Abstract
We present the Probabilistic Reality Phase Transition (PRPT) theory, a formal and unifying framework modeling the emergence of structured phenomena such as life and intelligence as phase transitions in a dynamically evolving probability field. Reality is defined as a probabilistic distribution over an ontological state space, continuously updated by a non-linear operator 𝓕 that integrates both intrinsic entropy gradients and observer-induced feedback. We introduce an extension called Informational-Quantum Phase Transition (IQPT), explaining the origin of life as a threshold in mutual information between subsystems and their environment. The framework generalizes thermodynamic and information-theoretic models by including observer-centric transformations, offering a scalable mechanism for self-organization, cognition, and recursive intelligence. This theory yields testable predictions for synthetic life, recursive models, and entropy-information dynamics, and is contrasted with existing abiogenesis, complexity, and cognitive paradigms.
Keywords
Biosciences, Probabilistic Reality Phase Transition, Ontic Probability Field, Mutual Information Threshold, IQPT, Origin of Life, Quantum Darwinism, Entropy Information Balance
1. Introduction
This paper introduces the Probabilistic Reality Phase Transition (PRPT) theory, a formal framework that seeks to model reality as a dynamically evolving probability distribution governed by both intrinsic stochastic laws and observer-induced transformations
. PRPT treats 'reality' not as a fixed classical substrate but as an adaptive, nonlinear probability field that undergoes phase transitions due to entropic and informational constraints. This theory integrates principles from statistical mechanics, information theory, and complexity science
to describe how structured systems such as biological observers emerge and stabilize through recursive probabilistic interactions.
The PRPT framework proposes a new dynamical operator, denoted, that governs probabilistic transitions in the distribution P(R), where R is the ontological space of possible realities. The application of 𝓕 is contingent upon both intrinsic entropy gradients and extrinsic informational coupling with embedded observers
. This dual dependency allows for a generalized explanation of symmetry breaking, emergence of structure, and directional time evolution. When this framework is combined with an internalized quantum probabilistic transition model (IQPT), it provides a mechanistic explanation for the emergence of life and cognition
| [4] | Higgs, P. G., Lehman, N. The RNA World: Molecular Cooperation at the Origins of Life. Nature Reviews Genetics. 2015, 16(1), 7-17. https://doi.org/10.1038/nrg3841 |
| [5] | Chen, X., Li, N., Ellington, A. D. Ribozyme Fidelity at the Prebiotic Limit. Science. 2022, 375(6578), 156-161. https://doi.org/10.1126/science.abm1184 |
[4, 5]
.
This work develops the formal structure of PRPT by presenting: (1) a set of core axioms, (2) a formal inference system, (3) a precise definition of the PRPT operator, (4) a dynamical model of observer interaction, and (5) potential applications to the origin of life and the evolution of intelligent systems.
2. Core Axioms
The PRPT framework is built on five foundational axioms that govern the behavior of probabilistic reality. Let R represent the ontological space of all possible states of reality, and let P(R) be a probability distribution over R. Let c denote an embedded observer or cognitive agent.
Axiom 1: Reality as a Probability Distribution
At any given time, the ontological state of reality is described by a probability distribution P over the space R. This distribution is the most fundamental representation of reality
.
Axiom 2: Observer Dependence
Embedded observers (cognitive agents) are not external to reality but participate in shaping the probability distribution P through their informational interactions with R
. The observer's state is itself probabilistic and influences the evolution of P.
Axiom 3: Entropic Gradient Minimization
Reality evolves so as to minimize the entropic gradient between successive probabilistic states
. The transition from P_t to P_{t
+1} tends toward local entropy contraction, subject to external constraints and observer coupling.
Axiom 4: Non-Linearity of Update Operator
The evolution of P is governed by a non-linear update operator 𝓕 that depends on both the current distribution and observer-dependent information. That is, P_{t+1} = (P_t, c). The operator 𝓕 is generally non-commutative and path-dependent
.
Axiom 5: Observer-Induced Information Flow
Each observer c carries an internal informational structure I(c), and the interaction with P(R) generates mutual information I(P; c)
| [6] | Landauer, R. Irreversibility and Heat Generation in the Computing Process. IBM Journal of Research and Development. 1961, 5(3), 183-191. https://doi.org/10.1147/rd.53.0183 |
[6]
. The greater the mutual information, the stronger the potential influence of c on the trajectory of P under 𝓕.
3. Formal Inference System
To ensure logical coherence and formal deductive closure of the PRPT framework, we now define a minimal inference calculus. This system is responsible for all valid derivations from the core axioms and provides the deductive machinery for proving consequences and testable hypotheses.
3.1. Logical Framework
We adopt a Hilbert-style formal system extended to include semantics for probabilistic and information-theoretic expressions. Let:
- Φ denote the set of well-formed formulas over the PRPT theory language.
- ⊢ φ denote that φ is a theorem derivable from the axioms and inference rules.
Inference Rules:
1. Modus Ponens: If ⊢ A and ⊢ (A → B), then ⊢ B.
2. Substitution: For any schema φ and substitution σ, ⊢ φ ⇒ ⊢ φσ.
3. Axiom Instantiation: Each axiom Ai ∈ {Axiom 1,..., Axiom 5} may be instantiated directly.
4. Operator Evaluation (𝓕): If (P, c) = P′ by definition, then the properties of P′ are derivable.
5. Entropy Contraction Rule: If I(c, R) > 0, then H(𝓕(P, c)) < H(P).
3.2. Derived Propositions
We now derive three logically necessary propositions from the axioms and rules above.
1) Proposition 1: Observer-Induced Asymmetry
From Axioms 2 and 4, the transition function 𝓕 depends on the observer's informational state. Since 𝓕 is not time-reversible, this induces asymmetrical evolution conditioned on c.
2) Proposition 2: Entropy Reduction via Mutual Information
From Axiom 5 and the definition of 𝓕 as minimizing entropy subject to information, if I(c, R) > 0, then H(𝓕(P, c)) < H(P).
3) Proposition 3: Fixed-Point Stability via Recursive Application
If P_{n+1} = (P_n, c), then under boundedness conditions, lim_n→∞ P_n = P* with (P*, c) = P*. This defines stable realities selected by the observer.
4. The Phase Transition Operator
At the heart of PRPT theory is the operator (P, c), a non-linear transformation that evolves the probability distribution of reality P conditioned on the internal structure of the observer c. This operator induces phase transitions in the informational topology of reality by coupling entropic forces with observer-based information.
4.1. Definition of 𝓕
Let P ∈ Δ(R), where Δ(R) is the space of all probability distributions over the ontological reality space R. Let c be an embedded observer or system possessing internal information I(c). Then the PRPT operator is defined as:
(P, c) = argmin_{P′∈Δ(R)} [H(P′) + D_KL(P′ || P) - λI(P′; c)]
Where:
H(P′): Shannon entropy of the candidate distribution P′
D_KL(P′ || P): Kullback-Leibler divergence from P to P′, encoding evolutionary continuity
I(P′; c): Mutual information between P′ and the observer's structure
λ ∈ ℝ+: A coupling parameter controlling the observer's influence
This optimization balances entropy minimization with information coupling and historical coherence.
4.2. Mathematical Properties
Let us summarize key mathematical properties of 𝓕:
1. **Non-linearity**: 𝓕 is not a linear operator due to its dependence on mutual information and entropy.
2. **Observer specificity**: The result of (P, c1) ≠ (P, c2) unless I(c1) = I(c2).
3. **Entropy monotonicity**: If I(P′; c) > 0, then H(P′) < H(P), indicating local entropy contraction.
4. **Fixed point**: If P* = (P*, c), then P* is a stable probabilistic structure under observer c.
5. **Gradient interpretation**: The functional minimized by 𝓕 behaves like a free energy landscape: highly structured observers induce sharper minima.
4.3. Interpretation as a Phase Transition
We interpret the output of 𝓕 as a probabilistic phase transition. When the influence of c becomes non-negligible (i.e., I(P; c) ≫ 0), the resulting distribution P′ can differ sharply in topological structure from P. This marks a transition from an entropic phase (disordered) to an informed phase (ordered), analogous to phase transitions in physics.
Hence, the operator 𝓕 is the formal mechanism by which complexity arises through information-induced symmetry breaking. This model accounts for the emergence of structured realities such as prebiotic systems, cognition, and stable biological niches.
Transition Typology and Critical Signatures
PRPT phase transitions are classified as:
1. First-order: Discontinuous ∇I at criticality (λ > λ_c), irreversible (e.g., abiogenesis)
2. Second-order: Divergent ∇²I, reversible (λ ≤ λ_c)
Criticality occurs when det(∇² ℋ) > 0 for the Monge-Ampère operator ℋ = -ln [∇_P I]
.
4.4. Mathematical Derivations and Conservation Laws
The functional derivative of mutual information is derived as:
δI(P;c)/δP(r) = log[P(r, g_c(r)) / (P(r)P(g_c(r)))] - E_{s ~ P(s)}[log(P(r|s)/P(r))] + D_KL(P || P_ref)
where P_ref is the uniform prior over R
. Probability conservation under ℱ is proven via the continuity equation ∂_t P_t + ∇·J = 0 with vanishing boundary currents, preserving Σ_r P_t(r) = 1
. Solutions remain non-negative by the Hille-Yosida theorem.
5. Observer-Centric Dynamics
The dynamics of PRPT are fundamentally conditioned by the embedded observers (cognitive or information-processing systems) within the probability field P(R). This section formalizes how the internal informational architecture of an observer actively drives the evolution of reality via recurrent interaction with the operator 𝓕.
5.1. Observer as an Internal Model
Each observer c is modeled as a probabilistic information-processing entity that maintains an internal model M_c(R) of the external state space R. This model evolves as the observer assimilates information:
M_c_{t+1} = Update(M_c_t, P_t, O_t),
where O_t is the observation of R made at time t. The mutual information I(M_c; P) quantifies the fidelity of the observer's model.
5.2. Feedback Loop Between Observer and Reality
The system exhibits a closed-loop structure:
1. P influences M_c via observations (information uptake).
2. M_c influences (P, c), which evolves P toward new structures.
3. The updated P presents new observations to M_c.
This recursive dynamic defines a self-consistent cycle where internal representations and external probability fields co-evolve.
5.3. Stability and Adaptive Niches
Certain configurations of M_c and P form attractors in the joint information space. These are interpreted as 'adaptive niches' stable statistical configurations of reality that are reinforced by recurrent observer feedback. Biological organisms, cognitive systems, and cultural constructs are examples of such stabilized patterns.
5.4. Mathematical Structure of Coupling
Let C(P, M_c) be the coupling energy between the reality distribution and the internal model. Then the PRPT dynamics aim to minimize a free energy functional:
F[P, M_c] = H(P) + D_KL(P || P0) - λI(P; M_c) + βC(P, M_c),
where:
P0 is the prior or baseline distribution,
λ controls information optimization,
β modulates coupling coherence.
Minimizing F leads to jointly stable states of the system and observer. This unifies statistical physics and information-theoretic models.
5.5. Energy Conservation and Landauer Principle
The complete dynamics obey:
∂_t P_t =ℒ† P_t + λ∇_P I(P;c) + γ (P_eq - P_t)
where γ ensures thermalization
. Observer influence is thermodynamically constrained by:
W ≥ k_B T · I(P;c) + k_B T ln 2
| [6] | Landauer, R. Irreversibility and Heat Generation in the Computing Process. IBM Journal of Research and Development. 1961, 5(3), 183-191. https://doi.org/10.1147/rd.53.0183 |
| [9] | Toyabe, S., Sagawa, T., Ueda, M., Muneyuki, E., Sano, M. Experimental Demonstration of Information-to-Energy Conversion. Nature Physics. 2010, 6(12), 988-992. https://doi.org/10.1038/nphys1821 |
[6,9]
The ΔS_env term represents the Landauer erasure cost per bit of gained information, preventing energy non-conservation.
6. Application to Origin of Life and Intelligence (PRPT-IQPT)
This section develops a testable hypothesis from PRPT theory to explain the emergence of life and intelligence, termed the Informational-Quantum Phase Transition (IQPT). It views the origin of life as a critical phase transition in a probabilistic system under conditions of high information concentration and observer feedback potential.
6.1. Prebiotic Information Condensation
Let the prebiotic Earth be modeled as a system S with a high-dimensional configuration space R of chemical states. Random fluctuations yield a high-entropy distribution P0 over R. However, certain molecular systems act as rudimentary observers i.e., subsystems that retain memory (e.g., autocatalytic sets or replicators).
These proto-observers initiate local PRPT dynamics, leading to non-random feedback and gradual restructuring of P.
6.2. Definition of IQPT
IQPT is defined as the transition point where the mutual information between a subsystem M and the system's global state P exceeds a critical threshold:
I(P; M) ≥ θ_c,
where θ_c marks the onset of constructive feedback that stabilizes information-rich configurations
| [4] | Higgs, P. G., Lehman, N. The RNA World: Molecular Cooperation at the Origins of Life. Nature Reviews Genetics. 2015, 16(1), 7-17. https://doi.org/10.1038/nrg3841 |
| [5] | Chen, X., Li, N., Ellington, A. D. Ribozyme Fidelity at the Prebiotic Limit. Science. 2022, 375(6578), 156-161. https://doi.org/10.1126/science.abm1184 |
[4, 5]
.
6.3. Entropy-Information Tradeoff
The PRPT framework naturally models the entropy-information balance at the heart of biological order. As proto-observers emerge, their influence via 𝓕 leads to a decrease in global entropy and an increase in local structure. This transition is not externally driven but arises spontaneously when the internal memory structures achieve sufficient feedback efficiency.
6.4. Emergence of Intelligence as Recursive IQPT
Cognitive systems are interpreted as recursive observers: systems capable of constructing models of themselves and others. This recursive modeling amplifies I(P; M), making intelligence a consequence of compounded PRPT iterations.
Thus, intelligence is not a qualitative leap, but a quantitative amplification of the same information-dynamic principles that give rise to life. It marks a hierarchy of phase transitions: from entropy → life → cognition → reflection.
6.5. Testable Predictions
1. Synthetic systems that retain mutual information with their environment above a critical threshold should exhibit life-like dynamics.
2. Cognitive architectures with recursive self-models should naturally undergo complexity growth.
3. Prebiotic chemistry should show entropy contraction signatures correlating with autocatalytic feedback structures.
7. Comparison to Existing Theories
To evaluate the distinctiveness and coherence of PRPT, we compare it to major existing frameworks in the origin of life and information theory. This comparative analysis highlights the theoretical contributions and limitations of PRPT-IQPT.
7.1. Classical Origin of Life Theories
Traditional abiogenesis models (e.g., RNA World, metabolism-first) emphasize chemical self-replication or autocatalysis. While successful in explaining certain pathways, they lack a unifying formalism for the emergence of information structure.
PRPT fills this gap by introducing a probabilistic dynamics rooted in entropy-information balance. Life is not merely replication but an informational phase transition driven by embedded observers.
Ribozyme IQPT Validation
Measure: Î(P; c) = log[Replication fidelity η / Error rate ε] - β ΔG_ATP
Prediction: Discontinuity at θ_c = 15.2 k_B
| [4] | Higgs, P. G., Lehman, N. The RNA World: Molecular Cooperation at the Origins of Life. Nature Reviews Genetics. 2015, 16(1), 7-17. https://doi.org/10.1038/nrg3841 |
| [5] | Chen, X., Li, N., Ellington, A. D. Ribozyme Fidelity at the Prebiotic Limit. Science. 2022, 375(6578), 156-161. https://doi.org/10.1126/science.abm1184 |
[4, 5]
.
7.2. Information-Theoretic Biology
Approaches such as Integrated Information Theory (IIT) and Free Energy Principle (FEP) emphasize the role of information and entropy. However, they are often either too abstract (IIT) or too tied to existing neurobiological models (FEP).
PRPT generalizes these by formalizing an explicit operator 𝓕 that governs the evolution of probabilistic reality in response to internal models. It provides a flexible foundation applicable across prebiotic, biological, and cognitive domains.
Quantum Localization Test
Protocol: Optomechanical interferometry at 4K/300K. Measure Γ = d/dt⟨Δx²⟩. Validate Γ ∝ T
.
7.3. Complex Systems and Thermodynamic Models
Thermodynamic views of life (e.g., dissipative structures, maximum entropy production) explain life's tendency to increase entropy flow. However, these frameworks generally lack observer-dependence and fail to model the information-processing aspect of life.
PRPT incorporates thermodynamic principles via entropy and Kullback-Leibler terms, but explicitly adds mutual information to explain the rise of order and specificity. It models how feedback from information-processing entities shapes statistical structure.
Neural Agency Criterion
Define agency as I_agency = I(state; goal | tool) > θ_c
. Test: RL agents solving ≥3/5 novel puzzles.
7.4. Quantum and Cosmological Analogies
Some speculative theories propose quantum coherence or cosmological inflation-like analogies for life's emergence. PRPT does not rely on quantum mechanics per se but shares structural similarities in treating the universe as an evolving wavefunction-like probability field.
The observer-centric updating via 𝓕 mirrors quantum measurement collapse but here generalized to macroscopic, probabilistic systems.
7.5. Summary of Contributions
PRPT theory distinguishes itself through:
1. Formal grounding in information-theoretic and probabilistic dynamics.
2. Introduction of a concrete evolution operator 𝓕 for reality structure.
3. Unification of entropy minimization and information feedback.
4. Scalability across life, intelligence, and cognitive systems.
5. Testable criteria for synthetic life and complexity transitions.
8. Limitations and Research Frontiers
1. Scalability: High-dimensional P requires neural surrogates (e.g., normalizing flows)
.
2. Quantum-Classical Bridge: Decoherence in biological systems remains unresolved
.
3. Empirical Grounding: Ribozyme threshold θ_c = 15.2 k_B requires cross-validation
| [4] | Higgs, P. G., Lehman, N. The RNA World: Molecular Cooperation at the Origins of Life. Nature Reviews Genetics. 2015, 16(1), 7-17. https://doi.org/10.1038/nrg3841 |
| [5] | Chen, X., Li, N., Ellington, A. D. Ribozyme Fidelity at the Prebiotic Limit. Science. 2022, 375(6578), 156-161. https://doi.org/10.1126/science.abm1184 |
[4, 5]
.
9. Conclusion
The Probabilistic Reality Phase Transition (PRPT) theory offers a rigorous, logically grounded framework that models the emergence of life and intelligence as a sequence of entropy-reducing, information-driven probabilistic phase transitions. Through the formal operator (P, c), we defined how observer-based feedback modifies the informational topology of reality, giving rise to adaptive niches and self-organizing structures. By extending this mechanism, the IQPT model applies PRPT to the origin of life and cognitive systems, offering a unified explanation that bridges thermodynamics, information theory, and systems biology. Future empirical tests in synthetic life systems and recursive observer architectures will serve to evaluate and refine the theoretical predictions outlined herein.
Abbreviations
PRPT | Probabilistic Reality Phase Transition |
IQPT | Information-Driven Quantum Phase Transition |
POVM | Positive Operator-Valued Measure |
KL | Kullback–Leibler (Divergence) |
MPO | Matrix Product Operator |
RG | Renormalization Group |
FEP | Free Energy Principle |
IIT | Integrated Information Theory |
RL | Reinforcement Learning |
M_c | Observer’s Internal Model Distribution |
P_t(r) | Ontic Probability Distribution at Time t |
λ (lambda) | Observer–System Coupling Strength |
θ_c | Observer Critical Threshold |
β | Inverse Temperature (1/k_B T) |
ΔG | Gibbs Free Energy Change |
S_env | Entropy of the Environment |
Φ (phi) | IIT Measure of Integrated Information |
ð ”½ (F) | PRPT Variational Free Energy Functional |
ð ’«(ð ‘Ÿ) | Probability Distribution over Microstates |
ψ(r) | Quantum State Function |
Ï | Density Matrix (Quantum State) |
H(P) | Shannon Entropy of Distribution P |
I(P; c) | Mutual Information Between System and Observer c |
∇ | Gradient Operator |
ξ (xi) | Correlation Length |
χ (chi) | Bond Dimension in Tensor Networks |
Author Contributions
Syed Hassan Masoom Alam Shah is the sole author. The author read and approved the final manuscript.
Conflicts of Interest
The author declares no conflicts of interest.
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APA Style
Shah, S. H. M. A. (2025). The Probabilistic Reality Phase Transition (PRPT) Theory. American Journal of Physics and Applications, 13(4), 68-73. https://doi.org/10.11648/j.ajpa.20251304.11
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Shah, S. H. M. A. The Probabilistic Reality Phase Transition (PRPT) Theory. Am. J. Phys. Appl. 2025, 13(4), 68-73. doi: 10.11648/j.ajpa.20251304.11
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Shah SHMA. The Probabilistic Reality Phase Transition (PRPT) Theory. Am J Phys Appl. 2025;13(4):68-73. doi: 10.11648/j.ajpa.20251304.11
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@article{10.11648/j.ajpa.20251304.11,
author = {Syed Hassan Masoom Alam Shah},
title = {The Probabilistic Reality Phase Transition (PRPT) Theory
},
journal = {American Journal of Physics and Applications},
volume = {13},
number = {4},
pages = {68-73},
doi = {10.11648/j.ajpa.20251304.11},
url = {https://doi.org/10.11648/j.ajpa.20251304.11},
eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajpa.20251304.11},
abstract = {We present the Probabilistic Reality Phase Transition (PRPT) theory, a formal and unifying framework modeling the emergence of structured phenomena such as life and intelligence as phase transitions in a dynamically evolving probability field. Reality is defined as a probabilistic distribution over an ontological state space, continuously updated by a non-linear operator 𝓕 that integrates both intrinsic entropy gradients and observer-induced feedback. We introduce an extension called Informational-Quantum Phase Transition (IQPT), explaining the origin of life as a threshold in mutual information between subsystems and their environment. The framework generalizes thermodynamic and information-theoretic models by including observer-centric transformations, offering a scalable mechanism for self-organization, cognition, and recursive intelligence. This theory yields testable predictions for synthetic life, recursive models, and entropy-information dynamics, and is contrasted with existing abiogenesis, complexity, and cognitive paradigms.},
year = {2025}
}
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TY - JOUR
T1 - The Probabilistic Reality Phase Transition (PRPT) Theory
AU - Syed Hassan Masoom Alam Shah
Y1 - 2025/07/21
PY - 2025
N1 - https://doi.org/10.11648/j.ajpa.20251304.11
DO - 10.11648/j.ajpa.20251304.11
T2 - American Journal of Physics and Applications
JF - American Journal of Physics and Applications
JO - American Journal of Physics and Applications
SP - 68
EP - 73
PB - Science Publishing Group
SN - 2330-4308
UR - https://doi.org/10.11648/j.ajpa.20251304.11
AB - We present the Probabilistic Reality Phase Transition (PRPT) theory, a formal and unifying framework modeling the emergence of structured phenomena such as life and intelligence as phase transitions in a dynamically evolving probability field. Reality is defined as a probabilistic distribution over an ontological state space, continuously updated by a non-linear operator 𝓕 that integrates both intrinsic entropy gradients and observer-induced feedback. We introduce an extension called Informational-Quantum Phase Transition (IQPT), explaining the origin of life as a threshold in mutual information between subsystems and their environment. The framework generalizes thermodynamic and information-theoretic models by including observer-centric transformations, offering a scalable mechanism for self-organization, cognition, and recursive intelligence. This theory yields testable predictions for synthetic life, recursive models, and entropy-information dynamics, and is contrasted with existing abiogenesis, complexity, and cognitive paradigms.
VL - 13
IS - 4
ER -
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