When Chaos Hides Meaning: The Paradox Formula Discovered by AI

Author: Andrei Leonov

Introduction

What if an AI could discover physical laws — not by being told the answer, but by evolving its own formulas? What if the result looked like pure nonsense — yet hid a physically perfect structure underneath?

This is the story of a formula that is at once absurd and brilliant. Generated by symbolic regression using genetic programming (GP), the expression appeared chaotic and unreadable. But after careful analysis, it revealed a deep, interpretable pattern — one that could be used in physics, geometry, and even memory modeling in AI systems.

The Raw Formula (a Monster)

X′ = neg(((tanh(x) / pi) * (1 + p1)))
Y′ = get_y(rot2d(sin(y), sqrt(get_x(grad((...)))), exp(tanh(...))))

The formula was one of many evolved by a symbolic GP system trained only to minimize MSE — no physics was built in.

At first glance: complete chaos. Deeply nested operations like grad(p1) (which evaluates to zero), rotations of constants, and layered trigonometric transforms.

The Simplified Truth

After simplification, all of the junk vanished:

X'(x) = K_1 \cdot anh(x)
Y'(y) = K_2 \cdot \sin(y)

Where:

Why This Matters

✅ Interpretable Physics

X′: a saturating drift toward the origin — common in systems with friction, dissipation, or magnetization.

Y′: pure standing wave dynamics — just like vibrations in materials or fields.

✅ Separability

The system is decoupled: what happens in x doesn't affect y. This is a common first-order approximation in many real physical systems.

✅ Gradient Field

This is the gradient of a scalar potential:

\phi(x, y) = K_1 \cdot \log\cosh(x) - K_2 \cdot \cos(y)

This means the vector field is conservative, and describes a smooth deformation of space.

Visualization

Here’s what the field looks like:

X direction: flows toward the center and flattens

Y direction: periodic oscillations

Real Applications

🎓 Physics: models center-seeking fields and wave behavior.

🧠 Neuroscience: resembles excitation + inhibition in neural membranes.

🌌 Geometry: local deformation in HyperTwist-like metric fields.

🤖 AI Interpretability: serves as a benchmark for explainable symbolic AI.

Why This Is a Paradox

The formula looks meaningless.

But it behaves like a law of nature.

This is the paradox. The AI didn’t know what a physical law was. It just minimized error. And yet, it rediscovered a classical behavior — hidden beneath layers of symbolic noise.

It’s a perfect example of emergent intelligence: complex behavior arising from simple rules.

Lessons Learned

🧠 Even chaotic outputs from AI can hold gold.

🛠 Add complexity penalties (complexity_penalty = 1e-4) to force parsimony.

🔍 Always analyze evolved formulas symbolically — don’t discard them based on looks.

🧭 Physics may emerge, even without being encoded explicitly.

Next Steps

I plan to:

Add this formula to a library of emergent symbolic laws

Use it as a φ-field in geometric models

Publish more case studies like this

You can follow my symbolic regression work, geometric models, and HyperTwist experiments.

If you're building AI that creates, this kind of emergent structure is what you want to watch for.

What do you see in this paradox?

Have you encountered symbolic chaos that turned out meaningful?
Share your thoughts — or your monsters — below.