Mathematical Foundations of the Theory of Entropicity (ToE) with Key Concepts—For the Reader in a Hurry 

The Theory of Entropicity (ToE), proposed by John Onimisi Obidi, posits entropy as the fundamental, dynamic field 

$S(x,t)$

that generates space, time, and gravity, with its mathematics centered on the Obidi Action and the Master Entropic Equation (MEE). This framework uses the Amari-Čencov

$\alpha$

-connection to map information geometry, specifically the Fisher-Rao metric (classical) and Fubini-Study metric (quantum), onto a unified, curved entropic manifold. 

Key Mathematical Components 

• Obidi Action & Master Entropic Equation (MEE): The core equation
  

  $G_{\mu \nu }\left[g\right(S\left)\right]=\eta T_{\mu \nu }^{\left(S\right)}$
  defines space-time curvature as a result of the entropy field
  

  $S\left(x\right)$
  .
• Entropic Manifold: The physical metric
  

  $g_{ij}^{\left(ToE\right)}$
  is an entropy-weighted deformation of the informational metric:
  

  $g_{ij}^{\left(ToE\right)}=e^{S/k_B}{g}_{ij}^{\left(IG\right)}$
  .
• Information Geometry: ToE bridges quantum and classical regimes by merging Shannon (via Fisher-Rao) and von Neumann (via Fubini-Study) entropies.
• Non-extensive Deformation: The relationship between entropy and geometry is linked by
  

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

  $q$
  represents non-extensive parameters.
• Entropy Potential Equation: Defines the flow and gradient of the scalar entropy field. 

The theory reformulates the speed of light, c

$c$

, as the maximum rate of entropic rearrangement. 

Appendix: Extra Matter 

The

Theory of Entropicity (ToE), proposed by John Onimisi Obidi, establishes entropy as a fundamental, dynamical scalar field

$S(x,t)$

that generates spacetime curvature and physical laws. It utilizes the Obidi Action, Master Entropic Equation (MEE), and information geometry (Fisher-Rao and Fubini-Study metrics) to unify gravity, quantum mechanics, and thermodynamics. 

Core Mathematical Components 

• The Master Entropic Equation (MEE): Acts as the central field equation, analogous to Einstein’s field equations, where the entropic curvature
  

  $G_{\mu \nu }\left[g\right(S\left)\right]$
  is determined by the stress-energy tensor
  

  $T_{\mu \nu }^{\left(S\right)}$
  .
• The Obidi Action: A variational principle that defines the dynamics of the entropy field, establishing it as the causal fabric of space, time, and matter.
• Entropic Manifold & Metric: The physical spacetime metric
  

  $g_{ij}^{\left(ToE\right)}$
  is derived from an entropy-weighted deformation of information-theoretic metrics:
  
  

  $g_{ij}^{\left(ToE\right)}=e^{S/k_B}{g}_{ij}^{\left(IG\right)}$
• Amari–Cencov
  

  $\alpha$
  -Connection: Used to replace the Levi-Civita connection to encode the affine asymmetry, representing the irreversible flow of entropy (the arrow of time).
• Non-Extensive Deformation: The link between entropy deformation and geometric asymmetry is defined by the relation
  

  $\alpha =2(1-q)$
  . 

Key Theoretical Implications 

• Entropy as Geometry: The theory posits that entropy gradients create the sensation of motion, gravity, and time, rather than entropy being just a result of disorder.
• Speed of Light (
  

  $c$
  ): Reformulated as the maximum rate of entropic rearrangement within the system.
• Quantum-Classical Unity: The framework integrates Fisher-Rao (classical/Shannon) and Fubini-Study (quantum/von Neumann) metrics to bridge quantum and relativistic scales. 

The

Theory of Entropicity (ToE) posits entropy as a fundamental scalar field

$S(x,t)$

that generates spacetime and gravity through its gradients, mathematically linking information, geometry, and thermodynamics via the Obidi Action and Master Entropic Equation (MEE). It utilizes non-extensive entropy (

$R\acute{e}nyi$

/Tsallis), information geometry (

$\alpha$

-connections), and iterative, self-correcting computational logic to unify physical laws as emergent properties of entropic flow.

 

Core Mathematical Components 

• The Obidi Action: A variational principle,
  

  $I=\int {\mathcal{L}}_E\sqrt{-g}{d}^{4}x$
  , that defines the dynamics of the entropy field, where
  

  ${\mathcal{L}}_E$
  is the entropic Lagrangian.
• Master Entropic Equation (MEE): An entropic analogue to Einstein's field equations, derived from the Obidi Action, mapping entropy evolution to spacetime curvature.
• Information Geometry: The framework employs Amari–Cencov
  

  $\alpha$
  -connections to link quantum (
  

  $Fubini-Study$
  ) and classical (
  

  $Fisher-Rao$
  ) probability distributions, treating physical systems as points on an information manifold.
• Constitutive Relation: Defined as
  

  $\alpha =2(1-q)$
  , linking non-extensive (Tsallis) entropy deformation parameter
  

  $q$
  to affine asymmetry, bridging irreversible information flow with geometric curvature.
• Universal Relation: A core unifying equation,
  

  $\hslash c=k_B{T}_S{\ell }_S$
  , relating quantum, thermodynamic, and geometric constants to bridge information, energy, and curvature.
• Entropic Geodesics: Derived paths representing the natural, non-linear trajectories of systems within the entropic manifold. 

Key Concepts 

• Entropy Field (
  

  $S(x,t)$
  ): Entropy is treated as a continuous, local field rather than just a statistical, macroscopic consequence of disorder.
• Non-Equilibrium Thermodynamics: The theory relies on non-extensive, non-equilibrium entropy formulations.
• Iterative Solution Methods: Unlike closed-form equations, the ToE utilizes iterative, self-correcting mathematical techniques to solve field equations.
• Speed of Light (
  

  $c$
  ): Reinterpreted as the maximum rate of entropic rearrangement. 

The ToE, often described by John Onimisi Obidi, proposes that the universe is a thermodynamic image of an informational continuum. 

The Theory of Entropicity (ToE) defines entropy as a fundamental, dynamic scalar field

$S(x,t)$

that generates space, time, and gravity through its gradients. It utilizes the Obidi Action—a variational principle based on Araki relative entropy—to derive the Master Entropic Equation (MEE), which serves as an entropic analogue to Einstein's field equations. 

Key mathematical foundations include: 

• Geometric Integration: Combines information geometry (Fisher-Rao/Fubini-Study metrics) with the Amari–Cencov
  

  $\alpha$
  -connection to connect non-extensive entropy (
  

  $S_q$
  ) with spacetime curvature.
• Constitutive Relation:
  

  $\alpha =2(1-q)$
  links entropy deformation (
  

  $q$
  -deformation) to affine asymmetry.
• Fundamental Constants: Relates quantum, thermodynamic, and geometric constants via
  

  $\hslash c=k_{B}T{\ell }_S$
  .
• Dynamics: Replaces traditional differential geometry with entropic geodesics and an iterative, self-correcting computational logic. 

References 


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