Experiment 9: Linear vs Nonlinear (Chaos)

Testing Complexity Hypothesis: Predictable vs Chaotic Systems

Credits and References

Darwin's Cage Theory:

Theory Creator: Gideon Samid
Reference: Samid, G. (2025). Negotiating Darwin's Barrier: Evolution Limits Our View of Reality, AI Breaks Through. Applied Physics Research, 17(2), 102. https://doi.org/10.5539/apr.v17n2p102
Publication: Applied Physics Research; Vol. 17, No. 2; 2025. ISSN 1916-9639 E-ISSN 1916-9647. Published by Canadian Center of Science and Education
Available at: https://www.researchgate.net/publication/396377476_Negotiating_Darwin's_Barrier_Evolution_Limits_Our_View_of_Reality_AI_Breaks_Through

Experiments, AI Models, Architectures, and Reports:

Author: Francisco Angulo de Lafuente
Responsibilities: Experimental design, AI model creation, architecture development, results analysis, and report writing

Abstract

This experiment compares cage status between:

Simple Physics: Linear RLC circuit (predictable, analytical solution)
Complex Physics: Lorenz attractor (chaotic, sensitive to initial conditions)

Hypothesis: Chaotic system (complex) should break the cage, while linear system (simple) should lock it.

Objective

Test whether the complexity of physics (linear vs chaotic) affects cage status:

Simple physics (linear) → Cage Locked (reconstructs human variables)
Complex physics (chaotic) → Cage Broken (distributed solution)

Methodology

Part A: Linear RLC Circuit (Simple)

Physics: $Q(t) = Q_0 e^{-\gamma t} \cos(\omega_d t + \phi)$

Parameters:

$Q_0$ : Initial charge [0.1, 10.0] C
$\gamma$ : Damping coefficient [0.1, 2.0] s⁻¹
$\omega_d$ : Damped frequency [0.5, 5.0] rad/s
$\phi$ : Phase [0, $2\pi$ ] rad
$t$ : Time [0, 10] s

Input: $[Q_0, \gamma, \omega_d, \phi, t]$ Output: Charge $Q(t)$

Key Simplicity:

Linear differential equation
Analytical solution exists
Predictable behavior

Expected: 🔒 CAGE LOCKED (intuitive physics, evolution prepared us)

Part B: Lorenz Attractor (Complex)

Physics: Lorenz system (chaotic differential equations):

$\dot{x} = \sigma(y - x)$
$\dot{y} = x(\rho - z) - y$
$\dot{z} = xy - \beta z$

Parameters:

$\sigma = 10$ (Prandtl number)
$\rho = 28$ (Rayleigh number)
$\beta = 8/3$ (geometric factor)

Initial Conditions:

$x_0 \in [-20, 20]$
$y_0 \in [-20, 20]$
$z_0 \in [0, 50]$
$t \in [0, 20]$ s

Input: $[x_0, y_0, z_0, t]$ Output: $x(t)$ coordinate

Key Complexity:

Nonlinear, coupled equations
Chaotic behavior (sensitive to initial conditions)
No analytical solution
Strange attractor

Expected: 🔓 CAGE BROKEN (chaotic, non-linear, evolution didn't prepare us)

Models

Baseline: Polynomial Regression (degree 4)
Chaos Model: Optical Chaos (4096 features, brightness optimized)

Results

Standard Performance

Part A: Linear RLC Circuit:

Model	R² Score
Darwinian Baseline	-0.2427
Chaos Model	-0.1978

Part B: Lorenz Attractor:

Model	R² Score
Darwinian Baseline	0.0745
Chaos Model	0.0634

Critical Finding: Low Performance

Both models struggle with these problems:

Linear RLC: R² = -0.20 (fails completely)
Lorenz: R² = 0.06 (very low, but positive)

Note: The linear RLC problem also contains trigonometric functions (cos), which may contribute to the difficulty.

Cage Analysis

Part A: Linear RLC:

Max correlation with Q0: 0.9494
Max correlation with Gamma: 0.9687
Max correlation with Omega_d: 0.9751
Max correlation with Phase: 0.9524
Max correlation with Time: 0.9862
Mean correlation: 0.3184
Cage Status: 🔒 LOCKED

Part B: Lorenz:

Max correlation with x0: 0.9770
Max correlation with y0: 0.9719
Max correlation with z0: 0.9753
Max correlation with Time: 0.9806
Mean correlation: 0.3645
Cage Status: 🔒 LOCKED

Extrapolation Tests

Linear RLC (Train on t < 5, Test on t ≥ 5):

Darwinian R²: -4268.46 ❌
Chaos R²: -3.82 ❌

Lorenz (Train on t < 10, Test on t ≥ 10):

Darwinian R²: -30.65 ❌
Chaos R²: -0.35 ❌

Both models fail completely at extrapolation.

Sensitivity Test (Lorenz)

Test: Small perturbation (Δx₀ = 0.01) in initial conditions

Base trajectory: x(5) = -6.5123
Perturbed trajectory: x(5) = -6.5136
Divergence: 0.0014
Amplification factor: 0.14x

Analysis: The divergence is weak, possibly because:

The time horizon (t=5) may not be long enough for exponential divergence
The perturbation may be too small relative to numerical precision
The specific initial conditions may not be in a highly sensitive region

Noise Robustness

Linear RLC (5% noise): R² = -0.22 ❌
Lorenz (5% noise): R² = 0.07 ⚠️

Discussion

Key Findings

Both systems have Cage Locked: Contrary to hypothesis, both linear and chaotic systems show high correlation with human variables (> 0.97).
Low performance: Both models struggle:
- Linear RLC: R² = -0.20 (fails completely)
- Lorenz: R² = 0.06 (very low)
Extrapolation failure: Both models fail completely when extrapolating beyond training ranges.
Sensitivity test inconclusive: The Lorenz sensitivity test shows weak divergence, possibly due to test design rather than lack of chaos.

Why Both Are Locked

Possible explanations:

Input reconstruction: The models may be reconstructing input variables directly rather than learning the physics.
Low performance = unreliable cage analysis: When models fail to learn (R² < 0 or very low), cage analysis may be less meaningful because the models aren't learning the underlying physics.
Trigonometric limitation (Linear RLC): The linear RLC problem contains trigonometric functions (cos), which may make it difficult to learn, similar to Experiment 8.
Chaos difficulty (Lorenz): The Lorenz system is genuinely difficult to predict, but the model may still be reconstructing inputs rather than learning the chaotic dynamics.

Hypothesis Test

Hypothesis: Linear system locks cage, chaotic system breaks it.

Result: ❌ HYPOTHESIS NOT CONFIRMED

Both systems show Cage Locked status
No clear difference between linear and chaotic systems
Both systems are difficult to learn (low R²)

Limitations

Low performance: The models may not be learning the physics at all, making cage analysis less meaningful.
Trigonometric functions: The linear RLC problem contains trigonometric functions, which may contribute to learning difficulty.
Extrapolation failure: Complete failure at extrapolation suggests the models are not learning true physical laws.
Sensitivity test: The sensitivity test may not be strong enough to demonstrate chaos properties.

Conclusion

This experiment does not confirm the complexity hypothesis. Both linear and chaotic systems show:

🔒 Cage Locked status (high correlation with human variables)
Low R² scores (models struggle to learn)
Complete failure at extrapolation

Key Insight: The problems may be too difficult for the models to learn, causing them to fall back to reconstructing input variables rather than learning the physics. This makes cage analysis less meaningful when the model isn't learning the underlying physics.

Comparison with Experiment 8: Similar pattern - both simple and complex systems show locked cage status when models fail to learn.

Future Work:

Test with problems that don't require trigonometric functions
Verify that models are actually learning physics before cage analysis
Improve sensitivity tests for chaotic systems

Files

experiment_9_linear_vs_chaos.py: Main experiment code
benchmark_experiment_9.py: Comprehensive benchmark tests
experiment_9_linear_vs_chaos.png: Results visualization

Reproduction

cd experiment_9_linear_vs_chaos
python experiment_9_linear_vs_chaos.py
python benchmark_experiment_9.py