How can the theory of entropicity that says everything comes from entropy now have an expression that is a function of space and time?

⭐ 1. “Everything comes from entropy” does NOT mean entropy has no arguments

When ToE says:

Entropy is the fundamental field from which geometry, matter, and dynamics emerge

it is making an ontological claim, not a coordinate claim.

In physics, a field can be fundamental and still be expressed as a function of coordinates.

For example:

• In General Relativity, the metric $g_{\mu \nu }(x)$ is fundamental, yet it is written as a function of spacetime coordinates.

• In Quantum Field Theory, the electron field $\psi (x)$ is fundamental, yet it is written as a function of spacetime.

• In classical electromagnetism, the electromagnetic field $A_{\mu }(x)$ is fundamental, yet it is written as a function of spacetime.

Being fundamental does not mean “not a function of anything.” It means “not derived from anything deeper.”

Entropy in ToE is the substrate, but the substrate still has a configuration over the manifold.

⭐ 2. ToE says spacetime emerges from entropy — so why does S(x) use x?

This is the subtle part, and it’s where ToE departs from classical thinking.

✔ In ToE, spacetime is not an independent background.

✔ The coordinates $x^{\mu }$ are labels for points on the entropic manifold.

✔ The manifold exists because the entropic field exists.

Think of it this way:

• In GR, spacetime is a differentiable manifold with a metric.

• In ToE, the manifold is a differentiable structure induced by the entropic field.

So when we write:

$$S(x^{\mu })$$

we are not saying:

“Entropy depends on spacetime.”

We are saying:

“The coordinates $x^{\mu }$ are the chart labels of the entropic manifold itself.”

In other words:

The manifold is the stage on which the entropic field is defined —

and the stage itself is generated by the entropic field.

This is not circular. It is exactly how General Relativity works:

• The metric determines curvature.

• Curvature determines geodesics.

• Geodesics define the structure of spacetime.

• Yet the metric is still written as $g_{\mu \nu }(x)$.

ToE simply pushes this logic one level deeper.

⭐ 3. Why S(x) is necessary in any field theory

A field must assign a value to each point of a manifold.

That’s what a field is.

If entropy is a field, then:

• It must have a value at each point.

• Those points must be labeled.

• The labels are the coordinates $x^{\mu }$.

This is not a dependency. It is a definition.

Writing $S(x)$ is simply the mathematical way of saying:

“Entropy has a value at each point of the entropic manifold.”

⭐ 4. ToE’s deeper interpretation: x is not prior to S — S induces x

This is the part that makes ToE unique.

In ToE:

• The entropic field $S$ is primary.

• The manifold $\mathcal{M}$ is secondary.

• The coordinates $x^{\mu }$ are tertiary.

The correct conceptual order is:

$$S\Rightarrow \text{Information Geometry}\Rightarrow \text{Manifold Structure}\Rightarrow x^{\mu }$$

So when we write:

$$S(x^{\mu })$$

we are not saying:

“Entropy depends on spacetime.”

We are saying:

“Spacetime is the coordinate representation of the entropic field.”

This is exactly analogous to:

• Writing the metric $g_{\mu \nu }(x)$ even though the metric defines spacetime.

• Writing the wavefunction $\psi (x)$ even though the Hilbert space structure is defined by the wavefunction itself.

⭐ 5. A simple analogy: A painting and its canvas

Imagine a painting that creates its own canvas.

• The painting is the entropic field.

• The canvas is the manifold.

• The coordinates are the grid lines drawn on the canvas.

The painting exists first. The canvas is the structure the painting induces. The grid is just how we describe it.

Writing $S(x)$ is like saying:

“Here is the color of the painting at coordinate (x,y).”

It does not mean the grid created the painting. It means the painting created the grid.

⭐ 6. The formal mathematical justification

In differential geometry:

• A field is a section of a fiber bundle over a manifold.

• The manifold is defined by its differentiable structure.

• The differentiable structure can be induced by the field itself.

ToE uses:

• Information geometry to induce the metric.

• Entropic curvature to induce the connection.

• Entropic distinguishability to induce the manifold.

Thus:

$$x^{\mu }\text{ are coordinates on a manifold generated by }S\mathrm{.}$$

So writing $S(x)$ is not a contradiction. It is the standard way to express a field whose own structure generates the manifold.

⭐ 7. The short, intuitive answer

Because:

• Entropy is the field.

• The manifold is the stage entropy creates.

• Coordinates are just labels on that stage.

So writing $S(x)$ is simply the mathematical way to describe:

“The value of the entropic field at the point labeled x.”

It does not imply that spacetime is more fundamental than entropy. It only implies that we need coordinates to describe the field.