Experiment A2: The Definitive Coordinate Independence Test

Final Report

Date: November 27, 2025
Experiment Type: Coordinate Independence with Proper Architecture
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 A2 tested coordinate independence using LSTM (proper temporal architecture) vs Polynomial regression on chaotic Double Pendulum dynamics in standard and twisted coordinates.

Result: BOTH models are coordinate-independent. This surprising finding challenges the Darwin's Cage hypothesis and reveals a deeper truth about mathematical representations.

Methodology

Physical System

• Double Pendulum: Chaotic system with 4 state variables
• Task: Predict next state from sequence of 20 previous states
• Data: 100 trajectories, 3,600 sequences

Coordinate Transformation

Non-linear "twist" mixing positions and momenta: $u_1 = \theta_1 + 0.5 \sin(\theta_2)$ $u_2 = \theta_2 + 0.5 \cos(\theta_1)$ $v_1 = \omega_1 + 0.5 \tanh(\omega_2)$ $v_2 = \omega_2 + 0.2 \theta_1 \theta_2$

Models

1. Polynomial (Degree 3): Ridge regression on last state
2. LSTM (2-layer, 128 units): Recurrent network on full sequence

Results

Performance Comparison

Key Findings

1. LSTM is Coordinate Independent

   • Excellent performance in both frames (R² ≈ 0.997)
   • Negligible gap (0.002)
   • Confirms proper architecture can learn invariant dynamics

2. Polynomial is ALSO Coordinate Independent

   • Excellent performance in both frames (R² ≈ 0.98)
   • Actually performed slightly better in twisted frame
   • Challenges the assumption that "human math" is coordinate-dependent

3. Both Models Succeed

   • No significant performance degradation in twisted coordinates
   • Both learn the underlying dynamics, not the coordinate representation

Critical Analysis

Why Are Both Models Coordinate Independent?

LSTM Success (Expected)

• Temporal Patterns: LSTM learns sequential dependencies, not explicit functions
• Universal Approximation: Can represent any smooth dynamical system
• Coordinate Agnostic: Internal hidden states adapt to any coordinate system

Polynomial Success (Surprising)

• Local Smoothness: For small time steps (dt=0.05s), dynamics are locally smooth
• Universal Approximation: Polynomials approximate any smooth function locally
• Coordinate Flexibility: Polynomial basis can represent twisted coordinates as well as standard ones

The Deeper Truth

This experiment reveals that coordinate independence is not about "breaking the cage"—it's about:

1. Smoothness: If the dynamics are smooth (differentiable), both polynomial and neural approaches work
2. Time Scale: Short prediction horizons make the problem locally linear/polynomial
3. Architecture Match: Both models are appropriate for this task

What This Means for Darwin's Cage

The Darwin's Cage hypothesis conflates several distinct concepts:

1. Representation Bias: Do we force specific variables (velocity, energy)?
2. Coordinate Dependence: Does performance degrade in twisted coordinates?
3. Architectural Suitability: Is the model appropriate for the task?

Experiment A2 shows:

• ✅ Both models are coordinate-independent (no performance gap)
• ✅ Both models use appropriate architectures (LSTM for sequences, Polynomial for smooth functions)
• ❌ Neither model is "biased" by human coordinates—they learn the underlying dynamics

Implications

1. Polynomial Regression is Underrated

• It is genuinely coordinate-independent for smooth dynamics
• The "Darwinian bias" narrative is misleading
• Polynomials are universal approximators, just like neural networks

2. The "Cage" is a False Dichotomy

• The question is not "human math vs AI"
• The question is "appropriate tool for the task"
• Both polynomials and neural networks are mathematical tools that transcend human bias

3. Architecture Matters More Than Philosophy

• Success depends on matching architecture to problem structure
• LSTM for temporal, Polynomial for smooth, CNN for spatial, etc.
• The "cage" narrative distracts from proper engineering

Comparison with A1

Key Lesson: A1 failed because of wrong architecture, not because of the "cage". A2 succeeds because both architectures are appropriate.

Conclusion

Verdict: The Darwin's Cage hypothesis is NOT SUPPORTED.

Key Findings:

1. Both Polynomial and LSTM are coordinate-independent (gaps < 0.01)
2. Success depends on architectural appropriateness, not "breaking human bias"
3. Mathematical tools (polynomials, neural networks) are universal—they transcend coordinate systems

Final Insight: The "cage" metaphor is misleading. Mathematics—whether "human-derived" (polynomials) or "AI-derived" (neural networks)—provides universal approximation tools that work across coordinate systems. The real question is not "Are we trapped in human concepts?" but rather "Are we using the right tool for the job?"

Recommendation: Future research should focus on:

1. Architectural Design: Matching models to problem structure
2. Generalization: Testing on truly out-of-distribution scenarios
3. Interpretability: Understanding what models learn, not just how well they perform

Files Generated

• experiment_A2_definitive_test.py: Main experiment script
• experiment_A2_results.png: Visualization of predictions
• README.md: Experiment overview
• EXPERIMENT_A2_REPORT.md: This report