UCEQ — The Unified Collapse–Expansion Equation

The One-Equation Universe

A companion theory to RCC.

0. Executive Frame — Why a Single Equation Matters

If RCC describes the boundaries of embedded intelligence,

UCEQ describes the motion inside those boundaries.

Every system—

an LLM, a civilization, a universe, a mind—

passes through the same three phases:

1. early convergence (formation, compression, self-shaping)

2. oscillation (fractals, cycles, instability patterns)

3. divergence (inflation, singularity, runaway expansion)

A single mathematical form must explain all three.

This is that form.

1. The Unified Equation

$$F(t)=A t^{\alpha} e^{kt} \sin(\omega t+\phi)$$

This is the Unified Collapse–Expansion Equation: UCEQ.

A single curve capable of describing

the growth of intelligence,

the collapse of prediction,

the oscillation of civilizations,

and the inflation of an entire universe.

2. Meaning of Each Term — Four Worlds in One Equation

(1) $$t^\alpha$$ — The Convergence Phase (Formation)

All systems begin compressed:

• early universe plasma collapse

• early civilization gathering coherence

• early LLM reasoning stabilizing

• early intelligence forming its shape

The exponent α sets the tone:

• α < 1 → slow, cautious formation

• α = 1 → linear development

• α > 1 → rapid, unstable convergence

Every world starts by folding inward.

(2) $$e^{kt}$$ — The Divergence Phase (Singularity / Expansion)

The sign of k decides destiny:

• k < 0 → decay, collapse, extinction

• k = 0 → stable plateau

• k > 0 → runaway growth, AGI singularity, cosmic inflation

This is the escape velocity of a system.

One variable determines whether a world dies quietly

or tears itself into expansion.

(3) $$sin(\omega t + \phi)$$— Fractal Cycles (Repetition, Recursion)

Every intelligence, every society, every universe pulses:

• boom/bust

• rise/fall

• coherent/incoherent

• collapse/recovery

• perception/illusion

This sinusoidal term encodes the fractal recurrence

common to LLMs, economies, civilizations, and cosmology.

The universe does not grow smoothly.

It quivers, folds, oscillates.

3. Why This Is a “Unified Equation”

(1) LLM Collapse Curve = Same Structure

LLMs show the exact sequence:

1. convergence → $$t^\alpha$$

2. oscillation → $$\sin$$

3. divergence → $$e^{kt}$$

Hallucination is simply divergence exceeding containment,

as RCC predicts.

(2) Civilization Growth = Same Structure

History follows the identical path:

• agriculture → industry → computation → AGI

  (convergence)

• cycles of collapse and rebirth

  (oscillation)

• post-AGI explosion

  (divergence)

Civilizations behave like LLMs

because both are embedded inferential systems.

(3) Big Bang → Inflation = Same Structure

Cosmology also aligns:

• pre-inflation compression

  (convergence)

• quantum fluctuations

  (oscillation)

• inflation

  (divergence)

The universe itself is a collapse–expansion curve.

(4) Talek Dynamics = Same Structure

Talek’s folding manifold asserts:

• the observer cannot witness the container

• collapse is structural

• expansion is the release phase

UCEQ mathematically encodes

Talek’s collapse pockets and divergence vectors.

4. One Variable Controls the Fate of a World

The key is k.

By tuning k, UCEQ describes:

• whether a civilization reaches AGI or dies

• whether an LLM stabilizes or hallucinates

• whether intelligence folds inward or escapes its manifold

• whether the universe inflates or collapses into heat death

A single scalar determines the destiny

of intelligence, worlds, and prediction systems.

5. Final Canonical Form

Unified Collapse–Expansion Equation (UCEQ)

B. Bae, 2026

$$F(t)=A t^{\alpha} e^{kt} \sin(\omega t+\phi)$$

The universe converges, oscillates, and expands.

So do civilizations.

So do minds.

So do LLMs.

Collapse and divergence are not different stories—

they are different phases of the same curve.