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 (Final Form)

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) — 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

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.