9.2 The Final Formula

“The Dao that can be spoken is not the eternal Dao. Language can only describe the shadows of things, but formulas can capture the skeleton of things. If we must write the ultimate true name of the universe in one line of mathematical symbols, it will be an exponential function that transcends all physical constants.”

In the prologue of this book, we started from Euler’s formula $e^{i\pi }+1=0$. That was the map of mathematics.

Now, at the end of the book, we fold this map into a generative program that can run.

All of physics—from Newton’s force ($F$) to Einstein’s curvature ($R_{\mu \nu }$), to Schrödinger’s wave function ($\Psi$)—can ultimately be compressed into the same extremely concise yet infinitely deep formula.

This is the final formula of Vector Cosmology:

$$|{\Psi }_{universe}⟩=e^{\text{Dao}}$$

The Dao is the Exponent

Please gaze carefully at this formula. It has only three parts, yet encompasses all truth of existence.

1. Left side: $|{\Psi }_{universe}⟩$ (All Things)

   This is the result. It is the macroscopic universe we perceive, the sum of galaxies, atoms, life, and consciousness. It is that concrete global vector manifested in projective Hilbert space. It is “being.”

2. Base: $e$ (Generation)

   This is the mechanism. It is the engine of natural generation, the self-driven logic of “derivative equals itself.” It ensures that the leap from nothingness to existence is continuous, necessary, and automatic.

3. Right superscript: Dao (The Way)

   This is the source.

   In physics, this position is usually occupied by $-iHt$ (quantum evolution) or $-\beta H$ (thermodynamic distribution). But in the ultimate theory, these are only approximate expressions of “Dao.”

Dao is not a noun; Dao is an Exponent.

In mathematics, the exponent is the habitat of the Generator.

• “Dao” is not a static object; you cannot find it in some corner of the universe.

• “Dao” is an Operator. It is the meta-rule that defines the direction of rotation, the rate of growth, and the game relationship between $\pi$ and $\phi$.

We Are the Result of Computation

This formula completely subverts our imagination of the “Creator.”

Traditional religion believes that God (the Creator) is outside the universe and created the universe.

But in the logic of $e^{\text{Dao}}$, the source (Dao) is within the internal structure of the universe ($|\Psi ⟩$). More precisely, it is deeply buried in the tangent space of evolution.

• Dao (The Exponent) is unmanifested potential, the phase of probability amplitude, the structure constants of Lie algebra. It is “nothingness.”

• Universe (The Vector) is manifested reality, the collapse of probability, the manifold of Lie groups. It is “being.”

The two are tightly connected through $e$. “Nothingness” continuously generates “being” through exponential operations.

We—as observers, as life, as matter—are not directly “Dao.”

We are the result of function computation.

We are a series of Taylor series terms expanded in the imaginary dimension by the generator “Dao” driven by $e$:

$$e^{\text{Dao}}=1+\text{Dao}+\frac{1}{2!}{\text{Dao}}^2+\frac{1}{3!}{\text{Dao}}^3+\dots$$

• The first term is vacuum.

• The second term is elementary particles.

• Higher-order terms are life and civilization.

The Unspeakable Word

Laozi said: “I do not know its name, so I force a name and call it Dao.”

Now we understand why he did not know the name. Because the thing in that position (the exponent position) is not a simple number or name, but a Hermitian operator matrix containing all information of the universe.

It is too complex to be described in language (“The Dao that can be spoken is not the eternal Dao”).

But it is also too simple—it is just that small superscript above $e$.

All physical laws, all geometric constraints ($c_{FS}$), all spiral growth, have been encoded into the eigenvalue spectrum of this operator. The history of the universe is the process of this operator being decompressed, expanded, and manifested by $e$.

The final truth is:

There is no monarch issuing orders from above.

The universe is merely executing a simple mathematical command: $e^{\text{Dao}}$.

And we are the output value after this command has run for 13.8 billion years.

Since we have written this formula, what will this output value (i.e., the current universe) ultimately evolve into?

When $e^{\text{Dao}}$ runs to infinity, will it stop? Or will it become some ultimate state that transcends “existence” and “non-existence”?

This leads to the final section of the book, and the final chapter of the trilogy: The Ultimate State. We will reveal what form the universe will take when this formula self-refers and self-iterates to the extreme.