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

Further Expositions on the Theory of Entropicity (ToE) and Ginestra Bianconi's Gravity from Entropy: How the Theory of Entropicity (ToE) Unifies Spectral Entropy with Tsallis, Rényi, Fisher–Rao, Fubini–Study, and Amari–Čencov Formalisms

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 ( $I_{LOA}$ ): This spacetime integral governs the differential field evolution of $S(x)$ , specifying the local interaction of entropy gradients with geometry.
The Spectral Obidi Action ( $I_{SOA}$ ): 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 $I_{SOA}$ .

1.3 Core Mathematical Framework: The Entropic Field Equations

The $I_{LOA}$ 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 $e^{S/k_B}$ which endows the entropy field with a geometric weight, coupling local entropy fluctuations directly to spacetime volume and curvature.

The variation of $I_{LOA}$ with respect to the spacetime metric $g_{\mu\nu}$ yields a modified Einstein equation, establishing how entropic stress-energy sources curvature:

G_{\mu\nu}[g]=\kappa T_{\mu\nu}^{(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 ( $I_{SOA}$ ) is defined via the entropic modular operator $\Delta$ as a spectral trace functional:

I_{SOA} = -Tr(\ln\Delta)

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

Action/Equation	Mathematical Form (Simplified)	Physical Role	Source
Local Obidi Action ( $I_{LOA}$ )	$\frac{1}{16\pi G}\int d^{4}x\sqrt{-g}R+\int d^{4}x\sqrt{-g}$	Governs local, differential entropic dynamics and geometric coupling.
Spectral Obidi Action ( $I_{SOA}$ )	$-Tr(\ln\Delta)$	Governs global, spectral constraints, and entropic geometry invariants.
Entropic Modular Operator ( $\Delta$ )	$Gg^{-1}$	Dynamical bridge comparing current ( $G$ ) and equilibrium ( $g$ ) geometries.
Modified Einstein Equation	$G_{\mu\nu}[g]=\kappa T_{\mu\nu}^{(S)}$	Defines how the entropic stress-energy tensor sources spacetime curvature.

II. The Spectral Obidi Action ( $I_{SOA}$ ): Governing Global Entropic Geometry

The $I_{SOA}$ 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 $I_{SOA}$ enforces global constraints that transcend the pointwise Euler-Lagrange equations derived from the $I_{LOA}$ .

2.1 Formal Structure and Dynamical Relative Entropy

The $I_{SOA}$ is a trace functional defined over the spectrum of the entropic modular operator $\Delta$ . This structure is reminiscent of the Araki relative entropy formalism, $S(\rho||\sigma)$ , 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 $\Delta$ 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 $I_{SOA}$ 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 $\Delta$ .

The Spectral Obidi Action is rigorously connected to the origin of the dark sector phenomena. When the eigenvalues $\lambda_i$ of $\Delta$ deviate from unity (the equilibrium state), they contribute an effective spectral energy density $E_{spec} \propto \sum (\lambda_i - 1)^2$ . 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 $I_{SOA}$ on the entropic field.

Furthermore, the emergence of a small, positive cosmological constant ( $\Lambda_{ent}$ ) is also tied to the $I_{SOA}$ . 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 $I_{SOA}$ . A tiny violation or relaxation of this global entropic equilibrium results in residual entropic pressure that acts as vacuum energy, yielding $\Lambda_{ent} > 0$ . 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: $\alpha$ -Connections and Entropic Metrics

The fundamental mathematical achievement of the $I_{SOA}$ 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 $\alpha$ .

3.1 The Entropic Index $\alpha$ : The Continuous Deformation Parameter

The index $\alpha$ serves as a continuous deformation parameter within ToE, dictating the information-geometric structure of the entropic manifold $M_S$ . Varying $\alpha$ 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, $\alpha$ is even promoted to a dynamical field $\alpha(x)$ , allowing the fundamental information principle itself to vary across spacetime.

3.2 Unification of Generalized Entropies (Tsallis $S_q$ and Rényi $H_\alpha$ )

ToE unifies the non-extensive Tsallis entropy $S_q$ and the generalized Rényi entropy $H_\alpha$ by relating their respective parameters to the entropic index $\alpha$ .

Tsallis entropy is naturally incorporated by setting $\alpha = q$ , where $q$ is the Tsallis index. The action functional $I_{LOA}$ incorporates this choice through measure factors like $e^{\alpha S/k_B}$ , which act as escort distributions, ensuring that the statistics of the entropic field fluctuations are intrinsically non-extensive when $\alpha \ne 1$ .

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 $Tr\Phi(\mathcal{D}_S)$ is constructed such that it is structurally akin to $\ln \sum p_i^{\alpha}$ , directly yielding the Rényi entropy formula $H_\alpha$ . 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 $\alpha$ when $\alpha \ne 1$ .

3.3 Amari-Čencov Formalisms and Entropic Irreversibility

The full geometry of the entropic manifold $M_S$ is governed by the family of Amari $\alpha$ -connections, $\nabla^{(\alpha)}$ . These connections are included explicitly in the curvature term $R(G_\alpha, \nabla^{(\alpha)})$ within the unified ToE action. Extremizing this action ensures that entropic variations follow $\alpha$ -geodesics when mapped to the information manifold.

A profound consequence arises when the index $\alpha$ deviates from zero. For $\alpha \ne 0$ , the dual connections $\nabla^{(\alpha)}$ and $\nabla^{(-\alpha)}$ 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 $M_S$ is endowed with a unified metric $G_\alpha(S)$ that simultaneously measures classical statistical uncertainty and quantum coherence.

The Fisher-Rao metric ( $G_{FR}$ ), which measures the infinitesimal distinguishability of nearby probability distributions, is recovered as the classical sector of $G_\alpha$ at $\alpha \to 0$ or $\alpha \to 1$ . It governs classical statistical fluctuations of the entropy field.

The Fubini-Study metric ( $G_{FS}$ ), the natural Riemannian metric on the space of pure quantum states, is incorporated as the quantum sector block of $G_\alpha$ . This inclusion ensures that ToE accounts for quantum coherence, entanglement, and phase information within its geometric framework. The unification of $G_{FR}$ and $G_{FS}$ within a single $\alpha$ -parameterized entropic metric $G_\alpha$ 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.

4.1 Bianconi's Action and the $\alpha=1$ Limit

Bianconi’s theory derives gravity from the quantum relative entropy $D_{KL}(g||g^m)$ between a spacetime metric $g$ and a matter-induced metric $g^m$ . This relative entropy structure is related to Araki's formalism and is equivalent to the classical Shannon-Fisher information measure in the limit of small metric perturbations.

ToE formally reduces to Bianconi's framework by imposing two conditions on the entropic field $S(x)$ :

Shannon/Fisher Limit ( $\alpha \to 1$ ): This choice ensures the entropic geometry is governed by the standard, extensive Shannon entropy and the Fisher-Rao metric, eliminating non-extensive and irreversible $\alpha$ -corrections.
Near-Equilibrium Expansion: This restricts the dynamics to small fluctuations $\phi(x) = S(x) - S_0$ around a constant background entropy $S_0$ , leading to a linearized field regime ( $\nabla S \ll 1$ ).

4.2 The Quadratic Expansion and Formal Correspondence

The lowest-order expansion of the $I_{LOA}$ kinetic term ( $\frac{\chi}{2}e^{S/k_B}(\nabla S)^2$ ) in the near-equilibrium regime yields a quadratic functional of the field perturbation $\phi$ :

I_{LOA}[\phi] \approx \frac{\chi e^{S_0/k_B}}{2} \int d^4x \sqrt{-g^{(0)}} g^{(0)\mu\nu} \partial_\mu \phi \partial_\nu \phi

ToE establishes a rigorous mathematical correspondence: this quadratic kinetic term is precisely the leading-order approximation of the Fisher information metric, which, for metric perturbations, becomes the quantum relative entropy $D_{KL}(g||g^m)$ that forms the basis of Bianconi's action. Therefore, Bianconi’s 'Gravity from Entropy' is demonstrated to be the linearized, weak-field, classical, and extensive ( $\alpha=1$ ) projection of the fundamentally nonlinear, entropic field dynamics described by the Obidi Actions.

4.3 Interpretation of Bianconi’s G-Field and Emergent Cosmological Terms

Bianconi introduced an auxiliary G-field, $G(x)$ , as a Lagrange multiplier to enforce consistency, which subsequently yielded an emergent cosmological constant $\Lambda$ and effective dark matter terms. ToE assigns a clear physical role to this mechanism: the G-field is the Lagrange multiplier necessary to enforce the global entropic constraint derived from the $I_{SOA}$ spectrum, ensuring the local entropy density couples proportionally to the spacetime volume element ( $e^{S/k_B} \propto \sqrt{-g}$ ).

This constraint mechanism provides the origin of the dark sector in Bianconi’s model:

A non-zero vacuum energy $\Lambda > 0$ arises from a tiny, consistent deviation from this global entropy equilibrium constraint, acting as a small residual entropic pressure.
The effective dark matter terms arise from the dynamical response of $G(x)$ to non-equilibrated spectral degrees of freedom ( $\Delta$ eigenvalues) on large scales, mimicking the clustering behavior of pressureless dust.

This chain of relationships establishes that the cosmological effects hinted at by Bianconi's formulation originate from the global spectral dynamics governed by the $I_{SOA}$ .

V. Structural Superiority and Empirical Distinctions

The comprehensive structure of ToE, involving both $I_{LOA}$ and $I_{SOA}$ , positions it as a generative field theory structurally superior to purely reconstructive models like holographic pseudo-entropy. The latter framework, developed by Takayanagi, Kusuki, and Tamaoka, provides a striking equivalence between pseudo-entropy variations and the linearized Einstein equation in de Sitter space ( $dS_3$ ) but is limited to boundary diagnostics and kinematic constraints.

5.1 ToE as a Generative Field Theory vs. Kinematic Reconstruction

Holographic pseudo-entropy is defined as a functional of non-Hermitian density matrices in a non-unitary conformal field theory ( $CFT_2$ ) and is reconstructed through the complexified area of bulk extremal curves. Its central dynamical relation is the Klein-Gordon (KG) equation satisfied by pseudo-entropy variations on the kinematic $dS_2$ space: $(\Box_{dS_2} - m^2) \delta S_{pseudo} = 0$ .

ToE demonstrates that this holographic result is the boundary-projected, linearized shadow of the full, nonlinear entropic field dynamics. The KG equation for pseudo-entropy variations is the linearized limit of the Master Entropic Equation (MEE) when restricted to the 2D kinematic boundary space.

The complexified geodesics used in the pseudo-entropy reconstruction are similarly shown to be special cases of ToE's entropic geodesics, arising when the entropic field $S(x)$ is restricted to a holographic, analytically-continued boundary slice. The pseudo-entropy framework, therefore, does not generate geometry; it merely reconstructs the linear response of geometry from boundary information. ToE, conversely, is a bulk-first theory that generates geometry intrinsically from the entropic field $S(x)$ .

5.2 Intrinsic Irreversibility and the Entropic Time Limit (ETL)

A key structural advantage of ToE is its fundamental inclusion of irreversibility. The nonlinear MEE, particularly due to its coupling constants (such as $\chi(\Lambda)$ ) and the underlying dualistic nature of the $\alpha$ -connections ( $\alpha \ne 0$ ), is intrinsically time-asymmetric. This inherent irreversibility establishes the dynamical arrow of time at the level of the fundamental field.

This entropic flow constraint leads to the formulation of the No-Rush Theorem, which places a universal, finite bound on the speed of entropic reconfigurations. This constraint, termed the Entropic Time Limit (ETL), governs all interactions from the smallest to the largest scales.

A remarkable empirical consequence arises in the quantum domain: the ETL predicts a finite, non-zero time required for the formation of quantum entanglement. This prediction, approximately $\Delta t_{ent} \approx 232$ attoseconds, is consistent with precise measurements in ultrafast quantum optics. This successful prediction links fundamental geometric asymmetry (via the $\alpha$ -connections) directly to observable quantum dynamics, a feat unachievable by the kinematical pseudo-entropy framework.

5.3 Phenomenological Predictions Beyond Linearized Gravity

ToE yields numerous phenomenological predictions that are inaccessible to linearized or boundary-based gravity models:

Gravitational Corrections

The entropic field $S(x)$ modifies gravitational trajectories by introducing an entropic force term in the geodesic equation. This leads to measurable nonlinear corrections to General Relativity :

Gravitational Lensing: The deflection angle $\Delta \phi$ receives an entropic correction $\Delta \phi_{ent}$ proportional to the line integral of the entropic gradient $I \propto \int \nabla_{\perp}S dl$ .
Perihelion Precession: Orbital dynamics are modified by an entropic force term $F_{ent}(u)$ in the Binet equation, predicting corrections to the perihelion shift beyond the standard GR prediction.

Dark Sector Mechanism

As discussed in Section II, ToE provides an intrinsic, unified explanation for the dark sector, avoiding the introduction of new particles or ad-hoc cosmological constants :

Dark Energy: The entropic vacuum energy, $\Lambda_{ent} \propto \langle (\nabla S)^2 \rangle$ , sourced by residual entropic field tension, naturally provides a small, positive, and dynamically evolving cosmological constant.
Dark Matter: The energy density derived from spectral deviations of the modular operator $\Delta$ , $E_{spec} \propto \sum (\lambda_i - 1)^2$ , behaves as pressureless dark matter.

Black Hole Microphysics

The Spectral Obidi Action ( $I_{SOA}$ ) predicts deviations from semiclassical black hole thermodynamics. Microstates are predicted to correspond to the product of the modular operator eigenvalues $N_{micro} \propto \prod \lambda_i$ , yielding corrections to the Bekenstein-Hawking entropy $S_{BH} = A/4 + \delta S_{ent}$ . This suggests that Hawking radiation will exhibit non-thermal corrections due to spectral broadening, making ToE testable via gravitational wave observations that probe near-horizon physics.

Table 3: Structural Comparison: ToE, Bianconi, and Pseudo-Entropy (Synthesized)

Feature	Theory of Entropicity (ToE)	Bianconi’s Gravity from Entropy	Holographic Pseudo-Entropy
Entropy Status	Ontological Field $S(x)$	Derived Quantity (Relative Entropy)	Boundary Diagnostic (Functional $S_{pseudo}$ )
Governing Principle	$I_{LOA} + I_{SOA}$ (Nonlinear, Field-Based)	$D_{KL}(g
Geometric Scope	Unified $G_\alpha$ (Fisher-Rao + Fubini-Study)	Metric Comparison (Pure Fisher limit)	Kinematic $dS_2$ / Complex Geodesics
Time Dynamics	Intrinsic, Irreversible $\alpha$ -Dynamics (ETL)	Time-Symmetric (Lacks explicit arrow)	Emergent, Kinematical Time
Dark Sector Origin	Spectral Deviations $Tr$	Requires auxiliary G-field	Absent (No mechanism)
Falsifiability	High (ETL, $\Lambda_{ent}(t)$ , GR corrections)	Limited (Only near-equilibrium)	Low (Purely holographic consistency)

VI. Conclusion: Unification, Structural Integrity, and Future Directions

The investigation into the Spectral Obidi Action ( $I_{SOA}$ ) confirms its role as the unifying backbone of the Theory of Entropicity. The $I_{SOA}$ rigorously links classical statistical mechanics, quantum information theory, and gravitational dynamics by enforcing global entropic constraints on the bulk field $S(x)$ .

6.1 The Synthesis of Formalisms via the Spectral Obidi Action

The $I_{SOA}$ unifies the target formalisms by establishing a coherent information-geometric structure for the entropic manifold:

Generalized Entropies (Tsallis, Rényi): Unified through the entropic index $\alpha$ , which controls both the non-extensive measure (Tsallis) and the spectral constraints (Rényi).
Information Geometry (Amari-Čencov, Fisher-Rao, Fubini-Study): Unified through the dynamically included $\alpha$ -connections and the composite entropic metric $G_\alpha$ , which merge classical statistical geometry and quantum state geometry into a single structure. The resulting $\alpha$ -geodesics dynamically encode the fundamental irreversibility of the universe.
Entropic Gravity (Bianconi): Rigorously derived as the $\alpha=1$ , weak-field, linearized approximation of the full $I_{LOA}$ and $I_{SOA}$ . The ambiguity of Bianconi's G-field is resolved by identifying it as the Lagrange multiplier enforcing the global spectral constraint from the $I_{SOA}$ .

6.2 Structural Integrity and the Post-Holographic Paradigm

ToE is established as a generative, bulk-first field theory. The crucial implication of this structural integrity is that holographic reconstruction methods, such as the pseudo-entropy framework, are successful precisely because they are sampling the linearized, boundary-projected shadows of the universal, nonlinear entropic field dynamics. ToE encompasses the limits of successful entropic gravity models but extends significantly into domains where they are silent, including: fundamental quantum time limits, the self-consistent dark sector mechanism, and nonlinear gravitational corrections. The theory fulfills the ambitious goal of establishing entropy as the fundamental field, generating all of geometry, quantum dynamics, and causal structure.

6.3 Future Directions and Open Mathematical Problems

Further research demands the full mathematical rigorization of the nonlinear dynamics. Key challenges include proving the existence, uniqueness, and stability of solutions for the highly nonlinear Master Entropic Equation. Canonical quantization of the field $S(x)$ is necessary to complete the description of its predicted bosonic and fermionic excitations. On the empirical front, future work will focus on designing specific experimental verification of the unique predictions of ToE, particularly the Entropic Time Limit (ETL) via attosecond probes and the spectral dark matter signatures in high-resolution astronomical data. This ongoing program aims to move the Theory of Entropicity from a strong theoretical framework to a fully tested and validated foundation of physics.