What are Curvatures and Curvature Constants and Invariants in the Theory of Entropicity (ToE)?

The Theory of Entropicity (ToE) defines curvature as a manifestation of the universal constant of irreversibility,

$\mathrm{\alpha <span\; style="font-family:\; XITSMath-Regular;\; font-size:\; 1px;">𝛼</span>}$

which links information flow to physical spacetime geometry. This concept is expressed through the Obidi Action and the resulting Master Entropic Equation (MEE), rather than a single standalone "curvature invariant formula" like the well-known Riemannian invariants in General Relativity. 

Instead of a simple invariant formula, ToE uses a comprehensive geometric framework: 

• The Curvature Constant (
  

  $\alpha$
  ): The theory posits a universal constant,
  

  $\alpha$
  , which represents the "curvature of irreversibility" in the entropic field. This
  

  $\alpha$
  value is the core of the curvature concept, unifying the statistical (Tsallis/Rényi entropies, often denoted
  

  $q$
  ) and geometric (Amari–Čencov
  

  $\alpha$
  -connections) aspects of the theory. The constitutive relation linking the statistical
  

  $q$
  and geometric
  

  $\alpha$
  is given by
  

  $\alpha =2(1-q)$
  .
• The Obidi Action: The dynamics of the entropic field are governed by a variational principle called the Obidi Action, which is an analogue to the Einstein-Hilbert action in General Relativity. This action,
  

  $S_{Obidi}$
  , incorporates an entropic metric and potential:
  
  

  $S_{Obidi}=\int d^{4}x\sqrt{\left|g\right|}\mathcal{L}(S,\nabla S,g_{\mu \nu })$
  
  where
  

  $\mathcal{L}$
  is the Lagrangian density for the entropic field
  

  $S\left(x\right)$
  . The specific form of the Lagrangian includes terms for the kinetic energy of the entropy field and its coupling to the spacetime metric, effectively generating curvature from entropy gradients.
• The Master Entropic Equation (MEE): Varying the Obidi Action with respect to the metric and the entropic field yields the Master Entropic Equation, the ToE's analogue of Einstein's field equations. The MEE governs how the entropic scalar field
  

  $S(x,t)$
  evolves and couples to matter and geometry. It is a highly nonlinear and iterative equation that describes the continuous self-adjustment of spacetime geometry based on entropy flow. 

In essence, ToE describes curvature not as a fixed property of spacetime itself, but as an emergent, dynamic feature resulting from the flow and gradients of the fundamental entropy field, characterized by the universal constant

$\alpha$

. The curvature is embedded within the complex, iterative solutions of the Master Entropic Equation rather than a single, simple invariant formula.