The Omega Framework

9.7 Ontological Geometry: All Things Are Circles

After deriving the generalized Euler identity, we face an ultimate ontological question: If we strip away all holographic projections and ignore all measurement illusions, what does the most fundamental “entity” of the universe actually look like?

Based on the mathematical structure of Omega Theory, we arrive at a conclusion that is intuitively shocking but logically inevitable: At the ontological level, all things are circles. The linear time, three-dimensional space, and discrete matter we perceive are essentially holographic projections of great circles in high-dimensional complex planes onto low-dimensional tangent spaces.

9.7.1 Normalization of Existence: The Closure of Circles

Why must it be a circle? This stems from the most fundamental axiom of quantum mechanics—probability conservation. Any physical entity, whether an electron, a person, or the entire universe, its quantum state $|\psi ⟩$ in Hilbert space must satisfy the Normalization Condition: $$⟨\psi |\psi ⟩=1$$

In complex geometry, all points $z$ with modulus 1 form the unit circle $S^1$ in the complex plane: $$z=e^{i\theta },\theta \in \mathbb{R}$$

This means that at the most fundamental level of existence:

• No straight lines: All motion is rotation.
• No beginning and end: All paths are closed (or tend to close).
• No size: All “existence” possesses the same ontological weight (modulus 1).

Objects that appear vastly different in the universe are essentially just different phase angles $\theta$ or different rotation frequencies $\omega$ on the same unit circle.

9.7.2 Projection Mechanism: The Illusion from Circle to Line

Then why do we live in a world full of straight lines, boxes, and linear time? This is because we live in Projection, not Ontology.

Mathematically, this transformation from circle to line can be described through Stereographic Projection or Cayley Transform. It maps compact circles to infinitely extending lines.

Consider a point moving uniformly in circular motion on a circle: $$z(t)=e^{i\omega t}$$ When we observe this circle from the side (i.e., take the real part), we no longer see rotation but a wave oscillating in linear space: $$x(t)=\text{Re}(z)=\cos (\omega t)$$

Theorem 9.7 (Linear Illusion Theorem): Any linear physical quantity (such as position $x$, momentum $p$, time $t$) is a projection cross-section of circular motion in high-dimensional phase space onto low-dimensional tangent bundles.

• Ontology: A closed string (circle).
• Phenomenon: Sound waves (linear waves) produced by string vibration filling space. The world we perceive is “music,” while Hilbert space is the “string.” We mistake the propagation of sound (linear) for the essence of the world, ignoring the structure of the sound source (circle).

9.7.3 Matter as Standing Waves: The Geometric Origin of Integers

This perspective perfectly explains the Discreteness of matter. In classical physics, particles are entities like billiard balls. But in Omega Theory, particles are Standing Wave Modes on circles.

For a wave to exist stably on a closed circle, its wavelength $\lambda$ must satisfy the Circular Closure Condition: $$2\pi R=n\cdot \lambda ,n\in \mathbb{Z}$$

• Integer $n$: This is the origin of quantum numbers. It represents the Winding Number.
• Non-integer: If the frequency is not an integer, the wave will have phase misalignment after one revolution, causing Destructive Interference. Such modes are unstable and cannot form matter.

Therefore, matter is quantized circles. Atoms, electrons, quarks—they are not solid little balls; they are Integer Harmonics on that eternal great circle.

9.7.4 Conclusion: Shadows of Holography

Here we complete the ultimate deconstruction of physical reality. We often ask: “What is the universe made of?” The answer now is: The universe is made of ‘frequencies.’

All existence, ultimately, is shadows cast on the spacetime screen by that unique, static, total-budget-conserving great circle through the infinite resolution of the Fibonacci operator $\varphi$.

• Time is the tangent to the circle.
• Space is the unfolding of the circle.
• Matter is the nodes of the circle.

Plato said in his allegory of the cave that we see only shadows on the wall. Omega Theory restates this truth in mathematical language: We are holographic projections of high-dimensional circles onto three-dimensional membranes. We do not possess independent linear lives; we are melodies on that eternal circumference.

$$\text{Being}=S^1\stackrel{\text{Projection}}{\to }\text{Becoming}={\mathbb{R}}^3\times \mathbb{R}$$