Experiment B1: Symmetry Discovery (Rotational Invariance)

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 whether an optical chaos model can discover that rotational kinetic energy is invariant under coordinate transformations, without being explicitly told about rotation symmetry. This tests Noether's theorem—the most fundamental principle in physics: every continuous symmetry corresponds to a conservation law.

Scientific Question: Can the model learn that E_rot is the SAME regardless of which direction we call the "x-axis" or "y-axis"?

If successful, this would be definitive evidence of cage-breaking via discovery of geometric symmetry principles.

Motivation

Why This Experiment?

After comprehensive analysis of Experiments 1-10, this experiment was designed to:

1. Test the Deepest Physics Principle: Noether's theorem (symmetry → conservation) is more fundamental than F=ma, Maxwell's equations, or even relativity
2. Fill Critical Gaps: Symmetry discovery has NEVER been tested in the series
3. Avoid Known Weaknesses: No division operations, no trigonometry, no transfer learning
4. Intermediate Dimensionality: Tests the untested range (40 inputs) between low-dim success (3 inputs) and high-dim failure (36 inputs)
5. Clear Falsifiability: Binary test—rotation invariance YES/NO with quantifiable variance

Expected Probability of Success

Based on architectural analysis of Experiments 1-10:

• 80% probability of R² > 0.90 (successful learning)
• 70% probability of cage-breaking (discovering emergent features)
• 90% probability of scientifically valid results (regardless of outcome)

Physics Background

Noether's Theorem

Emmy Noether's Theorem (1918): Every differentiable symmetry of the action of a physical system has a corresponding conservation law.

Rotational Symmetry → Angular Momentum Conservation

In our 2D system:

• Symmetry: Physics laws don't change if we rotate our coordinate system
• Conservation: Angular momentum L_z is conserved
• Consequence: Rotational kinetic energy E_rot = L²/(2I) is rotation-invariant

The Physics Formula

E_rot = L_z² / (2I)

where:
  L_z = Σ m_i × (x_i × vy_i - y_i × vx_i)  [Angular momentum, z-component]
  I = Σ m_i × (x_i² + y_i²)                 [Moment of inertia]

Key Property: E_rot computed in ANY rotated coordinate frame gives the SAME value.

This is computed in the center-of-mass frame to eliminate translation effects.

Experimental Design

System Specification

Physics System: 10 point masses in 2D space

• N = 10 particles with random masses (0.1-10 kg)
• Random spatial configurations: circular, elliptical, scattered, clustered
• Random velocity patterns: rotating, expanding, random, stationary

Input Specification (40 dimensions):

X = [x₁, x₂, ..., x₁₀,    # x-coordinates (10 values)
     y₁, y₂, ..., y₁₀,    # y-coordinates (10 values)
     vx₁, vx₂, ..., vx₁₀,  # x-velocities (10 values)
     vy₁, vy₂, ..., vy₁₀]  # y-velocities (10 values)

Critical Detail: Coordinate frame is RANDOMLY ROTATED for each sample!

Output Specification (1 scalar):

y = E_rot = L_z² / (2I)  [Rotational kinetic energy, rotation-invariant]

Novel Aspect: Rotation Test

The KEY TEST that distinguishes this experiment:

1. Generate base configuration (e.g., 10 particles in specific positions/velocities)
2. Apply 10 different random rotations to the SAME configuration
3. Model predicts E_rot for each rotated version
4. Success: All 10 predictions are nearly identical (variance < 5%)
5. Failure: Predictions change significantly with rotation angle

This directly tests if the model discovered rotation symmetry.

Dataset

Training Set

• Size: 4,000 samples
• Rotation angles: θ ∈ [0, 2π] uniform random
• Configurations: Diverse (circular, elliptical, random, clustered)
• Velocities: Diverse (rotating, expanding, random, stationary)

Test Set

• Size: 1,000 samples
• Same distribution as training

Special Test Sets

1. Rotation Invariance Test: 500 base configs × 10 rotations each
2. Rotation Extrapolation: Train θ ∈ [0, π/4], test θ ∈ [π/4, 2π]
3. Configuration Extrapolation: Train circular, test elliptical/random
4. Noise Robustness: 5% Gaussian noise added

Models

Optical Chaos Machine

Architecture:

1. Random Projection: 40 inputs → 4096 optical features (fixed random matrix)
2. FFT Mixing: Simulates wave interference in frequency domain
3. Intensity Detection: |FFT|² (magnitude squared)
4. Nonlinear Activation: tanh(intensity × brightness)
5. Ridge Readout: Linear regression on optical features

Key: Reservoir layer is FIXED (no backprop). Only readout trains.

Hyperparameters:

• n_features: 4096 (optical reservoir size)
• brightness: 0.001 (tuned via validation)
• alpha: 0.1 (Ridge regularization)

Darwinian Baseline

Architecture:

• Polynomial features (degree 3)
• Ridge regression

Purpose: Verify problem is learnable, compare performance

Benchmark Test Suite

Test 1: Standard Accuracy

• Metric: R² score on held-out test set
• Pass: R² > 0.90

Test 2: Rotation Invariance ⭐ THE KEY TEST

• Protocol: 500 configs × 10 rotations each = 5000 predictions
• Metric: Fraction of configs with relative std < 5%
• Pass: > 85% of configs pass variance criterion
• Interpretation: If PASS, model discovered rotation symmetry!

Test 3: Rotation Magnitude Extrapolation

• Protocol: Train θ ∈ [0°, 45°], test θ ∈ [45°, 360°]
• Metric: R² on extrapolation set
• Pass: R² > 0.80
• Interpretation: Tests if invariance generalizes to unseen angles

Test 4: Configuration Extrapolation

• Protocol: Train on one config type, test on others
• Metric: R² on extrapolation set
• Pass: R² > 0.70
• Interpretation: Tests if learned representation generalizes

Test 5: Noise Robustness

• Protocol: Add 5% Gaussian noise to inputs
• Metric: R² with noisy inputs
• Pass: R² > 0.80
• Interpretation: Tests stability of learned representation

Test 6: Cage Analysis ⭐ CRITICAL INTERPRETATION

• Protocol: Correlate internal optical features with:
  • Cartesian coords (x, y, vx, vy) → If high correlation (>0.9): LOCKED
  • Emergent features (r², v², L_z) → If high correlation (>0.7): BROKEN
• Pass (Cage Broken): max(Cartesian) < 0.5 AND max(emergent) > 0.6
• Interpretation: Did model reconstruct coordinates or discover geometry?

Success Criteria

✅ PASS - Cage-Breaking Confirmed

Requirements:

1. Standard R² > 0.90
2. Rotation invariance pass rate > 0.85
3. Max correlation with Cartesian (x,y) < 0.5
4. Max correlation with emergent (r², L_z) > 0.6
5. At least 2 extrapolation tests R² > 0.70

Interpretation:

• Model discovered rotational symmetry WITHOUT being told
• Learned emergent geometric features (r², angular momentum)
• Did NOT reconstruct Cartesian coordinates
• STRONGEST evidence of cage-breaking in entire series

⚠️ PARTIAL - High Performance, Locked Cage

Requirements:

1. Standard R² > 0.95
2. Rotation invariance pass rate > 0.90
3. Max correlation with Cartesian (x,y) > 0.9

Interpretation:

• Model learned physics accurately
• Discovered rotation invariance
• But did so via coordinate reconstruction
• Still valuable: validates 40D learning capability
• Suggests 40D is transitional threshold

❌ FAIL - Negative Result

Requirements:

1. Standard R² < 0.70 OR
2. Rotation invariance pass rate < 0.50

Interpretation:

• 40 dimensions exceeded architectural threshold
• Energy calculation too complex
• Mitigation: Reduce to N=5 particles (20D)

Scientific Value: Even failure provides valuable data about dimensionality limits

Predicted Outcomes

Scenario A: Success (70% probability)

Expected Results:

• Standard R²: 0.92 - 0.98
• Rotation invariance: 89% pass rate
• Cage: BROKEN (max Cartesian = 0.42, max emergent = 0.78)
• Extrapolation R²: 0.82, 0.74

Impact: Strongest cage-breaking evidence via symmetry discovery

Scenario B: High Performance, Locked Cage (20% probability)

Expected Results:

• Standard R²: 0.97
• Rotation invariance: 94% pass rate
• Cage: LOCKED (max Cartesian = 0.93)

Impact: Refines dimensionality hypothesis, shows 40D is borderline

Scenario C: Failure (10% probability)

Expected Results:

• Standard R²: 0.62
• Rotation invariance: 45% pass rate

Impact: Identifies dimensionality threshold, guides future experiments

Comparison with Previous Experiments

*Only within training distribution

Key Differences:

• B1 is first to test symmetry discovery explicitly
• B1 has intermediate dimensionality (40D) - untested range
• B1 has structured high-D data (10×4) vs. flat 36D in Exp 10
• B1 uses emergent target (total energy) like successful 2-body case

Implementation Details

File Structure

experiment_B1_symmetry/
├── experiment_B1_symmetry.py      # Main experiment (~550 lines)
├── benchmark_experiment_B1.py     # 6 benchmark tests (~450 lines)
├── README.md                      # This file
└── results/                       # Generated at runtime
    ├── experiment_B1_main_results.png
    ├── rotation_invariance_test.png
    ├── cage_analysis.png
    ├── metrics.json
    └── benchmark_results.json

Key Functions

Physics Simulator:

• calculate_rotational_energy() - Computes E_rot = L²/(2I)
• apply_rotation() - Rotates coordinates by angle θ
• generate_base_configuration() - Creates particle system
• generate_sample() - Generates (X, y) with random rotation
• generate_dataset() - Produces full training/test sets

Models:

• OpticalChaosMachine - FFT-based chaos model
• DarwinianModel - Polynomial baseline

Validation:

• Energy invariance check: |E_before - E_after| / E_before < 1e-9
• No NaN/Inf in outputs
• Baseline learnability check: R² > 0.7

How to Run

Prerequisites

pip install numpy matplotlib scikit-learn scipy

Basic Execution

# Run main experiment
python experiment_B1_symmetry.py

# Run comprehensive benchmark suite
python benchmark_experiment_B1.py

Expected Runtime

• Main experiment: ~2-3 minutes (5000 samples)
• Benchmark suite: ~5-7 minutes (includes rotation invariance test with 5000 predictions)

Outputs

1. Console Output: Detailed progress and results
2. Visualizations: Saved to results/ directory
3. Metrics: JSON files with quantitative results

Interpretation Guide

How to Read Results

If R² > 0.90 AND Rotation Invariance Pass Rate > 0.85:

1. Check cage analysis correlations
2. If max(Cartesian) < 0.5 and max(emergent) > 0.6:
   • ✅ CAGE-BREAKING CONFIRMED
   • Model discovered symmetry without being told!
3. If max(Cartesian) > 0.9:
   • ⚠️ HIGH PERFORMANCE, LOCKED CAGE
   • Model learned via coordinate reconstruction

If R² > 0.70 but < 0.90:

• Moderate success
• Check if performance improves with reduced dimensionality (N=5)

If R² < 0.70:

• Failure mode activated
• 40D likely exceeded threshold
• Recommendation: Retry with N=5 particles (20D)

Significance of Rotation Invariance Test

Pass rate > 85% = Model discovered that:

• Physical laws don't depend on coordinate choice
• Energy is the same in all rotated frames
• Rotation is a SYMMETRY of the system

This is deeper than learning a formula—it's learning a structural principle.

Significance of Cage Analysis

BROKEN cage = Model learned:

• r² (radial distance squared) - geometric feature
• v² (speed squared) - geometric feature
• L_z (angular momentum) - rotation-invariant quantity

WITHOUT reconstructing:

• x, y (Cartesian positions)
• vx, vy (Cartesian velocities)

This means the model discovered a representation of physics different from human coordinates.

Scientific Impact

If PASS (Cage-Breaking Confirmed)

Immediate Impact:

• First demonstration of symmetry discovery without human guidance
• Strongest cage-breaking evidence in experimental series
• Validates optical chaos model for geometric learning

Future Directions:

1. Test other symmetries (translational, scaling, gauge)
2. Test with N=3 (12D) to see if even lower dimensionality breaks cage
3. Extend to 3D systems
4. Apply to real physics problems (molecular dynamics, astrophysics)

If PARTIAL (High Performance, Locked Cage)

Immediate Impact:

• Refines dimensionality hypothesis
• Shows 40D is transitional threshold
• Validates physics learning at intermediate dimensionality

Future Directions:

1. Test N=15 (60D) to find exact breaking point
2. Compare with N=5 (20D) to establish gradient
3. Investigate hybrid approaches (structured + chaos)

If FAIL (Poor Performance)

Immediate Impact:

• Identifies architectural limits
• Provides boundary for dimensionality range

Future Directions:

1. Immediate retry with N=5 (20D)
2. Simplify to total KE instead of rotational KE
3. Test alternative architectures

All outcomes provide scientifically valuable information.

Validation Checklist

Before trusting results, verify:

Physics Validation

• Energy invariant under rotation (error < 1e-9)
• Energy range spans 2-3 orders of magnitude
• No NaN/Inf in generated data
• Baseline R² > 0.7 (problem is learnable)

Code Quality

• Fixed random seeds (reproducibility)
• All functions have docstrings
• Rotation matrix tested independently
• Ground truth validated

Architecture

• Brightness tuned (0.0001, 0.001, 0.01, 0.1 tested)
• Ridge alpha verified (0.1 default)
• MinMaxScaler fit on train only (no data leakage)

References

Theoretical Background

1. Noether, E. (1918). "Invariant Variation Problems." Göttinger Nachrichten.
2. Samid, G. (2024). "Darwin's Cage: The Trap of Human-Defined Variables in AI."
3. Angulo, F. (Agnuxo1) (2024). "Physics vs. Darwin: Experimental Validation Series."

Related Experiments

• Experiment 2 (Einstein's Train): Best previous cage-breaking evidence
• Experiment 10 (N-body): Dimensionality effect on cage status
• Experiment 1 (Newtonian): Example of locked cage with high performance

Conclusion

Experiment B1 tests the most fundamental principle in physics: that physical laws are independent of coordinate system choice. If the optical chaos model discovers this WITHOUT being told, it would be definitive proof that AI can learn physics in a fundamentally different way than humans—through emergent geometric features rather than explicit coordinate reconstruction.

This experiment was carefully designed to:

• Avoid all known architectural weaknesses (no division, no trig)
• Fill the biggest gap in the experimental series (symmetry discovery)
• Test the critical intermediate dimensionality range (40D)
• Provide clear, falsifiable predictions
• Deliver scientifically valuable results regardless of outcome

Predicted probability of obtaining clear, interpretable results: 90%

Regardless of whether the cage breaks or locks, we advance understanding of how AI learns physics.

Last Updated: November 27, 2025 Authors: Francisco Angulo (Agnuxo1) & Claude Code Status: Ready for execution