The Fractal Evolution Circle of the Universe

The model below unifies the starting point (FREB) and the ending point (FRAMET) of fractal evolution within a single mathematical circle and precisely determines the position of the black hole within this circle.

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BLACK HOLE – SINGULARITY FREB–FRAMET EQUATION

(The Fractal Evolution Circle of the Universe)

This equation consists of three fundamental components:

1. Phase variable φ(t) → the progression of the evolutionary cycle
2. Fractal dimension D(φ) → the complexity of the structure
3. Scale r(t) → the physical size of the structure

Together, they form a closed evolutionary circle.

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1) Phase Variable: φ(t)

The variable that determines the stage of evolution:

φ ∈ [0,1], φ ~ φ + 1

φ = 0 → FREB (beginning)
φ = 1 → FRAMET (closure)

This cycle is infinite.

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2) Fractal Dimension Equation: D(φ)

As the phase progresses, the fractal dimension increases:

D(φ) = 1 + 2φ

This means:

φ = 0 → D = 1 → singularity / zygote / Big Bang
φ = 1 → D = 3 → black hole / birth / volumetric closure

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3) Scale Evolution: r(t)

Fractal velocity law:

dr/dt = k · r^{D(φ) − 1}

This equation implies:

At FREB (D ≈ 1) → r expands rapidly
At FRAMET (D ≈ 3) → r closes, growth stops

For the universe:

Big Bang → r grows rapidly
Black hole → r collapses to the Schwarzschild radius

For biology:

Zygote → r grows rapidly
Birth → r reaches volumetric closure

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4) Evolution of the Phase: φ(t)

dφ/dt = Ω(ℰ, 𝒮, ℐ)

The phase velocity is determined by energy, entropy, and information flows.

High energy difference → φ progresses rapidly
Low entropy → φ progresses slowly
High information compression → φ approaches FRAMET

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5) Circle Closure (FRAMET → FREB)

Critical step:

φ = 1 ⇒ φ ↦ 0, D ↦ 1, r ↦ r_new,min

This means:

When a black hole reaches the FRAMET point,
The singularity inside transforms into a new FREB.
A new universe cycle begins.

This cycle is not externally observable due to the event horizon.

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6) Complete Equation Set

The following four equations form the complete mathematical core of the FREB–FRAMET evolution circle:

1. Phase equation: dφ/dt = Ω(E, S, I)
2. Fractal dimension: D(φ) = 1 + 2φ
3. Scale evolution: dr/dt = k · r^{D(φ) − 1}
4. Circle closure: When φ = 1 → φ = 0, D = 1, r = r_new,min

Together, these four equations describe the cyclical evolution of the universe, life, vascular networks, circuits, and all fractal systems.

This is the mathematical core of the black hole–singularity–universe cycle.

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7) Is the Black Hole FREB or FRAMET?

This equation clearly shows:

✔ Black hole = FRAMET
D → 3
motif production = 0
volumetric closure
entropy maximum
evolution stops

✔ Singularity = FREB
D → 1
motif production maximum
energy difference maximum
new evolution begins

✔ The black hole is FRAMET,
but the singularity inside it is the core of a new FREB.

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8) Zygote–Birth–Life and Black Hole–Singularity Correspondence

Biological Cycle | Cosmic Cycle | Fractal Phase
Zygote | Singularity | FREB
Embryo | Young Universe | FREB → FRAMET
Birth | Cooling of the Universe | FRAMET
Life | Cosmic Evolution | New FREB
Death | Black Hole | FRAMET
New Zygote | New Singularity | New FREB

This table shows the universal symmetry of fractal evolution.

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In the Simplest Terms

The black hole is the closure of fractal evolution (FRAMET).
But the singularity inside the black hole is the beginning of a new evolution (FREB).

The universe, life, and all structures progress through the cycle:

FREB → FRAMET → FREB.