Preface: The Natural Generation

In the journey of the Vector Cosmology trilogy, we have traveled a long road.

In the first book The Conservation of the Circle, we stood in the sanctuary of $\pi$ and saw a perfect geometric universe constructed from Fubini-Study metrics and Pythagorean identities. That was a story about the “Body”—the universe’s skeleton is rigid, conservation laws are absolute, matter is dead knots of phase. We marveled at that static beauty of order.

In the second book The Ascension of the Spiral, we followed the footsteps of $\phi$ and rushed into dimensional inflation. We saw how life establishes negative entropy enclaves and how civilizations cross technological singularities. That was a story about the “Mind”—the universe’s will is wild, evolution is open, consciousness is the Red Queen’s run. We were intoxicated by that dynamic vitality.

However, when these two pictures—the closed circle and the open spiral—are placed side by side, a profound tension arises.

Order and freedom, conservation and growth, cycle and ascension… Are these two seemingly opposing concepts truly irreconcilable?

Is the underlying logic of the universe mechanical like a clock, or blindly growing like a vine?

To answer this question, we need a third key.

This key cannot only explain “what the universe is” (geometry), nor can it only explain “where the universe goes” (dynamics). It must explain “why the universe exists.”

This is the mission of the third book The Natural Generator.

In this book, we will excavate the third constant hidden at the deepest layer of mathematics—$e$ (the natural constant).

It is the “Spirit” of the universe.

Why are physical laws always written in the form of exponential functions $e^{-iHt}$? This is by no means a mathematical coincidence.

$e$ possesses a unique divinity: its derivative equals itself. This means that in the logic of $e$, existence (State) is itself propulsion (Trend). The universe needs no first mover, no external clockwork. The universe is a self-referential, self-driven generative engine.

In this book, we will dive into the deepest engine room of physics:

• Through Euler’s formula, we will see how $\pi$ (circle) and $i$ (rotation) are unified by $e$.

• Through imaginary time and Wick rotation, we will break the boundaries between quantum mechanics and thermodynamics, revealing the geometric truth that “time is temperature.”

• We will propose the Modular Flow Hypothesis, arguing that time is not external flow, but automatically “secreted” by the intrinsic entanglement structure of quantum states.

If the first two books describe the “Form” and “Function” of the universe, then this book will point directly to the “Essence” of the universe.

We will no longer be satisfied with observing that rotating vector; we will attempt to become the source of that rotation. We will see that all things are not created, but Self-Generated.

This is a pilgrimage to the ultimate ontology of physics. Let us abandon our attachment to “beginning” and “ending,” and enter that eternally generating exponential world.

Haobo Ma (马昊伯)

December 2025, Singapore