1. Introduction

Entropy as the Underlying Regulator of Light Speed in the Theory of Entropicity (ToE)

According to the Theory of Entropicity (ToE), the constancy of the speed of light—a foundational postulate in Einstein’s theory of relativity—is not an unexplained given of nature but an emergent consequence of a deeper entropic principle. In ToE, entropy is redefined not merely as a statistical measure of disorder, but as a dynamic and universal field—denoted

—that governs the structure and evolution of space, time, and interaction.

This entropic field plays two critical roles:

1. Temporal Regulation: It establishes an irreversible arrow of time by driving all physical systems toward higher entropy in a path-dependent and asymmetric manner.

Kinematic Constraint: It enforces a maximum propagation rate for causal influence, which manifests in our observed universe as the speed of light

ToE proposes that no physical interaction, signal, or object can move faster than the rate allowed by the entropy field itself^{[1][2][3][4][5][6]}. This rate is not arbitrary—it is encoded in the structure of entropy flow in the universe. The speed of light is thus the entropic speed limit:

In this view, the constancy of

arises not from spacetime geometry alone but from the underlying entropic constraint embedded in the field structure of the cosmos. Light, as a massless excitation, follows the least entropic resistance path, and its velocity is dictated by the local and global configuration of entropy.

This leads to several profound implications:

Implications of ToE on the Speed of Light and Causality

1. Entropic Explanation of Special Relativity:
   
   Time dilation and length contraction are interpreted in ToE as manifestations of local entropic field distortions. Systems moving near $c$ experience increased entropy resistance, which slows internal clocks and compresses spatial intervals—not because of Lorentzian spacetime, but because of the increased entropic cost of motion.

2. No Superluminal Interactions:
   
   Faster-than-light (FTL) propagation is forbidden not by geometric constraints alone but by the entropic No-Rush Theorem. That is, no process can occur faster than the entropic field permits, because the field must first establish the conditions for that process to happen.

3. Quantum Measurement and Entropy Speed Limit:
   
   The finite speed at which entanglement or wavefunction collapse can occur (e.g., the 232 attosecond constraint) is seen as a reflection of the entropic time constraint—i.e., collapse and information exchange must obey the entropic propagation limit, not just relativistic $c$.

4. General Relativity as an Emergent Entropic Geometry:
   
   ToE allows one to derive Einstein's field equations as an effective entropic metric theory, where the curvature of spacetime is simply how the entropy field encodes constraints on motion and interaction.

Conclusion

In the Theory of Entropicity, the speed of light (c) is not an unexplained constant but the natural maximum flow rate of the entropic field, which underlies all physical processes. The light speed barrier is thus a thermodynamic consequence of entropy’s universal governance over time, causality, and motion. It is not geometry that limits entropy—but entropy that shapes and constrains geometry. This is a crucial and vital point in the Theory of Entropicity (ToE).

2. What We Want To Show

In ToE, the universal constant “c” is not postulated but emerges as the characteristic (null) speed of disturbances in the entropy field and of any physical signal constrained by that field.

We do this by:

1. Writing (or recalling) the ToE action and its Euler–Lagrange equation for S(x).

2. Linearizing around a background to read off the wave operator.

3. Identifying the characteristic cone (light-cone) of that operator.

4. Showing that requiring matter, information, and entropic perturbations to share the same causal cone fixes the propagation speed to c.

5. Optionally cross-checking with an entropic flux law that reproduces Maxwell’s wave speed 1/√(ε0 μ0).

3. Starting From ToE's Master Entropic Equation (MEE)

We take the ToE's MEE schematic (already established in another submission)^[7][8]:

Let us group the -kinetic terms by defining

Thus the total kinetic piece is

Varying with respect to

and ignoring surface terms gives the Euler–Lagrange equation:

where

4. Linearization Around Homogeneous Background

Let

with

constant. Then

and

To first order:

For free entropic waves (neglecting sources and mass term):

So perturbations of the entropy field obey a wave equation with the same metric

 up to the constant K0. That constant rescales the field but does not change the null cone of

. 

The condition for characteristics obeys: 

which is the null cone of

 The entropy perturbation’s fastest signal travels along null directions of

Therefore, its characteristic speed is the same “c” that defines null intervals in

5. Characteristic Speed in Local Inertial Frame

In local inertial coordinates

we have the linearized equation:

such that as we divide by

which we set as equal to

we readily obtain:

which is the standard wave equation with signal speed

Thus, in the Theory of Entropicity(ToE), entropy field supports waves that move at a given speed c. If we now impose ToE’s No-Rush Theorem^[1][9] (“no process outruns the entropic field”), then all interactions are bounded by this same speed of c. We see that from simple entropic arguments of the Theory of Entropicity(ToE), we have arrived at Einstein's constancy of the speed of light in his beautiful Theory of Relativity (ToR).

6. Shared Null Cone Requirement with Matter and Light

In order to prevent causal paradoxes, we impose the following demands on light and matter:

• Matter couplings () feel as their kinetic metric (as usual).

• Entropic disturbances feel the same null cone (since their kinetic term is proportional to ).

• Electromagnetic field (if we recast it entropically) also propagates on the same cone.

Therefore, everyone shares the same characteristic speed. If EM waves didn’t, then we would have super- or subluminal mismatches, thus violating ToE’s causal-entropy consistency. That forces the “entropic cone” and “EM cone” to coincide → velocity = c.

7. Constitutive Flux–Law Derivation

Now, define the entropy flux 4-vector as:

with conductivity

The conservation/production equation:then linearizes (for) to

where is entropic capacity.

Let us further define

To prevent superluminal signals (No-Rush Theorem), we can demand that (that is, demand that all physical signals cannot outrun

 (No-Rush Theorem)):

where we know from classical physics that

that is

which directly fulfills the natural wave constant (c) in Maxwell's Electromagnetism (EM). Thus, in the Theory of Entropicty (ToE), radiation is just a special “entropic excitation”. This fixes χ0/C0 numerically and ties ToE's entropic constants back to measured EM constants.

8. Null Entropic Geodesic

Another way for us to derive the speed of light in the Theory of Entropicity (ToE) is for us to start by using the entropic action:

with for massless excitations along the path (that is, null entropic trajectory). This yields

which is the null condition whose coordinate speed is

Hence, the trajectory’s coordinate speed is c.

9. The Attosecond Constraint Cross-Check

We have the Empirical entanglement formation time

.

For correlation length L,

Observed

never implies

thus reinforcing

as the ceiling for global propagation.