Toward Experimental Confirmation of the Theory of Entropicity (ToE): Core Concepts, Mathematical Foundations, Scientific Status, Time’s Arrow, Entropic Inevitability, Quantum Entanglement, and Attosecond Electron Dynamics

1. Introduction

The Theory of Entropicity (ToE) is a recent and conceptually radical framework in theoretical physics, initiated primarily by John Onimisi Obidi beginning in 2025. In contrast to conventional statistical and thermodynamic treatments, where entropy is regarded as a derived measure of disorder or missing information, ToE promotes entropy to the status of a fundamental dynamical field. In this view, entropy is not merely a bookkeeping device for microscopic configurations but a primary ontological entity that underlies and organizes all physical phenomena.

Within this framework, the universe is modeled as being permeated by an entropic field, typically represented as an ontological scalar field \( S(x) \) defined over spacetime. This field encodes the local and global structure of what may be called entropic potential, and physical processes are interpreted as manifestations of the evolution and redistribution of this field. Traditional notions such as spacetime geometry, gravity, and even the speed of light are reinterpreted as emergent or effective properties arising from the dynamics of the entropic field rather than as primitive inputs.

The central ambition of ToE is to provide a unifying description of fundamental physics by deriving familiar laws and structures from the dynamics of this entropic field. In particular, ToE seeks to explain the emergence of gravitational interaction, the structure of spacetime, the observed arrow of time, and the behavior of quantum systems as consequences of a single underlying entropic dynamics. This requires a rigorous mathematical formulation, including a well-defined action functional for the entropic field, a corresponding set of field equations, and a clear prescription for how matter and information couple to the entropic substrate.

A distinctive feature of ToE is its insistence that irreversibility is not merely a statistical tendency but a structural property of the universe. The theory introduces specific mathematical constructs, such as the Vuli–Ndlela Integral, to encode intrinsic irreversibility into the evolution of quantum states. It also posits that the speed of light \( c \) should be understood as the maximal rate at which the entropic field can reorganize, rather than as an arbitrary universal constant. In this way, ToE attempts to unify kinematical limits, thermodynamic behavior, and quantum dynamics within a single entropic framework.

The present exposition focuses on the core conceptual and mathematical structure of ToE, its current scientific status, its treatment of the arrow of time and entropic inevitability, and its proposed experimental program, particularly in the domain of attosecond electron dynamics and quantum entanglement. The goal is to present a coherent and technically rigorous account of the theory as it stands in early 2026, emphasizing its internal logic, its points of departure from standard physics, and the concrete predictions that render it experimentally testable.

2. Core Concepts of the Theory of Entropicity

At the conceptual core of the Theory of Entropicity is the assertion that entropy is an ontological scalar field rather than a derived statistical quantity. This field, denoted by \( S(x) \), is defined over a manifold that, at the effective level, is identified with spacetime. The value of \( S(x) \) at a point encodes the local entropic density or entropic potential, and its gradients and higher derivatives govern the dynamics of physical systems.

In this framework, the familiar structures of spacetime and gravity are emergent. The metric structure of spacetime is interpreted as a manifestation of the configuration of the entropic field, and gravitational curvature is reinterpreted as the curvature of the entropic field rather than of a purely geometric manifold. Thus, what is traditionally described by the Einstein field equations in General Relativity is, in ToE, a derived effective description of deeper entropic dynamics.

A central organizing idea is that physical processes correspond to trajectories in the configuration space of the entropic field that minimize a suitable functional, often interpreted as an entropic action. The motion of bodies is then described not as geodesics in a purely geometric spacetime but as entropic geodesics that minimize an appropriate measure of entropic resistance. This entropic resistance quantifies the “cost” of reconfiguring the entropic field along a given trajectory and plays a role analogous to the action in classical mechanics and field theory.

Another key conceptual element is the No-Rush Theorem. This principle asserts that no physical interaction, information transfer, or entropic reconfiguration can occur instantaneously. Instead, every process requires a finite, non-zero duration. In ToE, this is not merely a practical limitation but a fundamental structural constraint. The theorem implies that the universe possesses a finite processing rate or reconfiguration rate, which is directly related to the maximal rate at which the entropic field can change. The conventional speed of light \( c \) is then interpreted as the emergent upper bound on this reconfiguration rate.

Within this conceptual architecture, the arrow of time is not a secondary statistical artifact but a direct consequence of the structure and dynamics of the entropic field. The directionality of time is associated with the monotonic evolution of entropic configurations, and the impossibility of perfectly reversing physical processes is encoded in the fundamental equations of motion rather than arising solely from coarse-graining or probabilistic arguments.

3. Mathematical Foundations and Key Components

The mathematical structure of the Theory of Entropicity is built around several central constructs: the Obidi Action, the Master Entropic Equation (MEE), the Vuli–Ndlela Integral, and the use of information geometry to connect statistical and physical structures.

The Obidi Action is the fundamental action functional governing the dynamics of the entropic field. In general form, it may be written schematically as \[ \mathcal{A}_{\text{Obidi}}[S, g, \Psi] = \int \mathcal{L}(S, \nabla S, g, \Psi)\, \mathrm{d}^4 x, \] where \( S \) is the entropic field, \( g \) denotes an effective metric or geometric structure, and \( \Psi \) represents matter or quantum fields coupled to the entropic field. The Lagrangian density \( \mathcal{L} \) encodes the kinetic and potential terms for the entropic field, its coupling to geometry, and its interaction with matter and information degrees of freedom. The precise form of \( \mathcal{L} \) is theory-specific and is designed to reproduce known physics in appropriate limits while introducing new entropic effects at relevant scales.

Variation of the Obidi Action with respect to the entropic field yields the Master Entropic Equation (MEE), which serves as the primary field equation for \( S(x) \). In symbolic form, this can be expressed as \[ \frac{\delta \mathcal{A}_{\text{Obidi}}}{\delta S} = 0, \] leading to a differential equation of the general type \[ \mathcal{E}[S, g, \Psi] = 0, \] where \( \mathcal{E} \) is a functional operator encoding the dynamics of the entropic field and its coupling to other fields. The MEE plays a role analogous to the Einstein field equations in General Relativity or the Klein–Gordon equation for scalar fields, but it is specifically tailored to the entropic ontology of ToE.

A distinctive mathematical innovation of ToE is the introduction of the Vuli–Ndlela Integral. This is an entropy-weighted reformulation of the Feynman Path Integral of Quantum Field Theory (QFT), and is a mathematical construct that is designed to incorporate intrinsic irreversibility into the evolution of quantum states. In standard quantum mechanics, the evolution of a wave function \( \psi \) under the Schrödinger equation is unitary and time-reversal symmetric. The Vuli–Ndlela Integral modifies this structure by adding a non-zero term that cannot be eliminated by unitary transformations and that encodes a directional flow of entropic potential. Schematically, one may write a modified evolution equation of the form \[ i \hbar \frac{\partial \psi}{\partial t} = \hat{H} \psi + \mathcal{I}_{\text{VN}}[\psi, S], \] where \( \hat{H} \) is the usual Hamiltonian operator and \( \mathcal{I}_{\text{VN}} \) denotes the contribution from the Vuli–Ndlela Integral, which depends on both the quantum state and the entropic field. This additional term breaks exact time-reversal symmetry and ensures that quantum processes possess an intrinsic directionality consistent with the macroscopic arrow of time.

ToE also makes extensive use of information geometry, particularly structures such as the Fisher–Rao metric, Fubiny-Study metric, and the Amari-Čencov α-connections to connect statistical descriptions with physical geometry. In information geometry, probability distributions are treated as points on a differentiable manifold, and the Fisher–Rao metric provides a natural measure of distance between distributions. In the context of ToE, such geometric structures are used to relate the configuration space of probability distributions to the configuration space of the entropic field. This allows the theory to interpret statistical and quantum states as embedded within, and constrained by, the geometry of the entropic field.

Through these mathematical components, the Theory of Entropicity constructs a unified formalism in which entropy is a dynamical field, irreversibility is built into the fundamental equations, and the familiar structures of spacetime and quantum mechanics emerge as effective descriptions of deeper entropic dynamics.

3.1 Amari–Čencov α-Connections

The theory of Amari–Čencov α-connections arises within the framework of information geometry, where families of probability distributions are treated as differentiable manifolds endowed with geometric structures derived from statistical properties. On such a statistical manifold, the central geometric object is the Fisher–Rao metric, which provides a Riemannian metric encoding the local distinguishability of probability distributions. Beyond this metric structure, information geometry introduces a family of affine connections, known as α-connections, that capture the dualistic and asymmetric aspects of statistical inference.

Let a statistical model be represented as a smooth manifold \( \mathcal{M} \) whose points correspond to probability distributions \( p(x;\theta) \) parametrized by coordinates \( \theta = (\theta^1, \dots, \theta^n) \). The Fisher–Rao metric on \( \mathcal{M} \) is defined by \[ g_{ij}(\theta) = \mathbb{E}_{\theta} \left[ \frac{\partial \log p(x;\theta)}{\partial \theta^i} \frac{\partial \log p(x;\theta)}{\partial \theta^j} \right], \] where \( \mathbb{E}_{\theta}[\cdot] \) denotes expectation with respect to the distribution \( p(x;\theta) \). This metric is invariant under sufficient statistics and forms the unique (up to a constant factor) Riemannian metric satisfying certain natural monotonicity conditions.

In addition to the metric, one introduces a family of affine connections \( \nabla^{(\alpha)} \), indexed by a real parameter \( \alpha \in \mathbb{R} \), called the Amari–Čencov α-connections. These connections are torsion-free but, in general, not metric-compatible. Their Christoffel symbols in local coordinates are defined by \[ \Gamma^{(\alpha)}_{ijk} = \mathbb{E}_{\theta} \left[ \frac{\partial^2 \log p(x;\theta)}{\partial \theta^i \partial \theta^j} \frac{\partial \log p(x;\theta)}{\partial \theta^k} \right] + \frac{1 - \alpha}{2} \mathbb{E}_{\theta} \left[ \frac{\partial \log p(x;\theta)}{\partial \theta^i} \frac{\partial \log p(x;\theta)}{\partial \theta^j} \frac{\partial \log p(x;\theta)}{\partial \theta^k} \right], \] where indices are lowered using the Fisher–Rao metric. Equivalently, one often writes the connection in mixed-index form \[ \Gamma^{(\alpha)k}_{\;\;ij} = g^{kl} \Gamma^{(\alpha)}_{ijl}, \] with \( g^{kl} \) the inverse of the Fisher–Rao metric.

A particularly important feature of the α-connections is the existence of a dualistic structure. For each value of \( \alpha \), there is a dual connection \( \nabla^{(-\alpha)} \) such that the pair \( (\nabla^{(\alpha)}, \nabla^{(-\alpha)}) \) is dual with respect to the Fisher–Rao metric. This duality is expressed by the condition \[ X \bigl( g(Y,Z) \bigr) = g\bigl( \nabla^{(\alpha)}_X Y, Z \bigr) + g\bigl( Y, \nabla^{(-\alpha)}_X Z \bigr) \] for all vector fields \( X, Y, Z \) on the statistical manifold. The case \( \alpha = 0 \) corresponds to the Levi-Civita connection of the Fisher–Rao metric, which is the unique torsion-free and metric-compatible connection. The values \( \alpha = 1 \) and \( \alpha = -1 \) correspond, respectively, to the exponential connection and the mixture connection, which play central roles in the geometry of exponential families and mixture families of probability distributions.

The Amari–Čencov α-connections provide a unified geometric framework for understanding different types of statistical inference and divergence measures. For example, the α-divergences, which generalize the Kullback–Leibler divergence, are naturally associated with the α-connections and encode the geometry of statistical models under different inference schemes. The curvature properties of these connections, and their dual pairs, reveal deep structural information about the underlying statistical manifold, including notions of flatness for exponential and mixture families.

In summary, Amari–Čencov α-connections extend the Riemannian structure given by the Fisher–Rao metric by introducing a one-parameter family of affine connections that capture the dualistic and asymmetric aspects of information geometry. They form a cornerstone of modern information geometry and play a crucial role in the geometric analysis of statistical models, divergence functions, and entropic or information-theoretic field theories.

5. The Arrow of Time and Entropic Inevitability

One of the most significant conceptual contributions of the Theory of Entropicity is its treatment of the arrow of time. In conventional physics, the asymmetry between past and future is typically attributed to the Second Law of Thermodynamics, which states that entropy tends to increase in closed systems. However, the underlying microscopic equations of motion in classical mechanics and standard quantum mechanics are time-reversal symmetric. This leads to the well-known tension between reversible microdynamics and irreversible macrodynamics.

In ToE, this tension is resolved by elevating irreversibility to a fundamental structural feature of the universe. The arrow of time is not merely a statistical tendency but a direct consequence of the dynamics of the entropic field. The key idea is that every physical interaction consumes a portion of the local entropic potential, and this consumption is encoded in the evolution equations themselves. The Vuli–Ndlela Integral plays a central role in this context by introducing a non-zero, intrinsically directional term into the evolution of quantum states. This term prevents quantum wave functions from perfectly “undoing” themselves, even in principle, thereby breaking exact time-reversal symmetry at the fundamental level.

In this framework, the past and future are not symmetric because the entropic field evolves in a way that monotonically depletes or redistributes entropic potential along physically realized trajectories. Each interaction leaves an irreversible imprint on the entropic field, and this imprint cannot be erased without further entropic cost. The arrow of time is thus identified with the directed flow of the entropic field, and the progression of time is associated with the cumulative consumption and redistribution of entropic potential.

ToE further proposes that time itself can be interpreted as the rate of change of the entropic field. If the entropic field were to cease evolving—if entropy stopped changing—then, in this view, time as experienced and measured by physical processes would effectively cease to exist. This leads to a conception of time as an emergent parameter derived from the dynamics of the entropic field rather than as an independent background coordinate.

The No-Rush Theorem reinforces this picture by asserting that information and entropic reconfigurations cannot occur instantaneously. The universe possesses a finite entropic processing speed, and this finite rate imposes an ordering on events that cannot be arbitrarily reversed. Once entropic potential has been expended in a given sequence of interactions, that sequence is effectively “locked in” behind a boundary of spent entropic potential. This provides a structural explanation for why the past is fixed and the future open, grounded in the dynamics of the entropic field rather than in purely probabilistic arguments.

6. Experimental Program: Attosecond Electron Dynamics and Quantum Effects

A central requirement for any fundamental physical theory is the ability to generate clear, testable predictions. The Theory of Entropicity addresses this requirement by proposing specific experimental signatures that should be observable at extremely short time scales and in regimes where entropic effects are expected to be significant. In particular, ToE focuses on attosecond electron dynamics and related quantum processes as promising arenas for empirical investigation.

An attosecond is \( 10^{-18} \) seconds, a time scale at which the dynamics of electrons in atoms, molecules, and solids can be directly probed using ultra-short laser pulses. In standard quantum mechanics, certain processes, such as quantum jumps between energy levels, are often treated as effectively instantaneous transitions, or at least as transitions whose duration is not fundamentally constrained by an underlying entropic structure. The Theory of Entropicity, by contrast, predicts that such transitions should exhibit a finite, measurable duration associated with the reconfiguration of the entropic field.

Within ToE, attosecond electron dynamics are interpreted as processes in which the entropic field must reorganize to accommodate changes in the quantum state of the system. The theory predicts the existence of a small but finite entropic lag, a micro-delay between the initiation of a transition and its completion. This lag is not an artifact of measurement or instrumentation but a fundamental consequence of the finite reconfiguration rate of the entropic field, as constrained by the No-Rush Theorem and the entropic interpretation of the speed of light.

More concretely, ToE posits that transitions traditionally modeled as instantaneous should, upon sufficiently precise measurement, reveal a characteristic time scale that can be calculated from the local entropic density and the structure of the Obidi Action. If the theory is correct, high-precision attosecond experiments should detect systematic deviations from the predictions of the Standard Model and conventional quantum mechanics, in the form of non-zero transition durations that correlate with entropic parameters.

In addition to transition times, ToE makes predictions regarding quantum decoherence. In standard quantum theory, decoherence arises from the interaction of a quantum system with its environment, leading to the suppression of interference terms in the system’s density matrix. The rates of decoherence are typically computed from the system–environment coupling and the spectral properties of the environment. In the Theory of Entropicity, decoherence rates are further influenced by the local structure of the entropic field. In regions of high effective gravitational field, where the entropic field is interpreted as being “thicker” or more densely structured, ToE predicts that decoherence may occur at rates that differ slightly from those predicted by the Schrödinger equation alone.

This leads to a class of experimental tests in which decoherence rates are measured in different gravitational or entropic environments and compared with standard predictions. Any systematic deviations that correlate with entropic density, as defined by ToE, would provide evidence in favor of the theory’s entropic field interpretation.

The theory also has implications for quantum entanglement. Entangled systems are characterized by non-local correlations that cannot be explained by classical local hidden variables. In ToE, such correlations are mediated and constrained by the global structure of the entropic field. While the theory does not deny the existence of quantum entanglement, it asserts that the establishment, propagation, and maintenance of entangled correlations are constrained by the finite reconfiguration rate of the entropic field. Consequently, the Theory of Entropicity (ToE) rejects the notion that entanglement updates occur with zero temporal duration; instead, all entanglement dynamics unfold over a non‑zero, entropically regulated timescale [as per the No-Rush Theorem (NRT), Entropic Accessibility (EA), Entropic Cost (EC), Entropic Constraint Principle (ECP), and the Entropic Accounting Principle (EAP)]. [Refer to this Section for details.] This leads to subtle but measurable modifications in entanglement dynamics, particularly over large separations or in strong‑field regimes, because ToE asserts that entangled correlations cannot update instantaneously; instead, their propagation and stability are regulated by entropic gradients and the finite reconfiguration rate of the entropic field, making these effects, in principle, experimentally accessible.

7. Comparative Perspective: Standard Physics vs. Theory of Entropicity

The conceptual and structural differences between standard physics (as represented by General Relativity and Quantum Mechanics) and the Theory of Entropicity can be summarized in a comparative manner. The following table highlights key contrasts in how each framework treats the arrow of time, time reversibility, spacetime, and the speed of light.

This comparative perspective underscores that ToE is not a minor modification of existing theories but a re-architecting of foundational concepts. It replaces geometric primacy with entropic primacy, reinterprets time and causality in terms of entropic dynamics, and seeks to derive familiar physical structures as emergent phenomena.

8. Outlook and Implications

The future development of the Theory of Entropicity will depend critically on both theoretical refinement and experimental investigation. On the theoretical side, further work is required to specify the exact form of the Obidi Action, to derive explicit solutions of the Master Entropic Equation, and to establish detailed correspondences with known results in General Relativity, Quantum Field Theory, and Statistical Mechanics. The internal consistency of the theory, including issues of renormalization, stability, and compatibility with established symmetries, must be rigorously examined.

On the experimental side, the most immediate opportunities lie in the domain of attosecond physics, ultrafast spectroscopy, and precision measurements of quantum decoherence and entanglement dynamics. If systematic entropic lags or deviations in decoherence rates are observed that align with the predictions of ToE and cannot be accounted for within standard frameworks, this would constitute strong evidence in favor of the entropic field paradigm. Conversely, the absence of such effects within the sensitivity of current and future experiments would impose stringent constraints on the theory’s parameter space and possibly necessitate its revision.

In summary, the Theory of Entropicity proposes a profound re-interpretation of entropy as a fundamental dynamical field, introduces new mathematical structures to encode irreversibility and time asymmetry, and offers concrete experimental predictions at the frontiers of ultrafast and quantum physics. Its ultimate viability will be determined by its ability to withstand theoretical scrutiny and to produce empirical signatures that distinguish it from established theories. Regardless of its final status, ToE provides a rich and technically sophisticated framework for exploring the deep connections between entropy, information, time, and the structure of physical reality.