Chapter 2: Geometry of the Spiral

We have seen through the illusion of synchronous inflation and heard the Red Queen’s ultimatum. Now, we must delve deep into the interior of this wildly growing cosmic machine to find its core code.

If the universe is not merely maintaining conservation but pursuing growth, then the geometric logic it follows must change.

In the first book, $\pi$ was the supreme monarch. It defined the closure of circles, the existence of particles, and the topological counting of Levinson’s theorem. But in the second book, we introduce a new challenger. It is not merely a number; it is a fundamentally different geometric philosophy.

It is $\phi$—the golden ratio.

This chapter will reveal that cosmic history is not a calm river, but a 13.8-billion-year war between these two constants: $\pi$ wants to return to the origin, while $\phi$ wants to go to uncharted territories.

2.1 $\pi$ is Memory, $\phi$ is Computation

“$\pi$ is a perfect record, recording all melodies that have occurred and attempting to replay them forever. $\phi$ is an improvising hand, refusing to repeat any note, only to explore unknown chords.”

The Geometry of Memory: The Closure of the Circle

Why does physics rely so heavily on $\pi$?

In the Schrödinger equation, in general relativity, in electromagnetic field theory, $\pi$ is everywhere. This is because the foundation of physics is built on “stability”.

From the perspective of Vector Cosmology, stability means geometric closure.

An electron can exist stably because its wave function forms a standing wave around the atomic nucleus.

• The condition for a standing wave is: the phase change must be exactly an integer multiple of $2\pi$.

• Only when head meets tail can the wave avoid self-interference and annihilation.

Therefore, $\pi$ is the constant of memory.

It is an instruction on “how to return to the origin.” It is responsible for maintaining structure, making yesterday like today, ensuring information is not erased in the long river of time.

If there were no $\pi$ in the universe, there would be no atoms, no matter, only a fleeting chaos. Through $\pi$, the universe locks the budget of $c_{FS}$ into closed loops, thus achieving “storage”.

The Geometry of Computation: The Unfolding of the Spiral

But if the universe only wanted storage, it would have reached heat death long ago. The universe also wants computation, wants generation.

For a system seeking new possibilities, returning to the origin is the worst outcome. If you step on old footprints with every step, you can never explore new territories.

This is when $\phi$ (1.61803…) takes the stage.

In number theory, $\phi$ is called “The Most Irrational Number”. This means it is the most difficult number to approximate with fractions.

In dynamical systems theory, this property has extraordinary physical significance:

• If a system’s frequency ratio contains $\pi$ or rational numbers, it easily produces resonance and falls into periodic cycles.

• If a system’s driving frequency ratio is $\phi$, it is anti-resonant. Its trajectory in phase space never repeats.

$\phi$ is the constant of computation.

It is an instruction on “how to avoid the past.”

On a sunflower’s seed disk, plants use $\phi$ angles to arrange seeds to ensure each new seed falls in an unoccupied gap, maximizing sunlight utilization.

In Hilbert space, the universe uses $\phi$ to drive vector rotation to ensure that every quantum state at each moment is orthogonal to any moment in history.

Through $\phi$, the universe transforms the budget of $c_{FS}$ into an exploration path that never looks back, thus achieving “growth”.

The War of Constants

Thus, we see the deep tension in cosmic evolution.

• The microscopic world belongs to $\pi$:

  Atoms, elementary particles, force fields. They need absolute stability, properties unchanged for billions of years. So they strictly follow the circle equation $v_{ext}^2+v_{int}^2=c_{FS}^2$ and run on $\pi$’s orbit. They are the universe’s hard drive.

• Macroscopic evolution belongs to $\phi$:

  Life, galactic spiral arms, cosmic expansion. They need adaptation, mutation, occupying new ecological niches. So they exhibit logarithmic spiral forms and continuously break old equilibria under $\phi$’s guidance. They are the universe’s CPU.

What we call “time passing” is essentially $\phi$ eroding $\pi$’s territory.

Every quantum leap, every genetic mutation, is a tiny crack that $\phi$ tears in the perfect circle. The universe is attempting to switch from pure “repeater” mode to “generative” mode.

Conclusion: The Universe’s Operating System

We live in a dual-core universe.

The underlying firmware is written in $\pi$, ensuring the universality and conservation of physical laws, giving us a solid stage.

The upper software runs on $\phi$, driving the emergence of complexity and dimensional inflation, allowing us to perform scripts that never repeat on the stage.

We often feel pain because we, as life (subjects of $\phi$), are trapped in bodies made of atoms (prisoners of $\pi$). We yearn for infinite ascension but must face material decay.

But this is not a tragedy; it is the driving force of iteration.

It is precisely because the circle tries to close and the spiral tries to unfold that this geometric tension between them generates a continuous stream of evolutionary momentum.

Since we have identified these two geometric masters, the next question is: In this war, can $\pi$ really hold its ground forever? Will the material world we consider eternally unchanging eventually loosen under the spiral’s powerful pull?

This is precisely the theme of the next section: The Loosening of Levinson’s Knot. We will see that in the distant future, even protons themselves cannot escape.