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

In John Onimisi Obidi’s Theory of Entropicity (ToE), Rényi entropy is used as a foundational mathematical building block to redefine entropy from a statistical measure into a fundamental physical field. 

Key applications of Rényi entropy in Obidi's work include: 

• Integration into the Obidi Action: Rényi entropy is incorporated into the Spectral Obidi Action (SOA). This action principle allows entropy to be treated as a dynamical field variable with its own equations of motion, rather than just a secondary byproduct of particle behavior.
• Constitutive Geometric Bridge: Obidi uses the Rényi–Tsallis
  

  $\alpha$
  -
  

  $q$
  formalism to link informational geometry with physical spacetime. Specifically, the relation
  

  $\alpha =2(1-q)$
  connects the non-extensive deformation of entropy (represented by Rényi and Tsallis orders) to the affine asymmetry of spacetime curvature.
• Unification of Scales: The theory uses Rényi entropy as a "limiting case" or sub-regime within a single "entropic manifold". By tuning parameters like the entropic order
  

  $q$
  or the
  

  $\alpha$
  -connection, the framework can recover different physical regimes, uniting thermodynamic, informational, and quantum geometries.
• Defining the Fabric of Reality: Rényi formulations are encoded into the potential
  

  $\psi \left(\theta \right)$
  of the Entropic Metric. This metric serves as the "new fabric of reality," where what was previously seen as informational curvature (how we process data) is reinterpreted as the physical curvature of the universe. 

Appendix: Extra Matter

 John Onimisi Obidi utilized Rényi Entropy as a foundational element in his Theory of Entropicity (ToE), a framework that proposes entropy, rather than spacetime geometry, as the fundamental, generative field of reality. 

Within this theory, Rényi entropy is used in the following ways: 

• Part of a Unified Entropic Framework: Obidi integrates Rényi and Tsallis entropic orders alongside the Amari–Cencov α-connections to create a singular, unified framework that merges thermodynamic, informational, and quantum geometries.
• Spectral Obidi Action (SOA): Rényi entropy is incorporated into the Spectral Obidi Action, a variational principle that defines the dynamics of the entropy field. In this context, it acts as part of a "spectral backbone" that underpins the structure and evolution of the universe.
• Deformation of Metrics: The Rényi-Tsallis
  

  $\alpha -q$
  formalism is used to mathematically link non-extensive entropy deformation to affine asymmetry, allowing for the transformation of statistical metrics (like Fisher-Rao) into physical metric-affine geometries.
• Defining Entropic Potentials: Obidi uses Rényi (and Tsallis) formulations within the potential
  

  $\psi \left(\theta \right)$
  of the entropic metric to describe how entropy drives curvature and motion. 

In summary, Obidi uses Rényi entropy not just as a statistical measure of uncertainty, but as a core component of a deeper, ontological, and dynamic field theory that derives physical laws from entropy itself.