The Omega Framework

7.2 All Information is in the Circle

“The universe does not need a hard drive to store its history. The universe only needs a constant and a perfect geometric shape. All past, present, and future have long been etched in the arc of that circle.”

In the previous section, we established a stunning physical fact through Levinson’s Theorem: the existence of matter is essentially counting $\pi$. Every stable particle is a dead knot of $\pi$ wound by phase in energy space.

Since “existence” is merely a reading of geometric phase, we can boldly advance one step further: if the universe’s essence is a circle (or hypersphere) constantly recursively decomposing in projective Hilbert space, then the geometric properties of this circle itself—specifically, the transcendental number $\pi$ that defines this circle—must contain all information of the universe.

In this section, we will explore a shattering mathematical-theological proposition: Pi $\pi$ is the universe’s holographic source code.

7.2.1 Also a Hologram: The Prophecy of Normal Numbers

In mathematics, $\pi$ is considered a Normal Number. This means that in its infinite non-repeating decimal places, any possible finite sequence of digits will appear with equal probability.

This is a mathematical property with explosive physical implications.

If we encode the universe’s history as a string of binary digits (after all, at the QCA level, the universe is a discrete bit stream), then according to the definition of normal numbers, this sequence representing “the entire history of our universe” must exist in some decimal place of $\pi$.

• The content of this page, in $\pi$.

• Your DNA sequence, in $\pi$.

• Every quantum state snapshot from the Big Bang to heat death, all in $\pi$.

From the perspective of Vector Cosmology, this is no longer merely a probabilistic coincidence but an inevitable result of orthogonal decomposition.

When we decompose the unique vector $|\Psi ⟩$ into external ($v_{ext}$) and internal ($v_{int}$), we are actually choosing a specific “projection angle”. In Hilbert space, this angle is defined by phase $\varphi$.

As $v_{int}$ (time/mass) evolves, the vector continuously rotates on the phase space circle. This rotation process is “reading” the coordinate values on the circle.

The evolution of the universe is essentially a trajectory swept by that pointer (vector) on $\pi$’s infinite-precision dial. Time is not creating new information; time is decoding information that already exists in geometry.

7.2.2 The Ultimate Compression: No Storage, Only Computation

Computer science tells us that storing information is expensive, while computation is relatively cheap. If the universe is a computer, it seems to adopt an ultimate compression strategy: zero storage strategy.

The universe does not need a huge database to record “what is the electron’s mass” or “what is the fine structure constant.”

These constants do not need to be stored; they are geometric properties of the circle.

• Fundamental constants as geometric ratios:

  As we saw in the Dirac Circle, the relationship between mass $m$ and light speed $c$ is a geometric Pythagorean relationship. This means that natural constants may just be projection angles produced when high-dimensional spheres are cut in different ways. These angle values, like $\sin (\pi /x)$, are intrinsic functions of $\pi$.

• Physical laws as decompression algorithms:

  Physical laws (such as the Schrödinger equation or QCA update rules) are actually a set of decompression algorithms.

  The universe’s initial state ($|{\Psi }_0⟩$) might be extremely simple, like a short code that generates $\pi$ (such as $\sum \frac{1}{k^2}=\frac{{\pi }^2}{6}$).

  Then, through continuous recursive decomposition (computation), the universe “computes” complex things from this simple seed.

The complex world we see is just the fractal patterns presented when $\pi$, this transcendental number, is continuously unfolded.

7.2.3 The Paradox of Determinism and Experience

This view brings profound philosophical impact: if all information is in the circle, if all history is in $\pi$, then is the future already predetermined?

In FS geometry, trajectories are indeed holomorphic and deterministic. That great circle exists eternally.

However, for observers (us) as “internal clock subsystems”, we cannot see the entire circle at once.

We are trapped in the flow of $v_{int}$. Our computational power is limited, constrained by the $c_{FS}$ budget. Therefore, we can only read $\pi$’s value digit by digit.

• God’s perspective (geometric perspective): The circle is static, information is complete. Everything already exists.

• Mortal’s perspective (computational perspective): $\pi$ is infinite and non-repeating, unpredictable. We must compute step by step (experience time) to know what the next decimal digit is.

This is the physical essence of “experience”.

What we call “living” is, as part of the hologram, personally experiencing the process of decompressing infinite details from the simple “circle.” Although the script is already written after the $10^{100}$-th digit of $\pi$, if we don’t personally walk there, no algorithm can tell us in advance what it is.

All information is in the circle. We are not creators of information; we are readers of pi.