar 4, \bar 5\}\}$      $(1,  1,  1,  1)$

150     $\{ {\rm R7} (9),  {\rm R9} (8),  {\rm R17} (3),  {\rm R24} (7)\}$       $\{\{\bar 1\}, \{\},\{\{\bar 2,\bar 3, \bar 4, \bar 5\}\}$      $(1,  2,  1,  1)$

151     $\{ {\rm R5} (9),  {\rm R7} (8),  {\rm R16} (3),  {\rm R20} (7)\}$       $\{\{\}, \{\},\{\bar 1, \{\bar 2,\bar 3, \bar 4, \bar 5\}\}$      $(1,  1,  1,  1)$

152     $\{ {\rm R7} (10),  {\rm R10} (2),  {\rm R11} (11),  {\rm R16} (5)\}$       $\{\{\}, \{\},\{\bar 1, \{\bar 2,\bar 3, \bar 4, \bar 5\}\}$      $(1,  1,  1,  1)$

153     $\{ {\rm R5} (10),  {\rm R11} (7),  {\rm R16} (3),  {\rm R26} (8)\}$       $\{\{\}, \{\},\{\bar 1, \{\bar 2,\bar 3, \bar 4, \bar 5\}\}$      $(1,  1,  1,  1)$

154     $\{ {\rm R3} (10),  {\rm R10} (7),  {\rm R11} (6),  {\rm R13} (5)\}$       $\{\{\}, \{\},\{\bar 1, \{\bar 2,\bar 3, \bar 4, \bar 5\}\}$      $(1,  1,  1,  1)$

155     $\{ {\rm R3} (10),  {\rm R9} (4),  {\rm R24} (11),  {\rm R26} (5)\}$       $\{\{\bar 1\}, \{\},\{\{\bar 2,\bar 3, \bar 4, \bar 5\}\}$      $(1,  1,  1,  1)$

156     $\{ {\rm R7} (8),  {\rm R15} (6),  {\rm R17} (9),  {\rm R20} (7)\}$       $\{\{\bar 1\}, \{\},\{\{\bar 2,\bar 3, \bar 4, \bar 5\}\}$      $(1,  1,  1,  1)$

157     $\{ {\rm R9} (4),  {\rm R15} (12),  {\rm R17} (11),  {\rm R26} (5)\}$       $\{\{\bar 1\}, \{\},\{\{\bar 2,\bar 3, \bar 4, \bar 5\}\}$      $(2,  1,  1,  1)$

158     $\{ {\rm R11} (7),  {\rm R15} (7),  {\rm R17} (10),  {\rm R26} (8)\}$       $\{\{\bar 1\}, \{\},\{\{\bar 2,\bar 3, \bar 4, \bar 5\}\}$      $(1,  1,  1,  1)$

159     $\{ {\rm R5} (14),  {\rm R17} (3),  {\rm R20} (7),  {\rm R26} (8)\}$       $\{\{\bar 1\}, \{\},\{\{\bar 2,\bar 3, \bar 4, \bar 5\}\}$      $(1,  1,  1,  1)$

160     $\{ {\rm R3} (15),  {\rm R10} (7),  {\rm R11} (6),  {\rm R18} (5)\}$       $\{\{\}, \{\},\{\bar 1, \{\bar 2,\bar 3, \bar 4, \bar 5\}\}$      $(1,  1,  1,  1)$

161     $\{ {\rm R3} (10),  {\rm R10} (7),  {\rm R11} (11),  {\rm R26} (5)\}$       $\{\{\}, \{\},\{\bar 1, \{\bar 2,\bar 3, \bar 4, \bar 5\}\}$      $(1,  1,  1,  1)$

162     $\{ {\rm R3} (14),  {\rm R20} (4),  {\rm R24} (11),  {\rm R26} (5)\}$       $\{\{\bar 1\}, \{\},\{\{\bar 2,\bar 3, \bar 4, \bar 5\}\}$      $(1,  1,  1,  1)$

163     $\{ {\rm R7} (10),  {\rm R9} (9),  {\rm R16} (5),  {\rm R24} (11)\}$       $\{\{\}, \{\},\{\bar 1, \{\bar 2,\bar 3, \bar 4, \bar 5\}\}$      $(1,  1,  1,  1)$

164     $\{ {\rm R5} (17),  {\rm R16} (3),  {\rm R20} (7),  {\rm R26} (8)\}$       $\{\{\}, \{\},\{\bar 1, \{\bar 2,\bar 3, \bar 4, \bar 5\}\}$      $(1,  1,  1,  1)$

165     $\{ {\rm R10} (7),  {\rm R11} (6),  {\rm R13} (15),  {\rm R17} (10)\}$       $\{\{\}, \{\},\{\bar 1, \{\bar 2,\bar 3, \bar 4, \bar 5\}\}$      $(1,  1,  1,  1)$

166     $\{ {\rm R7} (14),  {\rm R17} (9),  {\rm R20} (13),  {\rm R24} (6)\}$       $\{\{\bar 1\}, \{\},\{\{\bar 2,\bar 3, \bar 4, \bar 5\}\}$      $(1,  1,  1,  1)$

167     $\{ {\rm R11} (14),  {\rm R17} (10),  {\rm R24} (7),  {\rm R26} (15)\}$       $\{\{\bar 1\}, \{\},\{\{\bar 2,\bar 3, \bar 4, \bar 5\}\}$      $(1,  1,  1,  1)$

168     $\{ {\rm R15} (14),  {\rm R17} (17),  {\rm R20} (7),  {\rm R26} (8)\}$       $\{\{\bar 1\}, \{\},\{\{\bar 2,\bar 3, \bar 4, \bar 5\}\}$      $(1,  1,  1,  1)$

169     $\{ {\rm R10} (7),  {\rm R11} (21),  {\rm R17} (10),  {\rm R26} (15)\}$       $\{\{\}, \{\},\{\bar 1, \{\bar 2,\bar 3, \bar 4, \bar 5\}\}$      $(1,  1,  1,  1)$

170     $\{ {\rm R9} (14),  {\rm R17} (10),  {\rm R24} (21),  {\rm R26} (15)\}$       $\{\{\bar 1\}, \{\},\{\{\bar 2,\bar 3, \bar 4, \bar 5\}\}$      $(1,  1,  1,  1)$

List of MASs for 1-constituent CCRN $L=5$ (cntd.).

[1]
Tomas Veloz, Pablo Razeto-Barry, Peter Dittrich, and Alejandro Fajardo.
Reaction networks and evolutionary game theory.
Journal of mathematical biology, 68:181–206, 2014.

[2]
Eric Smith and Harold J Morowitz.
The origin and nature of life on earth: the emergence of the
fourth geosphere.
Cambridge University Press, 2016.

[3]
Irving R Epstein and John A Pojman.
An introduction to nonlinear chemical dynamics: oscillations,
waves, patterns, and chaos.
Oxford university press, 1998.

[4]
Peter Schuster.
What is special about autocatalysis?
Monatshefte für Chemie-Chemical Monthly, 150:763–775,

[5]
Wim Hordijk and Mike Steel.
Detecting autocatalytic, self-sustaining sets in chemical reaction
Journal of theoretical biology, 227(4):451–461, 2004.

[6]
Jakob L Andersen, Christoph Flamm, Daniel Merkle, and Peter F Stadler.
Defining autocatalysis in chemical reaction networks.
arXiv preprint arXiv:2107.03086, 2021.

[7]
Zhen Peng, Alex M Plum, Praful Gagrani, and David A Baum.
An ecological framework for the analysis of prebiotic chemical
reaction networks.
Journal of theoretical biology, 507:110451, 2020.

[8]
Nicola Vassena and Peter F Stadler.
Unstable cores are the source of instability in chemical reaction
arXiv preprint arXiv:2308.11486, 2023.

[9]
Abhishek Deshpande and Manoj Gopalkrishnan.
Autocatalysis in reaction networks.
Bulletin of mathematical biology, 76(10):2570–2595, 2014.

[10]
Martin Feinberg.
Necessary and sufficient conditions for detailed balancing in mass
action systems of arbitrary complexity.
Chemical Engineering Science, 44(9):1819–1827, 1989.

[11]
Alex Blokhuis, David Lacoste, and Philippe Nghe.
Universal motifs and the diversity of autocatalytic systems.
Proceedings of the National Academy of Sciences,
117(41):25230–25236, 2020.

[12]
Fritz Horn and Roy Jackson.
General mass action kinetics.
Archive for rational mechanics and analysis, 47(2):81–116,

[13]
Martin Feinberg.
Foundations of chemical reaction network theory.

[14]
Polly Y Yu and Gheorghe Craciun.
Mathematical analysis of chemical reaction systems.
Israel Journal of Chemistry, 58(6-7):733–741, 2018.

[15]
Wilhelm Ostwald.
Über autokatalyse.
Berichte über die Verhandlungen der
Königlich-Sächsischen Gesellschaft der Wissenschaften zu Leipzig,
Mathematisch-Physische Klasse, 42:189–191, 1890.

[16]
Zhen Peng, Klaus Paschek, and Joana C Xavier.
What wilhelm ostwald meant by “autokatalyse” and its significance
to origins-of-life research: Facilitating the search for chemical pathways
underlying abiogenesis by reviving ostwald's thought that reactants may also
be autocatalysts.
BioEssays, 44(9):2200098, 2022.

[17]
Jakob L. Andersen, Christoph Flamm, Daniel Merkle, and Peter F. Stadler.
Chemical transformation motifs—modelling pathways as integer
IEEE/ACM Transactions on Computational Biology and
Bioinformatics, 16(2):510–523, 2019.

[18]
Eric Smith and Supriya Krishnamurthy.
Flows, scaling, and the control of moment hierarchies for stochastic
chemical reaction networks.
Physical Review E, 96(6):062102, 2017.

[19]
Praful Gagrani and Eric Smith.
Action functional gradient descent algorithm for estimating escape
paths in stochastic chemical reaction networks.
Physical Review E, 107(3):034305, 2023.

[20]
Gheorghe Craciun.
Polynomial dynamical systems, reaction networks, and toric
differential inclusions.
SIAM Journal on Applied Algebra and Geometry, 3(1):87–106,

[21]
GAM King.
Chemical Society Reviews, 7(2):297–316, 1978.

[22]
Charles R. Johnson, Ronald L. Smith, and Michael J. Tsatsomeros.
Matrix Positivity.
Cambridge Tract s in Mathematics. Cambridge University Press, 2020.

[23]
Manoj Gopalkrishnan.
Catalysis in reaction networks.
Bulletin of mathematical biology, 73(12):2962–2982, 2011.

[24]
Uri Barenholz, Dan Davidi, Ed Reznik, Yinon Bar-On, Niv Antonovsky, Elad Noor,
and Ron Milo.
Design principles of autocatalytic cycles constrain enzyme kinetics
and force low substrate saturation at flux branch points.
Elife, 6:e20667, 2017.

[25]
Zhen Peng, Jeff Linderoth, and David A Baum.
The hierarchical organization of autocatalytic reaction networks and
its relevance to the origin of life.
PLOS Computational Biology, 18(9):e1010498, 2022.

[26]
Francesco Avanzini, Nahuel Freitas, and Massimiliano Esposito.
Circuit theory for chemical reaction networks.
arXiv preprint arXiv:2210.08035, 2022.

[27]
Personal conversations with Philippe Nghe.

[28]
Frank P Kelly.
Reversibility and stochastic networks.
Cambridge University Press, 2011.

[29]
Yu Liu and David JT Sumpter.
Mathematical modeling reveals spontaneous emergence of
self-replication in chemical reaction systems.
Journal of Biological Chemistry, 293(49):18854–18863, 2018.

[30]
Martin Feinberg.
Chemical reaction network structure and the stability of complex
isothermal reactors—ii. multiple steady states for networks of deficiency
Chemical Engineering Science, 43(1):1–25, 1988.

[31]
Michael R Garey and David S Johnson.
Computers and intractability, volume 174.
Freeman San Francisco, 1979.

[32]
Wolfgang Banzhaf and Lidia Yamamoto.
Artificial chemistries.
MIT Press, 2015.

[33]
Devlin Moyer, Alan R Pacheco, David B Bernstein, and Daniel Segrè.
Stoichiometric modeling of artificial string chemistries.
bioRxiv, 2020.

[34]
Jakob L Andersen, Christoph Flamm, Daniel Merkle, and Peter F Stadler.
A software package for chemically inspired graph transformation.
In Graph Transformation: 9th International Conference, ICGT
2016, in Memory of Hartmut Ehrig, Held as Part of STAF 2016, Vienna, Austria,
July 5-6, 2016, Proceedings 9, pages 73–88. Springer, 2016.

[35]
David F Anderson and Tung D Nguyen.
Prevalence of deficiency-zero reaction networks in an
erdös–rényi framework.
Journal of Applied Probability, 59(2):384–398, 2022.

[36]
Angel Garcia-Chung, Marisol Bermúdez-Montaña, Peter F Stadler,
Jürgen Jost, and Guillermo Restrepo.
Chemically inspired erd$\backslash$h $\{$o$\}$ sr$\backslash$'enyi
oriented hypergraphs.
arXiv preprint arXiv:2309.06351, 2023.

[37]
David A Baum, Zhen Peng, Emily Dolson, Eric Smith, Alex M Plum, and Praful
The ecology–evolution continuum and the origin of life.
Journal of the Royal Society Interface, 20(208):20230346, 2023.

[38]
Victor Blanco and Praful Gagrani.
Python Codes for the Generation of Minimal Autocatalytic
Subnetworks, 2 2023.
Available at

[39]
AA Balandin.
Structural algebra in chemistry.
Acta Physicochim, 1940.

[40]
Allen Hatcher.
Algebraic topology.

[41]
Mark I Dykman, Eugenia Mori, John Ross, and PM Hunt.
Large fluctuations and optimal paths in chemical kinetics.
The Journal of chemical physics, 100(8):5735–5750, 1994.

[42]
Christos H Papadimitriou.
On the complexity of integer programming.
Journal of the ACM (JACM), 28(4):765–768, 1981.

[43]
Hamdy A Taha.
Integer programming: theory, applications, and computations.
Academic Press, 2014.

[44]
Jakob L Andersen, Christoph Flamm, Daniel Merkle, and Peter F Stadler.
Maximizing output and recognizing autocatalysis in chemical reaction
networks is np-complete.
Journal of Systems Chemistry, 3(1):1–9, 2012.

4c$\#$ constituents

Type     1     2     3     4

I     100     100     100     100

II     100     100     100     100

The number of smallest minimal autocatalytic subnetworks for cluster species of maximum length $5$ classified by Blokhuis types.

4c$\#$ constituents

$\#$ reactions     1     2     3     4

2     9     30

3     11     51

4     0     1

5     0     0

$\#$ partitions

The number of MASs for cluster species of maximum length $5$.

Since the $\ell_3$ and $m_1$ are uniquely fixed by $i$ and $j$,
$\{\overline{i} + \overline{k_1} \to \overline{j} + \overline{k_2}, \overline{j} + \overline{k_3} \to 2\overline{i}\}$ or $\{\overline{i} + \overline{\ell_1} \to 2\overline{j},  \overline{j} + \overline{\ell_2} \to 2\overline{i}\}$, for some species $\overline{k_1}, \overline{k_2}, \overline{k_3}, \overline{\ell_1}$, and  $\overline{\ell_2}$ different from $\overline{i}$ and $\overline{j}$.
In the first set of reactions, $k_1$ and $k_2$ are fixed by the conservation law to be $k_1 = 2j - i$ and $k_2=2i - j$. The set of equations is only feasible if $i=2j$ (because $i>j$), being then $k_1=0$ and $k_2=2i-j=3\, j$.
By the conservation laws, species $\overline{k_3}, \overline{\ell_1}$ and $\overline{\ell_2}$ are uniquely determined by $\overline{i}$ and $\overline{j}$ ($\overline{k_3}=\overline{\ell_2} = \overline{2i - j}$ and $\overline{\ell_1}=\overline{2j - i}$). The first set of reactions, once $\overline{i}$ and $\overline{j}$ are fixed, and either $\overline{k_1}$ or $\overline{k_2}$, the rest of the species and reactions involved in the MASs are  uniquely determined by the conservation law. On the other hand, the second set of reactions is only feasible if $i=2j$ (because $i>j$), being then $\ell_1=0$ and $\ell_2=2i-j=3\, j$, being all the species involved in the partition uniquely determined by $i$ and $j$. Thus, there is no other MASs with the same partition as $A$.