How does  the Theory of Entropicity (ToE) derive the Einstein Relativistic speed of light c from Entropy?

In the Theory of Entropicity (ToE), the speed of light $$c$$ is **not taken as a postulate but derived** as the **maximum propagation speed of entropic disturbances** in the universal entropic field $$S(x,t)$$. The position is that the usual relativistic structure of $$c$$—its constancy, invariance, and role as a causal limit—follows from the constitutive law and constraints of this entropic field [1][2][3].

### Strategy of the derivation

ToE starts from the **Master Entropic Equation (MEE)**, which governs the dynamics of the entropic field $$S(x,t)$$ via the **Obidi Action** and the associated entropy‑driven field equations [3][2]. When the MEE is linearized (for small perturbations), it yields a **wave‑type equation** whose principal part is proportional to the spacetime d’Alembertian $$g^{\mu\nu}\partial_\mu\partial_\nu$$, and the coefficient of that operator is identified with $$c^2$$ [1][2]. Crucially, ToE argues that the dimensionful combination of fundamental constants in that coefficient—$$\hbar, G, k_B$$—forces the characteristic speed of these entropic excitations to **numerically equal the measured $$c$$**, so that  

$$

   \text{entropic stiffness} / \text{entropic inertia} \sim c^2.

$$

Light is then interpreted as a **null entropic excitation**: a massless mode that propagates along the **causal cone defined by the entropic field**, and whose speed is fixed by this stiffness–inertia ratio [2][3].

### Why $$c$$ is constant for all observers

ToE also claims that the **constancy and observer‑independence** of $$c$$ follow from the structure of the entropic field and a **No‑Rush Theorem (NRT)**, which forbids instantaneous change or infinite propagation speed for entropic configurations [1][6]. Because:

- all matter and radiation are embedded in the same entropic field,

- the NRT imposes a **universal finite propagation speed** for entropic information,

- and measuring devices (clocks, rulers) are themselves entropic systems constrained by the same field,

the theory asserts that **all observers must measure the same $$c$$**, without assuming Lorentz invariance as a starting point [1][6]. The symmetry group that preserves the entropic null cones is then shown to coincide with the Lorentz group, so Einstein’s second postulate is recast as a **derivable consequence** of entropic causality rather than a primitive axiom [2][7].

### Conceptual summary

So, in ToE’s language:

- The **speed of light** is the **characteristic speed of null entropic waves** in the entropic field.

- The **value of $$c$$** is fixed by the **ratio of entropic stiffness to entropic inertia**, which involves $$\hbar, G, k_B$$.

- The **invariance of $$c$$** follows from the fact that the **entropic field and the No‑Rush Theorem** impose the same causal limit on all observers.

In other words, ToE reinterprets the “speed of light” as the **maximum rate at which entropy can be redistributed and shared causally** in the universe, and derives $$c$$ from that thermodynamic‑entropic structure rather than from pure geometry [1][2][3].