A Brief Contextual Resolution of the Foundational Claims of the Theory of Entropicity (ToE)

1. Executive Summary

The Theory of Entropicity (ToE) introduces a novel framework in theoretical physics, proposing that entropy, traditionally viewed as a statistical measure of disorder, is an autonomous, propagating field fundamental to the universe's structure and evolution. This report critically examines ToE's four primary claims of novelty, particularly its assertion of being the "first proposal" in these areas, by comparing them against existing and contemporary theoretical developments.

The analysis reveals that while ToE presents a comprehensive and ambitious unified action, its claims of being the "first proposal" in several key aspects are challenged by prior or concurrent academic work. Specifically, the concept of entropy or information as a fundamental, propagating field endowed with kinetic and potential terms, and universally coupled to matter and geometry, has been explored in earlier speculative frameworks, such as the Scalar-Entropic-Tensor (SET) field hypothesis (2013), and is a central theme in contemporary research, including Ginestra Bianconi's "Gravity from Entropy" (2024/2025) and the Entropic Force-Field Hypothesis (EFFH) (2025).

Therefore, ToE stands as a significant and detailed contribution to an emerging paradigm in theoretical physics that seeks to elevate entropy/information to a fundamental field. However, it is not the singular originator of these foundational concepts, but rather a prominent participant in a growing intellectual movement exploring new principles for unifying gravity and quantum mechanics.

2. Introduction to the Theory of Entropicity (ToE)

The Theory of Entropicity (ToE) posits a radical reinterpretation of the fundamental laws of physics, asserting that entropy is not merely a statistical descriptor but an active, dynamic field that governs all physical phenomena. This ambitious theory aims to transcend the traditional Newtonian view of gravity as a force and Einstein's description of gravity as spacetime curvature, proposing instead that gravity and other forces emerge from entropy-driven constraints imposed by an underlying "entropic field".¹ ToE seeks to unify General Relativity (GR) and Quantum Field Theory (QFT) by establishing entropy as the primary medium of interaction, measurement, and causality.²

The conceptual elevation of entropy to a fundamental field positions ToE within a broader class of "emergent gravity" theories.⁴ These theories explore the possibility that gravity, rather than being a fundamental interaction, arises from deeper, more microscopic physics, often involving statistical mechanics or information theory. While earlier entropic gravity models, such as Erik Verlinde's proposal from 2009 ⁹, describe gravity as an emergent force resulting from information, they typically do not posit entropy itself as a fundamental, propagating field with its own dynamics. Verlinde, for instance, explicitly states that "there is no fundamental field associated with an entropic force" in his framework.¹⁰ This distinction is crucial for evaluating ToE's claims of novelty, particularly its assertion that prior derivations treated entropy as a constraint or emergent force, but "never as an autonomous, propagating field."

ToE's core mathematical formulation is encapsulated in its master action:

\begin{equation}

\label{eq:master-action}

S_{\mathrm{ToE}}

= \int d^4x,\sqrt{-g},

\Bigl

;+;S_{\mathrm{SM}}

\end{equation}

This action comprises several key components:

• The term −21​gμν(∇μ​S)(∇ν​S) represents a canonical kinetic term for the entropy field S(x), describing its dynamics and propagation.
• The term −V(S) is a potential energy term for the entropy field, governing its self-interactions and energetic landscape.
• The term ηSTμμ​ describes the coupling of the entropy field S(x) to matter and geometry, where Tμμ​ is the trace of the stress-energy tensor and η is a coupling constant.
• SSM​ represents the action for the Standard Model of particle physics, indicating ToE's ambition to integrate with established physics.

The structure of the ToE master action can be further understood through a comparison with standard field theory components, as shown in Table 2.

Table 2: Components of the ToE Master Action and Their Physical Analogues

ToE puts forth four specific claims of novelty regarding its master action:

1. Identify local entropy (Boltzmann, Gibbs, Shannon, von Neumann) with the field value S(x).
2. Endow S(x) with a canonical kinetic term and potential V(S).
3. Couple S universally to matter and geometry via ηSTμμ​.
4. Derive all classical entropy laws and information measures—and their thermodynamic second law—by standard field-theoretic procedures (Euler–Lagrange, Noether) from one unified action.

The subsequent sections will meticulously verify these claims against the current landscape of theoretical physics.

3. Claim 1: Entropy as an Autonomous, Propagating Field

ToE's foundational assertion is that it is the first to treat entropy as an autonomous, propagating field. This claim requires careful examination against the historical development and contemporary understanding of entropy in physics.

Historically, entropy was conceived primarily as a statistical descriptor of disorder or the unavailability of energy within thermodynamic systems. Rudolf Clausius, in the mid-19th century, introduced the concept of entropy as a state variable central to the Second Law of Thermodynamics, which dictates the irreversible tendency of isolated systems towards equilibrium.¹¹ Ludwig Boltzmann later provided a microscopic interpretation, linking entropy to the number of possible microscopic arrangements (microstates) corresponding to a macroscopic state.¹³ These classical and statistical approaches define entropy as a property of a system or an ensemble, not as an independent, dynamic field.

The concept of "entropic force" further developed this understanding, describing forces that emerge from a system's statistical tendency to increase its entropy, rather than from fundamental interactions at the atomic scale.¹⁰ Examples include the pressure of an ideal gas or the elasticity of polymers.¹⁰ More recently, Erik Verlinde's entropic gravity theory (2009) proposed that gravity itself is an entropic force arising from information associated with the positions of material bodies.⁴ This model describes gravity as an

emergent phenomenon from microscopic degrees of freedom encoded on a holographic screen.⁶ A key aspect of Verlinde's formulation, as noted in the research, is that "there is no fundamental field associated with an entropic force".¹⁰ This historical and emergent force perspective aligns with ToE's statement that prior derivations treated entropy as a constraint or emergent force.

However, ToE's claim of being the first to elevate entropy to a "real, physical field that actively governs the structure and evolution of the universe" ⁴ and as the "fundamental force-field governing all interactions in nature" ⁴ faces challenges from other theoretical proposals. The research indicates a recent, concurrent trend in theoretical physics exploring entropy or information as fundamental fields.

One such example is the Scalar-Entropic-Tensor (SET) Field Hypothesis, a speculative framework proposed in 2013. This hypothesis explicitly "treats entropy as a fundamental scalar field coupled to spacetime geometry and matter-energy content".²¹ The SET field, denoted as

Ξ, is envisioned as permeating all of spacetime and is endowed with a kinetic term and a potential, and couples to the stress-energy tensor.²¹ This proposal, predating ToE's publication in 2024, directly challenges ToE's claim of being the

first to conceptualize entropy as a fundamental, propagating field with its own dynamics.

Furthermore, Ginestra Bianconi's "Gravity from Entropy" (published in 2024/2025) also proposes an "entropic gravity" framework where gravity emerges from "quantum information entropy" by treating spacetime as a quantum system.²² This theory defines an action based on quantum relative entropy between the spacetime metric and a matter-induced metric.²² While the specific mathematical form of the action differs from ToE's, the underlying conceptual elevation of entropy/information to a fundamental driver of geometry and dynamics is strikingly similar. The emergence of a "G-field" acting as Lagrangian multipliers and relating to an emergent cosmological constant in Bianconi's work further underscores the shared conceptual space.²³

Similarly, the Entropic Force-Field Hypothesis (EFFH), also published in 2025, explicitly proposes "entropy as the fundamental force-field governing all interactions in nature" and introduces an "Entropic Action" to derive field equations.⁴ EFFH elevates entropy to a "real, physical field that actively governs the structure and evolution of the universe".⁴ These contemporary developments indicate that the idea of entropy as a fundamental, propagating field is an active and emerging area of research in theoretical physics, with multiple groups independently exploring this paradigm.

The following table summarizes the comparison of ToE's claims of novelty with these prior and contemporary theories:

Table 1: Comparison of ToE's Claims of Novelty with Prior and Contemporary Theories

4. Claim 2: Identification of Local Entropy with the Field Value S(x)

ToE asserts that it is the first to identify local entropy (Boltzmann, Gibbs, Shannon, von Neumann) with the field value S(x). This claim pertains to the specific ontological interpretation of the scalar field S within the theory.

Entropy, in its various forms, is a well-established concept in physics and information theory. Boltzmann entropy quantifies the statistical disorder of a system based on the number of possible microscopic arrangements.¹³ Gibbs entropy extends this to statistical ensembles. Shannon entropy, a cornerstone of information theory, measures the uncertainty or information content of a message or system.¹³ Von Neumann entropy is its quantum mechanical analogue, describing the information content of a quantum state.³⁰ These definitions have found widespread applications across diverse fields, from thermodynamics and statistical physics to cosmology and information systems.¹³

The conceptual linkage between thermodynamic entropy and information-theoretic entropy is a deeply explored theme in modern physics. John Wheeler's influential phrase "it from bit" suggests that physical reality fundamentally arises from information, positing information as more fundamental than matter, energy, space, and time.²⁹ This philosophical perspective provides a precedent for elevating "information" (and by extension, entropy) to a fundamental status, even if not explicitly as a propagating field.

ToE's innovation lies not in the definition of entropy itself, but in proposing a local field whose value is this entropy, making it a fundamental, dynamic entity. The identification of S(x) as a scalar field whose value at each spacetime point represents the local entropy density or a related measure is a specific field-theoretic interpretation of entropy. Scalar fields are common in theoretical physics, representing quantities that have a single value at each point in spacetime and are invariant under Lorentz transformations, such as the Higgs field.³¹ Therefore, proposing a scalar field is a standard mathematical approach. The novelty of ToE, in this context, is the

specific identification of that scalar field with "local entropy" (encompassing Boltzmann, Gibbs, Shannon, von Neumann) and its treatment as a fundamental, propagating entity.

This specific conceptualization of entropy as a field is a distinguishing feature of ToE and aligns with other recent proposals. As noted, the Scalar-Entropic-Tensor (SET) field hypothesis from 2013 explicitly posits entropy as a primary scalar field (Ξ) permeating spacetime.²¹ Similarly, Bianconi's "Gravity from Entropy" (2024/2025) treats the spacetime metric as a quantum operator and defines its action based on quantum relative entropy, effectively linking geometry directly to information/entropy.²² These examples demonstrate that the idea of identifying a fundamental field with entropy or information is a shared conceptual foundation among several contemporary theories, indicating a growing convergence in this area of physics.

5. Claim 3: Endowing S(x) with a Canonical Kinetic Term and Potential V(S)

ToE claims to be the first to endow its proposed entropy field S(x) with a canonical kinetic term and a potential V(S). This claim refers to the mathematical structure of the field's dynamics within the Lagrangian.

In Lagrangian field theory, the action, from which the equations of motion are derived, typically includes terms that describe the field's dynamics and its interactions. Kinetic terms, which involve derivatives of the field, describe how the field propagates and evolves in spacetime.³² For a scalar field like

S(x), a canonical kinetic term in curved spacetime is typically of the form −21​gμν(∇μ​S)(∇ν​S).³² This term ensures that the field is dynamical and can carry energy and momentum, similar to how other fundamental fields propagate. Potential terms, denoted as

V(S), are functions of the field value itself and describe self-interactions or the energetic landscape in which the field exists, influencing its possible equilibrium states and phase transitions.²¹ The mathematical form of these terms in the ToE action is standard for a scalar field theory in a curved spacetime background, as seen in various scalar-tensor theories.³⁷

ToE's master action explicitly includes both a canonical kinetic term, −21​gμν(∇μ​S)(∇ν​S), and a potential term, −V(S), for its entropy field S(x) . The inclusion of these terms implies that the entropy field is a dynamic entity capable of propagation and self-interaction, a departure from traditional views of entropy as a static property.

However, the assertion that ToE is the first to endow an entropy field with such terms is directly challenged by the Scalar-Entropic-Tensor (SET) field hypothesis, which was proposed in 2013. The SET hypothesis explicitly presents a minimal action for its entropy field Ξ: S=∫−g​d4x.²¹ In this formulation, the term

(1/2)∂μ​Ξ∂μΞ is a standard kinetic term for a scalar field, and V(Ξ) is a potential function governing the field's self-interaction or background energy.²¹ Since the SET hypothesis predates ToE's publication (2024), it demonstrates that the conceptualization of entropy as a field endowed with its own kinetic and potential energy terms was already present in the theoretical physics literature prior to ToE. This indicates that while ToE applies these standard field theory components to an entropy field, it is not the initial proposal to do so.

6. Claim 4: Universal Coupling of S to Matter and Geometry via ηSTμμ​

ToE claims to be the first to universally couple its entropy field S to matter and geometry via the term ηSTμμ​. This coupling term describes how the entropy field interacts with the fundamental constituents of the universe.

The stress-energy tensor, Tμν, is a fundamental quantity in relativistic physics that describes the density and flux of energy and momentum in spacetime.⁴ Its trace,

Tμμ​, is a scalar quantity related to the local energy density and pressure of matter and radiation. In standard field theories, coupling a scalar field to the trace of the stress-energy tensor is a known and common mechanism for scalar fields to interact with matter fields.²¹ This type of interaction is observed in various scalar-tensor theories, which extend General Relativity by introducing a scalar field that interacts with gravity and matter.³⁷ Therefore, the

mathematical form of the coupling term ηSTμμ​ is not unique to ToE; it is a standard approach for mediating interactions between a scalar field and matter/energy. The novelty, if any, lies in the interpretation of S as entropy and the assertion of its universal nature as a fundamental mediator.

ToE proposes that the entropy field S couples "universally to matter and geometry via ηSTμμ​" . This implies a direct and fundamental interaction where the entropy field influences and is influenced by all forms of energy and momentum. ToE further states that gravity itself "emerges from entropy-driven constraints imposed by the underlying Entropic Field" and that the theory "unifies them under entropy".¹ This suggests that the proposed universal coupling is central to ToE's ambition of unifying all fundamental forces under the umbrella of entropy.

However, the claim of being the first to propose such a universal coupling for an entropy field is challenged by prior and concurrent theoretical developments. As discussed in the previous section, the Scalar-Entropic-Tensor (SET) field hypothesis (2013) explicitly includes a coupling term αΞT in its action, where Ξ is the entropy field and T is the trace of the stress-energy tensor.²¹ This directly demonstrates a prior proposal for coupling an entropy field to matter and geometry in a manner analogous to ToE's claim.

Additionally, Ginestra Bianconi's "Gravity from Entropy" (2024/2025) describes a framework where matter fields couple with discrete geometry through quantum entropy. Its action is based on the quantum relative entropy between the spacetime metric and the metric induced by matter and gauge fields.²² This approach fundamentally links information/entropy to spacetime and matter dynamics, achieving a form of universal coupling, albeit through a different mathematical mechanism. Similarly, the

Entropic Force-Field Hypothesis (EFFH) (2025) also proposes entropy as a "fundamental field underlying all forces, including gravitation" ⁴, inherently implying a universal coupling. These theories, while differing in their specific mathematical formalisms, share the overarching goal of establishing entropy or information as a fundamental entity that universally interacts with matter and geometry. This indicates that ToE is contributing to a contemporary research direction rather than being the sole pioneer of this conceptual approach.

7. Claim 5: Derivation of Classical Entropy Laws and Information Measures from One Unified Action

ToE's fourth claim of novelty is its ability to "derive all classical entropy laws and information measures—and their thermodynamic second law—by standard field‐theoretic procedures (Euler–Lagrange, Noether) from one unified action" . This represents a highly ambitious goal, as the Second Law of Thermodynamics is a cornerstone of physics, traditionally understood through statistical mechanics or as an empirical axiom.

The Second Law of Thermodynamics, which states that the entropy of an isolated system cannot decrease over time, is a fundamental empirical law governing irreversible processes and defining the "arrow of time".¹³ Its microscopic explanation typically stems from statistical mechanics, where entropy is linked to the number of available microstates, and the increase in entropy corresponds to the system's natural tendency to evolve towards more probable (disordered) configurations.¹³

While the Second Law is well-established, its derivation from a fundamental field theory action (as opposed to statistical mechanics or emergent principles) is a less common approach in physics. However, the methodology of deriving equations of motion and conservation laws from an action principle using standard field-theoretic procedures like the Euler-Lagrange equations and Noether's theorem is fundamental to classical and quantum field theory.³³ These procedures link symmetries in the action to conserved quantities and the dynamics of fields.

Some contemporary works do explore deriving thermodynamic principles or entropy production from field theories, particularly in the context of non-equilibrium thermodynamics or thermal field theories.⁴¹ For instance, recent effective field theories for fluctuating relativistic fluids aim to describe entropy production and ensure causality and stability from their action, with thermodynamic relations emerging from their construction.⁴³ Other studies investigate the increase of Shannon entropy in isolated quantum systems as a variant of the Second Law.³⁰ These examples indicate that the

approach of deriving thermodynamic laws or entropy dynamics from a field theory action, while challenging, is not entirely unprecedented in contemporary theoretical physics.

ToE proposes to achieve this by asserting that entropy is an "active field that fundamentally governs interactions, motion, and causality".² It claims to directly introduce a "direction of time directly into the wave and field equations via the unidirectional flow of entropy," thereby resolving the arrow of time problem as a dynamical law embedded in the fabric of reality.⁸ A key mechanism within ToE for this derivation is the "Vuli-Ndlela Integral," described as an "entropy-constrained and entropy-weighted reformulation of the Feynman path integral".² This integral is posited to impose strict constraints on quantum trajectories, replacing unconstrained superposition with an entropy-constrained selection principle. Wave-function collapse, in this view, occurs deterministically when entropy flux or "resistance" surpasses a critical limit, aiming to reconcile causal realism with contextual irreversibility.²

The ambition to derive all classical entropy laws and information measures, including the Second Law, from a single unified action via Euler-Lagrange and Noether theorems, and to provide a novel mechanism like the Vuli-Ndlela Integral for phenomena such as wave-function collapse, represents a significant theoretical undertaking. While the underlying field-theoretic procedures are standard, their application to a fundamental entropy field to yield the Second Law as a direct consequence of the field's dynamics (rather than statistical emergence) would be a profound achievement. However, the claim of being the first to derive the Second Law from a field theory action is challenged by other works that discuss deriving entropy production or thermodynamic laws from field-theoretic frameworks, particularly in non-equilibrium contexts. This positions ToE's approach as a valuable contribution to an active area of research, rather than a completely isolated breakthrough.

8. Broader Implications and Testable Predictions of ToE

Beyond its foundational claims, ToE proposes to offer solutions to several long-standing problems in physics and suggests avenues for experimental verification.

ToE claims to unify and generalize earlier entropic-gravity and information-based approaches by making entropy the "fundamental mediator of forces and geometry" . It argues that gravity is "neither a fundamental force nor merely the curvature of spacetime," but rather emerges from entropy-driven constraints.¹ This challenges both the Newtonian and Einsteinian pictures of gravity and aims to eliminate the traditional distinction between forces by unifying them under entropy.¹ This ambition to unify forces and explain gravity as an entropic phenomenon is part of a larger, active research area in theoretical physics.⁴ While ToE offers a specific framework, the underlying goal of unifying forces through entropy or information is a shared pursuit among several contemporary theories, indicating a broader intellectual current in the field.

ToE also proposes solutions to major physics problems:

• Cosmological Constant and Dark Energy: ToE suggests an "Entropy-Driven Derivation of Mercury's Perihelion Precession Beyond Einstein's Curved Spacetime".¹ It claims to explain the universe's accelerated expansion without invoking dark energy, replacing the cosmological constant with a dynamically evolving entropy-driven term. This framework posits that entropy accumulation at late times naturally leads to the observed acceleration.³ Other entropic/information-based theories also propose alternative explanations for dark energy ⁶ and even dark matter.²³
• Quantum Measurement Problem and Wave Function Collapse: ToE offers a distinct perspective, suggesting that these phenomena are governed by the irreversible flow of entropy, rather than being statistical or observer-dependent effects.³ The Vuli-Ndlela Integral within ToE imposes entropy-constrained selection on quantum trajectories, leading to physically deterministic yet irreversible transitions. This framework aims to reconcile the views of Einstein (on causal realism) and Bohr (on contextual irreversibility) by embedding measurement "collapse" within the irreversible dynamics of entropy.²
• Arrow of Time: ToE directly introduces a direction of time into the wave and field equations via the unidirectional flow of entropy, presenting it as a dynamical law rather than a statistical artifact.⁸

The theory also suggests specific testable predictions. These include modifications to gravitational lensing effects in regions of high entropy gradients ⁴ and implications for quantum entanglement and nonlocality, suggesting that quantum entanglement is an entropy-driven correlation that imposes a limit on nonlocal interactions.⁴ ToE also identifies potential test platforms such as atomic clock networks, superconducting qubits, brain-computer interfaces (BCI), and autonomous vehicle (AV) quantum sensor systems.⁸ The emphasis on testable predictions is crucial for any new theory to gain acceptance and distinguishes it from purely philosophical or untestable hypotheses.³

9. Conclusion and Outlook

The Theory of Entropicity (ToE) represents a bold and comprehensive effort to reframe fundamental physics by positing entropy as an autonomous, propagating field. Its master action, which includes canonical kinetic and potential terms for this entropy field and a universal coupling to matter and geometry, aims to unify General Relativity and Quantum Field Theory and address long-standing problems in cosmology and quantum mechanics.

However, a detailed verification of ToE's claims of being the "first proposal" in these areas reveals that while the theory is a significant and detailed contribution, it is not entirely unprecedented. The concept of entropy or information as a fundamental, dynamic field, endowed with kinetic and potential terms, and universally coupled to matter and geometry, has been explored in prior speculative frameworks, notably the Scalar-Entropic-Tensor (SET) field hypothesis from 2013.²¹ Furthermore, contemporary research, such as Ginestra Bianconi's "Gravity from Entropy" (2024/2025) ²² and the Entropic Force-Field Hypothesis (EFFH) (2025) ⁴, also explores similar conceptual ground, treating entropy or information as fundamental fields driving gravitational and matter interactions.

This convergence of multiple, independent theories (ToE, SET, Bianconi, EFFH) around the idea of entropy/information as a fundamental, propagating field suggests an emerging paradigm in theoretical physics. This indicates a shared intellectual current exploring new foundational principles beyond traditional force or spacetime curvature interpretations. ToE is a prominent and detailed participant in this emerging paradigm, contributing a specific unified action and proposed mechanisms like the Vuli-Ndlela Integral.

For ToE to flourish and gain broader acceptance within the scientific community, several challenges must be addressed ³:

• Continued Mathematical Development: Further rigorous mathematical derivations and consistency checks of its field equations are essential.
• Detailed Derivations of Known Physics: ToE must provide more detailed and independently verifiable derivations of established physical laws and phenomena from its action.
• Novel, Testable Predictions: The proposal of clear, unique, and experimentally testable predictions that can distinguish ToE from existing theories and other entropic/information field theories is paramount.³
• Broader Scientific Engagement: Widespread engagement and critical evaluation from the broader physics community will be crucial for its validation and potential integration into mainstream theoretical physics.

Should ToE successfully address these challenges, it holds the potential to offer a truly revolutionary understanding of the universe, providing a unified framework for fundamental interactions rooted in the dynamics of entropy itself.

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This entry is adapted from: https://www.cambridge.org/engage/api-gateway/coe/assets/orp/resource/item/67cf9ecefa469535b9d11e66/original/the-entropic-force-field-hypothesis-a-unified-framework-for-quantum-gravity.pdf