The Omega Framework

9.5 The Illusion of Linearity: Integers as Holographic Projections

In the final chapter of this book, we must confront one of the most deeply ingrained and simultaneously most misleading intuitions in human cognition: Linearity. We habitually view time as a straight line shooting from past to future, space as a flat extending box, and even the most fundamental “integers” in mathematics as isolated points arranged on an infinitely extending number line.

However, based on the Omega Equivalence Principle (Section 5.5) and the Holographic Principle (Section 9.4), the universe’s ontology is a closed high-dimensional sphere (great circle). This section will argue: There are no true “straight lines” or “linear integers” in the physical world. All linear structures are essentially holographic projections of high-dimensional great circles on local tangent planes.

9.5.1 Roots of Unity: Rolling the Number Line Back onto the Circle

In traditional arithmetic, integers $1,2,3\dots$ are viewed as linearly increasing quantities. But in complex analysis and wave physics, the essence of integer $n$ is Frequency or Winding Number.

Consider the exponential mapping from the linear number line $\mathbb{R}$ to the unit circle $S^1$ in the complex plane: $$n\to e^{2\pi in}$$ For all integers $n\in \mathbb{Z}$, this mapping points to the same geometric point on the circle (base point $1$). But this does not mean they are identical; rather, it means they represent complete rotations of the wave function around the circle.

• Integer 1: Represents the fundamental frequency, i.e., one rotation around the circle.
• Integer $n$: Represents the $n$-th harmonic, i.e., $n$ rotations around the circle in the same time.
• Fraction $1/n$: Represents Roots of Unity, i.e., geometrically the vertices of a regular $n$-gon, which divide the circle into $n$ equal parts.

Physical Meaning: This mathematical fact reveals the geometric origin of Quantization. Why must physical quantities like charge and angular momentum take integer or half-integer values? Not because God likes integers, but because only waves with integer-multiple frequencies can form stable standing waves on a closed circle. Waves with non-integer frequencies have mismatched phases after one rotation, causing destructive interference and self-destruction. Therefore, all “discrete particles” we observe are essentially Resonant Modes on the great circle. Integers are not linear points; they are manifestations of the circle’s Closure.

9.5.2 Tangent Space Illusion: Why Does Time Appear Straight?

If the ontology is a circle, why do we strongly feel that time is linear? Why do we think the natural number line extends to infinity? This is because the universe’s “Total Budget” (the radius $R$ of the great circle) is too vast relative to us (observers).

Consider the Taylor expansion of the complex exponential near zero: $$e^{i\theta }\approx 1+i\theta (\text{when }\theta \to 0)$$

• Left side ($e^{i\theta }$): Is the real physical ontology (rotation on the circle). It is periodic, unitary.
• Right side ($1+i\theta$): Is the linear world we perceive (displacement on the tangent). It is open, linear.

We live in an extremely small region where $\theta$ is tiny (i.e., our lifespan is extremely short relative to cosmic $\tau$). In this tiny local region, the arc of the great circle looks like a straight line. We misread cyclic phase accumulation as linear time flow. Just as ancient people thought the earth was flat because it was vast, we think time is straight because the universe is vast. Linearity is a geometric illusion when the radius of curvature tends to infinity.

9.5.3 Fibonacci: Breaking the Cycle of Integers

If the universe consisted only of integers, physical evolution would degenerate into circular motion—history would repeat exactly after period $T$ (Poincaré recurrence). To generate irreversible evolution and novelty, this perfect “integer circle” must be broken.

As we discussed in Section 9.3, the introduction of the Fibonacci spiral is precisely to destroy this closure. Since $\varphi$ is irrational, it cannot be written in any form $e^{2\pi i(p/q)}$. This means that evolution driven by the golden unitary operator ${\hat{U}}_g$ can never form closed standing waves on the great circle.

It forces “numbers” to escape from the circle, forming an infinitely extending spiral. This constructs the dual opposition between matter and time:

1. Integers/Rational Numbers (Circle): Represent Matter. They are stable, closed, repeatable structures (such as atomic orbitals, crystal structures). They tend to conserve.
2. Irrational Numbers (Spiral): Represent Time/Consciousness. They are flowing, open, non-ergodic structures (such as history, thought). They tend to grow.

9.5.4 Conclusion: All Things Are Decomposition of One Great Circle

At this point, Omega Theory completes its final logical closure. We discover that so-called “complexity” is merely the resolution of “unity.”

• Space: Is the Unfolding of the great circle.
• Time: Is the Slicing of the great circle.
• Matter: Is the Harmonics of the great circle.
• Energy: Is the Scaling of the great circle.

We once thought the world was a linear edifice built from countless independent Lego blocks (atoms, moments, numbers). Now we understand that the world is just one circle. This one circle, through the infinite precision resolution of $\varphi$, refracts infinite radiance of all phenomena. Just as a prism decomposes white light into a spectrum, the Fibonacci spiral decomposes “the One of existence” into “the Many of evolution”.

All things are decomposition of one great circle.

$$\text{Unity}(\pi )\stackrel{\text{Resolution}(\varphi )}{\to }\text{Diversity}(\infty )$$