Experiment A1: Coordinate Independence (The Twisted Cage)

Final Report

Date: November 27, 2025
Experiment Type: Coordinate Independence Test
System: Double Pendulum (Chaotic)

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

Executive Summary

Experiment A1 tested whether the Chaos model could learn physics in a "twisted" coordinate system where human-derived mathematical simplicity is destroyed. The hypothesis was that the Darwinian (polynomial) model would fail in twisted coordinates, while the Chaos model would remain robust.

Result: The hypothesis was REFUTED. The Darwinian model performed equally well in both coordinate systems (R² ≈ 0.98), while the Chaos model failed in both (R² ≈ 0.0).

Methodology

Physical System

• Double Pendulum: A chaotic system with 4 state variables: $(\theta_1, \theta_2, \omega_1, \omega_2)$
• Task: Predict next state $x_{t+1}$ given current state $x_t$
• Data: 100 trajectories, 19,900 state transitions

Coordinate Transformation

Applied a non-linear "twist" to scramble the coordinates: $u_1 = \theta_1 + 0.5 \sin(\theta_2)$ $u_2 = \theta_2 + 0.5 \cos(\theta_1)$ $v_1 = \omega_1 + 0.5 \tanh(\omega_2)$ $v_2 = \omega_2 + 0.2 \theta_1 \theta_2$

This transformation mixes positions and momenta in a highly non-linear way.

Results

Performance Comparison

Key Findings

1. Darwinian Model is Coordinate Independent

   • Performed excellently in both frames (R² ≈ 0.98)
   • Actually performed slightly better in the twisted frame
   • The polynomial basis is flexible enough to approximate the twisted dynamics

2. Chaos Model Failed Completely

   • Failed in the standard frame (R² = -0.04)
   • Failed in the twisted frame (R² = 0.01)
   • The reservoir architecture is unsuitable for this temporal prediction task

Critical Analysis

Why Did the Darwinian Model Succeed?

The polynomial model succeeded because:

1. Local Approximation: Polynomial regression is a universal approximator for smooth functions in a local region
2. Short Time Steps: The prediction horizon (dt = 0.05s) is small enough that the dynamics are locally polynomial
3. Coordinate Flexibility: Polynomials can represent twisted coordinates as well as standard ones

Why Did the Chaos Model Fail?

The Chaos model failed because:

1. Temporal Structure: The reservoir has no recurrent connections or memory, making it unsuitable for temporal dynamics
2. Random Projection: The fixed random matrix doesn't capture the sequential nature of the data
3. Wrong Architecture: This task requires a recurrent network (RNN/LSTM) or an iterative solver, not a static reservoir

The Fundamental Flaw

This experiment revealed a critical flaw in the experimental design: The Chaos model architecture is fundamentally unsuited for temporal prediction tasks. It was designed for static pattern recognition, not dynamical systems.

Implications for Darwin's Cage

What We Learned

1. Polynomial Regression is Underrated

   • It is genuinely coordinate-independent for smooth, short-term dynamics
   • The "Darwinian bias" is actually a strength, not a weakness

2. The Chaos Model is Not a Universal Learner

   • It excels at specific tasks (multiplicative relationships, phase extraction)
   • It fails at others (temporal prediction, division, high-dimensional linear tasks)

3. Architecture Matters More Than Philosophy

   • The question is not "Is the model biased by human concepts?"
   • The question is "Does the architecture match the problem structure?"

Revised Understanding

The "Darwin's Cage" hypothesis conflates two separate issues:

1. Representation Bias: Do we force the model to use human variables?
2. Architectural Suitability: Does the model architecture match the problem?

Experiment A1 shows that architectural suitability dominates. A well-designed "Darwinian" model (polynomial regression) outperforms a poorly-suited "Chaos" model, regardless of coordinate system.

Conclusion

Verdict: The Chaos model is NOT coordinate-independent. It failed in both standard and twisted frames because it lacks the architectural features (recurrence, memory) needed for temporal prediction.

Key Insight: The success of polynomial regression in twisted coordinates demonstrates that "human-derived" mathematical tools (polynomials, calculus) are not arbitrary biases—they are universal approximation tools that work across coordinate systems.

Recommendation: Future experiments should:

1. Use Recurrent Neural Networks or Neural ODEs for dynamical systems
2. Test coordinate independence on static tasks (not temporal prediction)
3. Acknowledge that different architectures are suited for different problems

Files Generated

• experiment_A1_coordinate_independence.py: Main experiment script
• experiment_A1_results.png: Visualization of predictions
• README.md: Experiment overview
• EXPERIMENT_A1_REPORT.md: This report