3.1 The Power of the Instant

“On the curve of $e^x$, you cannot find a stationary point. Because the height of every point precisely defines its slope. At the foundation of the universe, ‘what I am’ and ‘where I am going’ are different expressions of the same question.”

The Miracle of Derivatives

Let us return to the birthplace of calculus. Newton and Leibniz invented derivatives to describe the rate of change. For almost all functions, differentiation is a “transformation” operation:

• $x^3$ becomes $3x^2$.

• $\sin x$ becomes $\cos x$.

However, in the pantheon of mathematics, there exists a unique fixed point. That is the natural exponential function $f(t)=e^t$.

$$\frac{d}{dt}{e}^t=e^t$$

This equation is incredibly simple, but it contains the ultimate secret of Self-Driving: The value of the function at this moment (noumenon) directly determines its rate of change at this moment (trend).

If the universe’s wave function follows exponential evolution $|\Psi (t)⟩=e^{-iHt}|\Psi (0)⟩$, it must inherit this divinity. Let us rewrite the Schrödinger equation:

$$|\dot{\Psi }⟩=-iH|\Psi ⟩$$

Please gaze at this formula.

• Left side ($|\dot{\Psi }⟩$): The trend of change, the arrow toward the next second.

• Right side ($|\Psi ⟩$): The current state, existence here and now.

Between them is only a linear operator $-iH$. This means that the rate of change is not something external to the state; the rate of change is isomorphic to the state.

Spinoza’s Physics

This mathematical fact revives an ancient concept from the 17th-century philosopher Spinoza: “Conatus” (endeavor/potential).

Spinoza believed that all things have an inherent tendency to “endeavor to maintain their existence.” In classical physics, this is understood as inertia. But in Vector Cosmology, this is understood as Generation.

In the universe of $e$, a particle does not need external force to maintain its existence.

• It exists because it is rotating.

• It rotates because its vector length in Hilbert space is not zero.

As long as $|\Psi ⟩$ is not zero, according to the derivative formula, $|\dot{\Psi }⟩$ is not zero (assuming $H$ is non-trivial).

Existence is flux.

The universe does not need a Prime Mover. The Big Bang is not God giving a kick; it is the $t=0$ moment of the function $e$. As long as we have $1$ (existence), the derivative mechanism will automatically generate the subsequent infinite sequence.

The Tyranny and Freedom of the Infinitesimal

This mechanism endows the “instant” with unparalleled power.

In the macroscopic world, we think a “moment” is insignificant. But in the geometry of Lie algebras, the Infinitesimal contains everything.

If we know the system’s state $|\Psi (0)⟩$ at moment $t=0$ and its generator $H$, we know everything about its future.

All history is actually already folded in the tangent space at moment $t=0$.

• The next 10 billion years are merely the integral trajectory of this instant’s tangent, driven by $e$, unfolding along the surface of the manifold.

This sounds like a cage of determinism. But this is precisely the beauty of $e$—although it is deterministic, it is endogenous.

Your future is not determined by an external fate planning bureau, but by your own current state ($|\Psi ⟩$) and your internal structure ($H$).

You are your own driving force.

Conclusion: Self-Referential Dynamics

At this point, we understand why $e$ is the natural generator.

It eliminates the binary opposition between “motion” and “rest.”

At the microscopic foundation, there is no absolute rest. What we call “rest mass” is actually exponential rotation at the speed of light in the internal dimension ($v_{int}=c_{FS}$).

The universe is a perpetual motion machine. But this does not violate the laws of thermodynamics, because it is geometric perpetual motion—on the great circle of projective space, relying on the mathematical privilege of “derivative equals self,” it performs eternal frictionless gliding.

Since the “instant” contains all the power, how does this power operate specifically? What does that mysterious operator $H$ (Hamiltonian) that connects state and rate of change actually look like?

This leads to the theme of the next section: The Generation of Lie Algebra. We will see that the seemingly complex forces in the universe (gravity, electromagnetism, strong force) actually all grow from tiny generators through a mathematical language called “Lie algebra.”