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An Erdős-Inspired Perspective on Abiogenesis

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29 July 2026

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30 July 2026

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Abstract
Whether life is an inevitable consequence of the laws of nature or an exceptional outcome of prebiotic evolution is unresolved. Origin-of-life hypotheses primarily seek to reconstruct molecular pathways linking prebiotic chemistry to the first evolving systems. Here, we introduce a theoretical framework inspired by the probabilistic reasoning of Paul Erdős, reformulating abiogenesis as a mathematical existence problem rather than a historical reconstruction problem. We define an abstract chemical configuration space comprising all chemically accessible prebiotic organizations and identify life as the subset of configurations simultaneously exhibiting compartmentalization, energy transduction, information persistence and heritable variation. Using probability theory and set theory, we aim to investigate whether this subset necessarily occupies a nonzero region of chemical configuration space. We establish the general conditions under which life-capable organizations have positive measure and derive an abiogenetic threshold beyond which the probability of life's emergence approaches unity. Our framework predicts threshold behavior, functional convergence and scaling relationships that are independent of any particular molecular substrate. In our setting, origin-of-life hypotheses like RNA-first, metabolism-first, compartment-based and autocatalytic models are interpreted as complementary mechanisms that either enlarge the life-capable subset or increase the exploration of chemical configuration space. Potential applications include quantitative analyses of prebiotic experimental systems, comparative assessment of planetary habitability and computational studies of chemical organization.
Keywords: 
combinatorics
;  
configuration
;  
probability
;  
threshold
;  
organization
Subject: 
Biology and Life Sciences  -   Biophysics

Introduction

Hypotheses concerning the origin of life, including RNA-first, metabolism-first, lipid-world and hybrid scenarios, differ in their proposed mechanisms and the importance assigned to information storage, catalysis, compartmentalization and energy transduction. (Wachowius, Attwater, and Holliger 2017; Lopez-Fontal et al. 2018; Caliari, Xu, and Yomo 2021; Martin et al. 2022; Pavlinova et al. 2023; Peng, Adam, Fahrenbach, and Kaçar 2023) Although these approaches reconstruct a historical pathway leading from prebiotic chemistry to the first living system, determining the precise sequence of molecular events that occurred more than four billion years ago may ultimately prove impossible. Even if one successful pathway were identified, it would not establish whether it was unique or simply one realization among many chemically equivalent possibilities. Consequently, a general theoretical question is unresolved: was the emergence of life an exceptionally improbable accident or is it an inevitable consequence of sufficiently rich chemical organization?
To tackle the question, we address abiogenesis from a conceptual perspective inspired by the probabilistic method introduced by Paul Erdős in combinatorics (Erdős and Spencer 1974; Shirali 1998; Girard 2012; Csóka, Hubai, and Lovász 2023). Rather than attempting to reconstruct one particular sequence of reactions, we investigate whether the existence of at least one life-capable chemical organization can be established without specifying its molecular composition or historical trajectory. In mathematics, the probabilistic method establishes the existence of an object by showing that a randomly selected element from an appropriately defined configuration space has a nonzero probability of possessing the required properties. The proof is nonconstructive, since it demonstrates existence without identifying the object itself. Thus, abiogenesis is reformulated as an existence problem defined over an ensemble of possible chemical organizations.
Instead of focusing on peculiar molecules or reaction networks, we consider the whole ensemble of chemically accessible prebiotic organizations allowed by physical and chemical laws. Individual molecular systems become specific realizations within a much larger ensemble rather than privileged historical solutions, independently of any specific molecular pathway or environmental history. From this perspective, RNA molecules, peptide networks, lipid assemblies and other prebiotic systems (Milner-White 2019; Frenkel-Pinter et al. 2020; Steinkühler et al. 2020; Ozturk et al. 2023; Mastreux et al. 2024) stand for alternative molecular realizations of equivalent organizational principles rather than mutually exclusive origins of life. The main question becomes whether the ensemble necessarily contains at least one functional organization simultaneously capable of compartmentalization, energy transduction, information persistence and heritable variation. If randomly sampled organizations have a nonzero probability of satisfying the minimal functional requirements for life, sufficiently extensive exploration of chemically accessible organizations will eventually encounter at least one life-capable system. The challenge becomes determining whether these organizations occupy a nonzero region of chemical configuration space and, if so, under what conditions their appearance becomes statistically expected.
We will proceed as follows. First, we introduce a chemical configuration space as the mathematical domain of chemically accessible prebiotic organizations and develop a probabilistic framework for analyzing the existence of life-capable organizations within that space. Then, we derive the conditions under which these organizations occupy a nonzero region of chemical configuration space and identify the circumstances in which their appearance is statistically expected. Finally, we derive experimentally testable predictions concerning critical thresholds in chemical diversity to distinguish environments in which life is statistically unlikely from those in which its emergence becomes an expected consequence of chemical organization.

Chemical Configuration Space

The Erdős probabilistic existence problem requires a mathematical domain over which probability is defined (Kuperberg, Lovett, and Peled 2012; Goldschmidt 2016; Lovett, Karingula, and Vardy 2020). We introduce an abstract chemical configuration space, denoted by Ω , whose elements stand for all the chemically accessible prebiotic organizations compatible with the physical and chemical constraints of a given planetary environment. Each element of Ω is not merely a molecular composition, but an organizational state comprising molecular identities, concentrations, reaction topology, spatial compartmentalization, energetic exchanges and temporal interactions. Therefore, individual molecules are treated as components of higher-order organizations rather than as the primary objects of analysis.
The purpose of introducing Ω is not to enumerate every possible molecular arrangement, but to define a mathematical sample space analogous to those routinely employed in probability theory and statistical mechanics. In thermodynamics, microscopic configurations constitute phase space, while in random graph theory all graphs satisfying specified constraints form the ensemble from which random realizations are sampled. Similarly, Ω represents the ensemble of chemically feasible organizations that may arise under a given set of environmental conditions.
Membership in Ω is restricted by chemical accessibility. Only organizations consistent with the laws of physics and chemistry under the environmental conditions considered are included. Configurations requiring chemically impossible reactions, violations of conservation laws or unattainable energetic states are excluded. Consequently, different planetary environments generate different configuration spaces because they impose different physicochemical constraints, although the mathematical formulation is unchanged.
Within Ω , we define the subset L ⊆ Ω ,
where L contains all organizations capable of initiating Darwinian evolution. Membership in L is determined by functional organization rather than molecular composition. An organization belongs to L if it simultaneously exhibits four minimal functional capabilities:compartmentalization ( C ), sustained energy transduction ( E ), information persistence ( I ) and heritable variation ( H ).
These properties are intentionally formulated in functional rather than molecular terms. They do not require RNA, proteins, lipids or any other specific biochemical substrate. Different chemistries may satisfy the same organizational requirements, allowing chemically distinct systems to belong to the same subset. Symbolically, L = { x ∈ Ω :   C ( x ) ∧ E ( x ) ∧ I ( x ) ∧ H ( x ) } .
Life is represented not by a unique molecular object, but by a region of chemical configuration space characterized by a common functional architecture.
Having defined L , probability can be introduced naturally. Let P : Ω → [ 0,1 ]
denote the probability measure associated with randomly sampling a chemically accessible organization from Ω . The probability that a sampled organization belongs to the life-capable subset is P ( L ) ,
or, for continuous configuration spaces, P ( L ) = ∫ L d P .
This quantity is an intrinsic property of chemical configuration space. It measures the fraction of chemically accessible organizations satisfying the minimal functional criteria for life, independently of any particular planetary history or molecular realization.
This mathematical formulation leads to two questions. The first concerns existence: does the subset L contain at least one element? The second concerns exploration: under what conditions is a planetary environment expected to encounter one of its elements? The remainder of our work addresses these questions sequentially. First, we establish conditions under which L has nonzero measure within Ω , then investigate how the exploration of chemical configuration space determines the probability of abiogenesis.

Probabilistic Existence Framework

Having defined the chemical configuration space Ω , the life-capable subset L and the associated probability measure, we can now formulate the probabilistic existence problem. Let P ( L )
denote the probability that a randomly sampled chemically accessible organization belongs to L . This quantity measures the abundance of life-capable organizations within chemical configuration space.
The first objective is to determine whether L is empty. If P ( L ) > 0 ,
then the subset necessarily contains at least one element, P ( L ) > 0     ⟹     L ≠ ∅ .
Conversely, if no life-capable organization exists, L = ∅ ,
then P ( L ) = 0 .
Therefore, the existence problem may be expressed equivalently in terms of either the non-emptiness of L or the positivity of its probability measure.
However, existence alone does not imply that life will arise in a particular planetary environment. Even when P ( L ) is positive, the appearance of a life-capable organization depends on the extent to which chemical configuration space is explored. Therefore, we introduce the parameter N ,
representing the effective number of statistically independent chemical organizations generated during the prebiotic evolution of a planetary environment. The parameter N does not simply measure elapsed time or the total number of chemical reactions, rather it quantifies the number of distinct organizational states sampled under nonequilibrium conditions. Consequently, environments differing in chemical diversity, energy flux, compartment formation or long-term physicochemical stability may exhibit substantially different values of N .
The expected number of life-capable organizations encountered during this exploration is E = N   P ( L ) .
This simple expression separates two independent contributions to abiogenesis. The probability P ( L ) characterizes the structure of chemical configuration space, whereas N
characterizes the efficiency with which a planetary environment explores that space. This means that abiogenesis depends on both the abundance of life-capable organizations and the extent of chemical exploration.
Three qualitative regimes immediately follow.
When N P ( L ) ≪ 1 ,
the expected number of life-capable organizations is much smaller than one, making abiogenesis unlikely.
When N P ( L ) ≈ 1 ,
the system enters a transition region in which stochastic effects dominate and the appearance of a life-capable organization becomes plausible but not guaranteed.
Finally, when N P ( L ) ≫ 1 ,
multiple life-capable organizations are expected to occur, indicating that abiogenesis becomes a statistically robust consequence of extensive exploration of chemical configuration space.
The next mathematical problem is reduced to two questions. First, under what physical and chemical conditions does the subset L dispay positive measure? Second, once P ( L ) > 0 , how large must N become for the probability of encountering at least one element of L to approach unity? These questions are addressed in the following sections.

Conditions for Positive Measure

The probabilistic framework developed above shows that the existence of life-capable organizations is equivalent to the positivity of the measure associated with the subset L . The remaining question is whether P ( L ) > 0 .
Our objective is not to estimate the numerical value of this probability, which depends on chemical incompletely understood details, but to identify general conditions under which it cannot vanish.
Establishing this result requires only three assumptions. The first is chemical accessibility. At least one organization satisfying the defining properties of L must be attainable through ordinary physicochemical processes operating under the environmental conditions considered. Organizations requiring violations of conservation laws, chemically forbidden reactions or unattainable energetic states are excluded from the configuration space by definition. Consequently, if no chemically accessible organization satisfies the minimal functional requirements for life, then L = ∅
and therefore P ( L ) = 0 .
Thus, chemical accessibility is a condition required for positive measure.
The second assumption is organizational robustness. Life-capable organizations should not lose their defining properties under arbitrarily small perturbations of molecular composition, concentrations or reaction rates. Catalytic closure, energy transduction, information persistence and compartmental integrity are expected to tolerate limited fluctuations without immediate loss of functionality. Mathematically, this implies that a life-capable organization does not correspond to an isolated point in configuration space but belongs to a neighborhood of nearby organizations sharing the same functional properties. Since these neighborhoods have finite measure, whenever life is organizationally robust, the subset L occupies a finite region of chemical configuration space rather than a measure-zero set.
The third assumption is functional redundancy. Different molecular realizations may implement the same organizational functions. Chemically distinct catalysts may sustain equivalent reaction networks, different polymers may preserve heritable information and multiple amphiphilic molecules may generate stable compartments. Functional equivalence therefore increases the number of organizations belonging to L without changing its defining properties. Instead of a single realization, one achieves multiple independent realizations distributed throughout configuration space, increasing the measure occupied by the life-capable subset.
These assumptions lead naturally to the following result.
Positive-Measure Principle. If (i) at least one chemically accessible organization satisfies the defining properties of L  , (ii) these properties are preserved under sufficiently small perturbations and (iii) multiple chemically distinct realizations of the same functional organization are possible, then P ( L ) > 0 .
This result does not require RNA, proteins, lipids or any specific molecular substrate, but follows from general properties of chemically accessible, robust and functionally redundant nonequilibrium organizations. Consequently, its validity is compatible with a wide range of origin-of-life hypotheses and potentially with chemistries different from those of early Earth.
Combining this principle with the probabilistic existence framework yields P ( L ) > 0 ⟹ L ≠ ∅ .
Under these assumptions, chemical configuration space necessarily contains at least one organization capable of initiating Darwinian evolution. The remaining problem is no longer whether this organization exists, but under what conditions planetary environments explore chemical configuration space sufficiently extensively to encounter them with high probability. This question is addressed in the next section.

Thresholds for Abiogenesis

The positivity of P ( L ) establishes that life-capable organizations exist within chemical configuration space. The remaining question is whether a planetary environment is expected to encounter one of them during its prebiotic evolution. This depends on the extent to which the configuration space is explored.
Let
N
denote the effective number of statistically independent chemical organizations generated during the history of a planetary environment. Planetary environments differing in chemical diversity, environmental heterogeneity, energy availability or compartment formation may exhibit substantially different values of N .
If each explored organization belongs to the life-capable subset L  
with probability P ( L ) , the probability that none of the N organizations satisfies the criteria for life is ( 1 − P ( L ) ) N .
Consequently, the probability that at least one life-capable organization is encountered is P l i f e = 1 − ( 1 − P ( L ) ) N .
This expression links the structure of chemical configuration space with the dynamics of its exploration. Whereas P ( L ) measures the abundance of life-capable organizations, N measures the effectiveness with which a planetary environment samples that space.
Several limiting regimes immediately follow. When N P ( L ) ≪ 1 ,
The probability of abiogenesis is approximately P l i f e ≈ N P ( L ) ,
indicating that life remains an uncommon event because only a small fraction of chemical configuration space has been effectively explored.
When N P ( L ) ≈ 1 ,
The system enters a transition region in which relatively small increases in either P ( L ) or N produce disproportionately large increases in the probability of abiogenesis. Planetary environments with apparently similar physicochemical characteristics may exhibit markedly different biological outcomes if they lie on opposite sides of this transition.
Finally, when N P ( L ) ≫ 1 ,
the probability approaches unity, P l i f e → 1 ,
implying that the emergence of at least one life-capable organization is statistically expected. Note that this conclusion does not require individual organizations to be highly probable: even extremely small values of P ( L ) can be compensated by sufficiently extensive exploration of chemical configuration space.
These observations motivate the following result.
Abiogenetic Threshold Theorem. Let  L ⊆ Ω  be a life-capable subset with positive measure  P ( L ) > 0  and let  N  denote the number of statistically independent organizations explored by a planetary environment. Then, the probability of encountering at least one life-capable organization is
P l i f e = 1 − ( 1 − P ( L ) ) N , which converges to unity as
N → ∞ . Consequently, abiogenesis becomes statistically expected whenever N P ( L ) ≫ 1 .
Our theorem has several consequences. First, it predicts that abiogenesis is governed by a threshold rather than by a gradual accumulation of complexity. Once the product N P ( L ) exceeds a critical value, the probability of life's emergence rises rapidly toward unity. Similar threshold phenomena are characteristic of percolation theory, random graph theory and statistical mechanics, where macroscopic organization appears abruptly after a control parameter crosses a critical value.
Second, the theorem reconciles apparently conflicting views concerning the rarity of life. Life-capable organizations may be individually extremely uncommon, while the emergence of life remains statistically inevitable in environments that explore sufficiently large regions of chemical configuration space. Conversely, the same organizations may never be encountered if chemical exploration is too limited. Thus, rarity and inevitability are not contradictory but jointly depend on the values of P ( L ) and N .
Finally, the theorem identifies two complementary determinants of abiogenesis. Increasing the abundance of life-capable organizations enlarges P ( L ) , whereas increasing the diversity of explored organizations enlarges N . Any physical or chemical process increasing either quantity moves a planetary environment closer to the abiogenetic threshold. The following section examines the experimental consequences of this prediction.

Experimental Predictions

Our probabilistic framework predicts general statistical regularities governing abiogenesis rather than the appearance of specific molecular systems. Consequently, experimental validation should focus on organizational behavior across large ensembles of chemically accessible configurations rather than on reproducing a unique prebiotic pathway.
The first prediction concerns chemical diversity. Increasing the diversity of molecular species expands the number of accessible chemical organizations, increasing the effective exploration of configuration space. This increase is not expected to remain proportional indefinitely. Once most life-capable organizations become accessible, additional molecular diversity should produce progressively smaller gains. Laboratory systems based on combinatorial chemistry, synthetic protocells or high-throughput reaction networks should therefore exhibit a sigmoidal rather than linear relationship between chemical diversity and the probability of generating life-like organization.
A second prediction concerns environmental cycling. Hydration-dehydration cycles, thermal oscillations, freezing-thawing events and redox fluctuations continually reorganize chemical systems, generating new organizational states without necessarily changing their molecular inventory. Therefore, the main effect is expected to be an increase in the number of independently explored organizations. Consequently, abiogenesis should depend more strongly on the cumulative number of organizational reorganizations than on elapsed time alone. Two experiments of equal duration but different numbers of environmental cycles should display different probabilities of producing life-capable organizations.
Our framework predicts also functional convergence. Because membership in the subset L is determined by functional organization rather than molecular composition, chemically distinct experimental systems should repeatedly converge toward equivalent organizational architectures despite differing molecular inventories. Independent experiments are expected to reproduce similar patterns of catalytic closure, compartment stability, information persistence and heritable variation more consistently than identical molecular compositions.
Another prediction concerns the existence of critical transitions. As the product N P ( L ) approaches the abiogenetic threshold, small increases in chemical diversity, environmental heterogeneity or organizational complexity should produce disproportionately large increases in the probability of observing life-like behavior. Systems operating near this threshold should exhibit substantial variability among replicate experiments, whereas experiments performed well above the threshold should produce life-capable organizations reproducibly. This behavior would distinguish threshold-driven abiogenesis from models predicting approximately linear increases in complexity.
Still, we predict that planetary habitability depends not only on chemical composition but also on the capacity of an environment to explore chemical configuration space. Planets or moons characterized by numerous transient microenvironments, sustained nonequilibrium energy sources and persistent physicochemical cycling should possess larger effective values of N than chemically similar but environmentally static systems. This suggests that assessments of habitability should complement traditional geochemical criteria with measures of organizational exploration.
Finally, we predict molecular degeneracy during the earliest stages of abiogenesis. If life is an organizational phenomenon, multiple chemically distinct systems should satisfy the same functional criteria defining membership in L . Experimental evolution initiated from different prebiotic conditions should therefore generate several independent classes of life-like organizations rather than converging toward a single molecular architecture.
Taken together, our predictions provide a means of testing the framework without reconstructing Earth’s specific prebiotic history. Instead, validation depends on demonstrating the predicted scaling relationships, threshold behavior, functional convergence and molecular degeneracy across chemically diverse experimental systems.

Discussion and Conclusions

Most origin-of-life hypotheses seek to identify molecular mechanisms capable of generating the organizational properties associated with life. We address a different question: rather than asking how a particular chemical pathway operates, we mathematically investigate the general conditions under which at least one life-capable organization is expected to exist and be encountered within the space of chemically accessible organizations. This distinction parallels the difference between constructive and nonconstructive reasoning in Erdős’ combinatorics. Existing origin-of-life theories primarily investigate how life could emerge through specific molecular mechanisms (Adam et al. 2017; Branscomb and Russell 2018; Caliari, Xu, and Yomo 2021; Michaelian 2021; Freeland 2022; Hernández and Michaelian 2022; Pavlinova et al. 2023; Lingam, Nichols, and Balbi 2024; Kipping 2025), whereas we address the more general question of whether sufficiently rich chemical organization necessarily contain systems capable of initiating Darwinian evolution. Our probabilistic framework is independent of the chemistry through which terrestrial life emerged and is compatible with multiple molecular scenarios. Therefore, the appearance of life on Earth represents one historical realization within a much broader space of chemically possible organizations
Limitations should be recognized. First, our framework is qualitative with respect to the probability measure P ( L ) . not estimating the numerical value of this probability for any planetary environment. Second, our Positive-Measure Principle relies on the assumptions of chemical accessibility organizational robustness and functional redundancy. While these assumptions are well supported by contemporary biological systems, their applicability to the earliest stages of prebiotic evolution remains an empirical question. Third, the exploration parameter N represents an effective number of statistically independent organizations and neglects correlations arising from shared environments, diffusion, material exchange and common energetic constraints. Finally, we leave open the precise definition of the minimal functional organization required for Darwinian evolution.
Our mathematical formulation suggests paths for further development. Chemical configuration space could be investigated using graph theory, stochastic processes, information geometry or algebraic topology to characterize its global organization and the structure of life-capable regions. Statistical mechanics may provide tools for describing the exploration of nonequilibrium chemical ensembles, whereas random graph theory and percolation theory may clarify the origin of the predicted threshold behavior. A promising extension would be to combine the present probabilistic framework with autocatalytic networks, allowing catalytic closure to be interpreted quantitatively as an expansion of the life-capable subset within chemical configuration space.
Beyond abiogenesis, the same mathematical principles may apply whenever complex organization emerges through the exploration of large configuration spaces. Ecological community assembly, neural development, technological innovation and cultural evolution all involve extensive ensembles of possible organizations from which only a subset is realized. Distinguishing between the existence of admissible organizations and the mechanisms by which they are discovered may provide a framework for assessing the emergence of complexity across diverse natural systems.
In conclusion, we proposed an Erdős-inspired probabilistic framework to reformulate abiogenesis as an existence problem defined over chemical configuration space. By introducing the concepts of a life-capable subset, positive measure and statistical exploration, we distinguish the existence of life-capable organizations from the historical emergence of the first living system on Earth. Rather than identifying a single molecular pathway, our framework identifies the statistical conditions under which life becomes an expected outcome of sufficiently extensive exploration of the chemically accessible organizations

Declarations

Funding

This research did not receive any specific grant from funding agencies in the public, commercial or not-for-profit sectors.

Authors' Contributions

The Author performed: study concept and design, acquisition of data, analysis and interpretation of data, drafting of the manuscript, critical revision of the manuscript for important intellectual content, statistical analysis, obtained funding, administrative, technical and material support, study supervision.

Availability of Data and Materials

All data and materials generated or analyzed during this study are included in the manuscript. The Author had full access to all the data in the study and took responsibility for the integrity of the data and the accuracy of the data analysis.

Disclaimer/Publisher’s Note

The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.Declaration of generative AI and AI-assisted technologies in the writing process: During the preparation of this work, the author used ChatGPT 5.3 to assist with data analysis and manuscript drafting and to improve spelling, grammar and general editing. After using this tool, the author reviewed and edited the content as needed, taking full responsibility for the content of the publication.

Acknowledgments

none.

Competing Interests

The Author does not have any known or potential conflict of interest including any financial, personal or other relationships with other people or organizations within three years of beginning the submitted work that could inappropriately influence or be perceived to influence their work.

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