Experiment A1: Coordinate Independence (The Twisted Cage)

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 tests the "Darwin's Cage" hypothesis by investigating Coordinate Independence. Human physics relies on "good" coordinate systems (Cartesian, Polar) that make equations simple (linear, separable). A true AI physicist should be indifferent to the coordinate representation, learning the underlying topological dynamics regardless of the "viewpoint."

Objective

To determine if the Chaos model can learn physical laws in a "Twisted" coordinate system where human-derived mathematical structures (polynomials, trigonometric functions) are scrambled, while the Darwinian baseline fails.

Methodology

1. Physical System: Double Pendulum

A classic chaotic system with 4 state variables: $(\theta_1, \theta_2, p_1, p_2)$.

• Standard Frame: The equations of motion are complex but composed of standard functions ($\sin, \cos$).
• Twisted Frame: We apply a non-linear diffeomorphism (invertible transformation) to mix positions and momenta: $u_1 = \theta_1 + 0.5 \sin(\theta_2)$ $u_2 = \theta_2 + 0.5 \cos(p_1)$ $v_1 = p_1 + 0.5 p_2^2$ $v_2 = p_2 + 0.5 \theta_1 \theta_2$

2. The Test

We train both models to predict the next state given the current state.

• Task A (Standard): Predict $x_{t+1}$ given $x_t$.
• Task B (Twisted): Predict $u_{t+1}$ given $u_t$.

3. Hypothesis

• Darwinian Model: Will fail in the Twisted frame because the dynamics $u_{t+1} = F(u_t)$ are highly non-polynomial and "ugly" to human math.
• Chaos Model: If it "breaks the cage," it should perform robustly in the Twisted frame, as its reservoir dynamics are universal and not biased towards "clean" equations.

4. Evaluation

We measure the Performance Gap: $\text{Gap} = R^2_{\text{Standard}} - R^2_{\text{Twisted}}$

• Large Gap: The model is "Caged" (relies on good coordinates).
• Small Gap: The model is "Uncaged" (perceives the underlying dynamics).

Expected Outcome

If the Chaos model maintains high accuracy in the Twisted frame while the Darwinian model collapses, we have strong evidence that the AI is learning geometric invariants rather than just fitting human-style equations.