How Did Obidi Use Tsallis Entropy in the Mathematical Development of the Theory of Entropicity (ToE)?

In John Onimisi Obidi’s Theory of Entropicity (ToE), first introduced in 2025, Tsallis entropy is used as a parameterization for deforming informational geometry into physical spacetime geometry. 

Within this framework, Obidi utilizes the Rényi-Tsallis

$\alpha$

-

$q$

formalism to establish a mathematical bridge between information flow and physical curvature through the following methods: 

• Geometric Deformation: Obidi uses Tsallis entropy to transform statistical metrics—such as the Fisher-Rao (classical) and Fubini-Study (quantum) metrics—into physical metric-affine geometries.
• The Constitutive Relation: He introduces the mathematical link
  

  $\alpha =2(1-q)$
  , where
  

  $q$
  (the Tsallis non-extensivity parameter) is tied to
  

  $\alpha$
  (the Amari-Čencov affine connection). This relation maps non-extensive entropy deformation to affine asymmetry in spacetime.
• Ontological Scalar Field: Tsallis entropy is integrated into the Obidi Action, a variational principle where entropy is treated as an ontological scalar field
  

  $S(x,t)$
  . This results in the Master Entropic Equation (MEE), which describes how informational curvature becomes physical gravity and motion.
• Unification of Scales: By incorporating Tsallis entropy, the theory absorbs non-extensive statistical mechanics as a special case of a broader entropic manifold, allowing it to describe both local differential dynamics and global spectral constraints. 

Detailed expositions on these mathematical foundations can be found in Obidi's research on TechRxiv and the Encyclopedia of World Problems and Human Potential. 

Holography Interpreted in the Theory of Entropicity (ToE)

The Theory of Entropicity (ToE), formulated by John Onimisi Obidi, interprets holography not merely as a boundary-volume duality, but as a direct manifestation of a fundamental, dynamic entropic field, 

$S\left(x\right)$

, which underlies all reality. In this framework, the universe is not just "projected" from a boundary; it is an active, self-correcting computation continuously updating its informational structure through local entropy flow, with spacetime itself emerging from this process. 

Here is an analysis of how ToE interprets holography: 

1. Entropy as the Fundamental Field 

• From Boundary to Bulk: Traditional holography (AdS/CFT) suggests that
  

  $n$
  -dimensional gravity is encoded on an
  

  $(n-1)$
  -dimensional surface. ToE extends this by treating entropy
  

  $S\left(x\right)$
  as the fundamental physical field (an "entropic substrate") that generates curvature, motion, and time in the bulk.
• Fundamental vs. Secondary: ToE flips the conventional view where entropy is a statistical byproduct of established dynamics. Instead, entropy is the starting point from which physical laws emerge. 

2. Holography as Entropic Dynamics 

• Obidi Actions: ToE introduces the Local and Spectral Obidi Actions to define how the entropic field
  

  $S\left(x\right)$
  evolves. Holographic principles, such as holographic entropy bounds, are reinterpreted as specific boundary conditions within this broader entropic field theory.
• Emergent Geometry: In ToE, spacetime curvature is not an independent fabric but a derivative of entropic flow density. The "holographic" projection is interpreted as the way the entropic field expresses its own internal reconfiguration (curvature). 

3. Key Interpretations 

• Reinterpretation of
  

  $c$
  (Speed of Light): ToE proposes that the speed of light
  

  $c$
  is not just a geometric constant but the "heartbeat of existence"—the maximum rate at which the entropic field can rearrange information/energy. This defines the limit of holographic projection (how quickly information can update the 3D projection).
• Beyond Pseudo-Entropy: While some holographic models use "pseudo-entropy" to map boundary data to bulk gravity, ToE argues that these are just "holographic shadows." ToE provides a more comprehensive, nonlinear, and time-asymmetric framework that absorbs these models.
• "No-Rush" Theorem: ToE includes the "No-Rush" theorem, which states that entropic redistribution (information updating) cannot happen instantaneously. This provides a physical, rather than just geometrical, basis for causality in a holographic universe. 

4. Comparison with Other Theories 

• Verlinde's Entropic Gravity: ToE goes beyond Verlinde’s work by moving from "entropic force" to a more rigorous, foundational field theory based on the Obidi action, replacing the coarser thermodynamic approach with a detailed, fundamental, and unified framework.
• Ryu-Takayanagi Formula: ToE absorbs holographic entanglement entropy results, showing that these are consistent with the "Master Entropic Equation" derived from the entropic field. 

In essence, ToE views the universe as an "entropic conversation"—a vast, self-correcting computation—where the "holographic" nature of reality is just one of many ways the entropic field manages its internal, evolving structure.