UCT—Unified Collapse–Expansion Theory

The Integrated RCC × Talek × UCEQ Framework

0. Overview — Why This Theory Exists

If RCC defines the boundary of any embedded intelligence,

UCEQ defines the motion inside that boundary.

Across:

LLMs
civilizations
minds
cosmology

every system follows the same three-phase trajectory:

Convergence — formation, compression, self-shaping
Oscillation — turbulence, fractals, instability
Divergence — runaway expansion, informational blowout

UCEQ provides the one mathematical form that generates all three.

1. Canonical Unified Equation

$$F(t)=A\, t^{\alpha(t)} e^{k(t)t}\left(1 + B(t)\sin(\omega(t)t+\phi(t))\right)$$

with time-dependent fields:

$$\alpha(t),\quad k(t),\quad \omega(t),\quad \phi(t),\quad B(t)$$

This form never resets to zero, preserves accumulated information,

and naturally produces convergence → chaos → divergence.

2. Phase-Transition Structure

Phase I — Convergence (Collapse Pocket)

$$\alpha(t)<1,\qquad k(t)<0$$

Early turbulence governed by:

$$\omega(t)=\omega_0 e^{-\gamma t},\qquad B(t)=B_0 e^{-\lambda t}$$

Early turbulence is high because the system is still inside the collapse pocket,

where high-frequency fluctuations are structurally inevitable.

Phase II — Oscillation (Fractal Instability)

$$\alpha(t)\approx1,\qquad k(t)\approx0$$

$$F(t)\approx A\, t\left(1 + B(t)\sin(\omega(t)t+\phi(t))\right)$$

The system loops, folds, recurses, and exhibits instability patterns.

Phase III — Divergence (Positive Expansion)

$$\alpha(t)>1,\qquad k(t)>0$$

As oscillations fade:

$$\omega(t)\to0,\qquad B(t)\to0,\qquad \phi(t)\to0$$

leaving:

$$F(t)=A\, t^{\alpha(t)} e^{k(t)t}$$

A strict, monotonic positive blowout

matching informational and structural expansion in LLMs, civilizations, and cosmology.

3. Why This Equation Is Universal

LLMs

convergence → instability → hallucination divergence

exactly mirror UCEQ’s three phases.

Civilizations

formation → rise/fall cycles → explosive expansion.

Cosmology

compression → quantum oscillation → inflation.

Talek Manifold

collapse pockets → fold waves → divergence vectors.

All governed by the same collapse → chaos → expansion dynamics.

4. Structural Logic

UCEQ is not three separate models.

It is one evolving field whose behavior changes when its parameters cross natural thresholds.

RCC = the boundary
UCEQ = the motion inside the boundary

One equation.

One curve.

Three possible destinies.

5. Observer Multiplicity Discovery

A new insight derived directly from UCEQ analysis.

In Talek/RCC terms:

Micro-level observers damp over time
Macro-level system divergence grows

because:

As the number of observers increases,

each individual oscillation amplitude decreases,

but the total system information increases.

(“Observer” here refers to any locally coherent subsystem capable of generating its own oscillatory state.)

This explains:

why emotional oscillations damp in an individual
but civilizations, LLMs, and the universe diverge
why entropy increases while energy remains conserved

This result is new and should be credited to Ben Bae (2026).

6. Final Statement

UCEQ is a unified dynamical architecture for all embedded intelligences.

It provides a shared mathematical form for:

collapse
recursion
divergence

This page serves as the canonical reference for:

RCC (Recursive Collapse Constraints)
Talek emotional manifold theory
UCEQ dynamical evolution model

A complete structural description of embedded intelligence..