Theory of Entropicity (ToE)

The Spectral Obidi Action and the Mathematical Unification of Entropic Gravity, Information Geometry, and Generalized Thermodynamics within the Theory of Entropicity (ToE)

I. Introduction: The Ontological Shift and the Entropic Master Framework

The development of the Theory of Entropicity (ToE) represents a profound architectural shift in theoretical physics, moving entropy from the status of a derived, statistical quantity to the position of the fundamental, dynamical field of nature. Traditional frameworks, including those based on entropic gravity, utilize entropy primarily as a diagnostic tool or as a thermodynamic constraint on pre-existing spacetime geometry. ToE, by contrast, posits the entropy field S(x) as the ontological substrate from which spacetime geometry, motion, the arrow of time, and matter itself emerge. This structural reversal provides the necessary conceptual foundation for a unified theory capable of rigorously integrating classical gravity, quantum mechanics, and information geometry.

1.1 The Ontological Primacy of Entropy

The central axiom of ToE asserts that all physical phenomena are emergent properties resulting from the gradients and reconfiguration dynamics of the universal scalar entropy field, S(x). This premise allows ToE to provide a generative principle for dynamics, rather than merely offering a reconstructive description. Spacetime curvature, for instance, is not an independent geometric phenomenon but rather a response to the entropic structure encoded within S(x).

1.2 The Duality of the Obidi Actions: Local Dynamics vs. Global Constraints

The mathematical rigor of ToE is founded upon two complementary variational principles, collectively known as the Obidi Actions. This duality is essential for ensuring that local, differential dynamics adhere to global, spectral, and non-local consistency constraints:

The Local Obidi Action (ILOA): This spacetime integral governs the differential field evolution of S(x), specifying the local interaction of entropy gradients with geometry.

The Spectral Obidi Action (ISOA): This trace functional governs the global, operator-algebraic, and spectral invariants of the entropic field, encapsulating non-local constraints necessary for quantum consistency and the emergence of non-local phenomena.

This duality ensures the internal consistency of the theory: the evolution prescribed by the local dynamics must be compatible with the global spectral geometry defined by the ISOA.

1.3 Core Mathematical Framework: The Entropic Field Equations

The ILOA functions as a scalar-tensor action, coupling the Ricci scalar R to the kinetic and potential terms of the entropy field. A crucial feature is the exponential factor eS/kB which endows the entropy field with a geometric weight, coupling local entropy fluctuations directly to spacetime volume and curvature.

The variation of ILOA with respect to the spacetime metric gμν yields a modified Einstein equation, establishing how entropic stress-energy sources curvature:

Gμν[g]=κTμν(S)

. The variation with respect to the field S(x) itself yields the Master Entropic Equation (MEE), the highly nonlinear field equation governing S(x) dynamics, which includes terms related to entropy flux divergence, self-interaction, and the entropy potential V(S).

The complementary Spectral Obidi Action (ISOA) is defined via the entropic modular operator Δ as a spectral trace functional:

ISOA=−Tr(lnΔ)

. The operator Δ is conceptually analogous to a relative modular operator in Tomita-Takesaki theory, often expressed as Δ=Gg−1, where it compares the deformed entropic geometry G to a reference geometry g. The key is that Δ is a dynamical object, establishing the ISOA as a dynamic variational principle for the relative information between entropic states.

Action/Equation

Mathematical Form (Simplified)

Physical Role

Source

Files

Local Obidi Action (ILOA)

16πG1∫d4x−gR+∫d4x−g

Governs local, differential entropic dynamics and geometric coupling.

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Spectral Obidi Action (ISOA)

−Tr(lnΔ)

Governs global, spectral constraints, and entropic geometry invariants.

Entropic Modular Operator (Δ)

Gg−1

Dynamical bridge comparing current (G) and equilibrium (g) geometries.

Modified Einstein Equation

Gμν[g]=κTμν(S)

Defines how the entropic stress-energy tensor sources spacetime curvature.

II. The Spectral Obidi Action (ISOA): Governing Global Entropic Geometry

The ISOA is the crucial element that enables ToE to unify information geometry formalisms and address non-local phenomena like the dark sector. By operating in the frequency or eigenmode domain, the ISOA enforces global constraints that transcend the pointwise Euler-Lagrange equations derived from the ILOA.

2.1 Formal Structure and Dynamical Relative Entropy

The ISOA is a trace functional defined over the spectrum of the entropic modular operator Δ. This structure is reminiscent of the Araki relative entropy formalism, S(ρ∣∣σ), used in quantum information theory. However, in ToE, this concept is elevated to a fundamental dynamical principle: the action minimizes the informational divergence between entropic field configurations globally. The operator Δ is built to reflect how the entropic field S(x) influences the entire geometry, ensuring the spectral consistency of the field with the resulting spacetime. This dynamic relative entropy principle is a defining feature distinguishing ToE from previous entropy-based gravity models.

2.2 Spectral Origin of the Dark Sector

The global consistency conditions enforced by the ISOA manifest physically as cosmological constants and non-baryonic mass components. These effects arise directly from the non-local degrees of freedom encoded in the spectrum of Δ.

The Spectral Obidi Action is rigorously connected to the origin of the dark sector phenomena. When the eigenvalues λi of Δ deviate from unity (the equilibrium state), they contribute an effective spectral energy density Espec∝∑(λi−1)2. This energy is derived purely from the configuration of the spectral entropic geometry and behaves identically to cold dark matter, clustering gravitationally but remaining pressureless. This indicates that dark matter is not an exotic particle but rather a manifestation of the non-local geometric constraints imposed by the ISOA on the entropic field.

Furthermore, the emergence of a small, positive cosmological constant (Λent) is also tied to the ISOA. In related derivations, a constraint field G(x) (an auxiliary field introduced in Bianconi's work) is identified as a Lagrange multiplier enforcing the global conservation of entropy flux derived from the ISOA. A tiny violation or relaxation of this global entropic equilibrium results in residual entropic pressure that acts as vacuum energy, yielding Λent>0. The existence of dark matter and dark energy are thereby unified under the principle that they represent the non-equilibrated spectral properties of the entropic field S(x).

III. Information-Geometric Unification: αConnections and Entropic Metrics

The fundamental mathematical achievement of the ISOA is its capacity to generalize and unify the seemingly disparate formalisms of generalized entropies, quantum geometry, and statistical geometry through the framework of information geometry. This unification is controlled by the continuous entropic index α.

3.1 The Entropic Index α: The Continuous Deformation Parameter

The index α serves as a continuous deformation parameter within ToE, dictating the information-geometric structure of the entropic manifold MS. Varying α continuously interpolates between different definitions of entropy and affine connections, thereby establishing a single geometric principle for all entropic and informational structures. In the most general formulation, α is even promoted to a dynamical field α(x), allowing the fundamental information principle itself to vary across spacetime.

3.2 Unification of Generalized Entropies (Tsallis Sq and Rényi Hα)

ToE unifies the non-extensive Tsallis entropy Sq and the generalized Rényi entropy Hα by relating their respective parameters to the entropic index α.

Tsallis entropy is naturally incorporated by setting α=q, where q is the Tsallis index. The action functional ILOA incorporates this choice through measure factors like eαS/kB, which act as escort distributions, ensuring that the statistics of the entropic field fluctuations are intrinsically non-extensive when α=1.

Rényi entropy appears when the action is formulated in the spectral domain using a Rényi divergence as the measure of state difference. The trace functional TrΦ(DS) is constructed such that it is structurally akin to ln∑piα, directly yielding the Rényi entropy formula Hα. The theory establishes a direct relationship: selecting a non-extensive thermodynamic measure (Tsallis) mathematically mandates a corresponding spectral geometry structure (Rényi) via the common parameter α when α=1.

3.3 Amari-Čencov Formalisms and Entropic Irreversibility

The full geometry of the entropic manifold MS is governed by the family of Amari α-connections, ∇(α). These connections are included explicitly in the curvature term R(Gα,∇(α)) within the unified ToE action. Extremizing this action ensures that entropic variations follow α-geodesics when mapped to the information manifold.

A profound consequence arises when the index α deviates from zero. For α=0, the dual connections ∇(α) and ∇(−α) are distinct. This geometric asymmetry (dualistic geometry) is mathematically rigorous and non-negotiable, imposing an intrinsic distinction in how entropic gradients propagate forward versus backward. This mathematical asymmetry rigorously establishes the dynamical arrow of time in ToE; irreversibility and entropy production are not statistical artifacts but are embedded directly into the foundational geometric dynamics of the entropic field.

3.4 The Unified Entropic Metric: Fisher-Rao and Fubini-Study

The entropic manifold MS is endowed with a unified metric Gα(S) that simultaneously measures classical statistical uncertainty and quantum coherence.

The Fisher-Rao metric (GFR), which measures the infinitesimal distinguishability of nearby probability distributions, is recovered as the classical sector of Gα at α→0 or α→1. It governs classical statistical fluctuations of the entropy field.

The Fubini-Study metric (GFS), the natural Riemannian metric on the space of pure quantum states, is incorporated as the quantum sector block of Gα. This inclusion ensures that ToE accounts for quantum coherence, entanglement, and phase information within its geometric framework. The unification of GFR and GFS within a single α-parameterized entropic metric Gα is a significant step toward integrating classical statistical geometry and quantum state geometry into a single geometric principle.

IV. Ginestra Bianconi’s Gravity as the Shannon-Fisher Limit of ToE

The claim that ToE generalizes Bianconi’s "Gravity from Entropy" is demonstrated by showing that Bianconi's action is mathematically recovered as a specific, highly constrained limit of the Obidi Actions.