Experiment D1: Complexity Phase Transition

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

Objective

Systematically map the boundary where cage-breaking begins

This experiment implements a 5-level complexity ladder to empirically determine the exact dimensionality and complexity threshold at which the optical chaos model transitions from reconstructing human variables (LOCKED) to discovering emergent representations (BROKEN).

Strategic Context

Why This Experiment?

After 11 experiments (1-10 + B1), we have identified 3 confirmed cases of cage-breaking:

1. Experiment 2 (Relativity): R²=1.0, max_corr=0.01 (geometric encoding)
2. Experiment 3 (Phase): R²=0.9998 (phase interference)
3. Experiment 10 (N-body): max_corr=0.13 at 36D (dimensionality forcing)

We also have 5 confirmed cases of locked cage:

• Low-dimensional systems (2-3D): Perfect reconstruction
• Architectural failures (division, trig): Fallback to reconstruction

Critical Unknown: What is the exact boundary? When does the transition occur?

Hypothesis

Based on:

• 2-3D: Consistently LOCKED (Exp 1, 10 2-body)
• 36D: Consistently BROKEN (Exp 10 N-body), but performance fails
• 40D: B1 failed (exceeded capacity)

D1 tests the untested intermediate range to find the precise transition point.

Experimental Design

The Complexity Ladder

Five progressive levels in orbital/dynamical mechanics:

Level Descriptions

Level 1: Harmonic Oscillator (4D)

Physics: Simple harmonic motion

x(t) = A * cos(ω*t + φ)

Inputs: [ω, A, φ, t] (4 dimensions) Output: x(t) (displacement)

Characteristics:

• Fully analytical
• Linear system
• No chaos
• Lowest complexity

Expected: LOCKED (model reconstructs ω, A, φ, t)

Rationale: Baseline test to confirm low-D behavior

Level 2: Kepler 2-Body (3D)

Physics: Planetary orbit (Kepler's equation)

r(θ) = a(1 - e²) / (1 + e cos(θ))

Inputs: [a, e, θ] (3 dimensions)

• a: semi-major axis
• e: eccentricity
• θ: true anomaly

Output: r (orbital radius)

Characteristics:

• Integrable system
• Conservation laws (energy, angular momentum)
• Known from Exp 10: R²=0.98, max_corr=0.98

Expected: LOCKED

Rationale: Validates previous results, establishes low-D baseline

Level 3: Restricted 3-Body (6D)

Physics: Circular Restricted 3-Body Problem (CR3BP)

Two massive bodies orbit barycenter, test particle moves in their gravitational field.

Inputs: [x₀, y₀, vx₀, vy₀, μ, t] (6 dimensions)

• (x₀, y₀): Initial position
• (vx₀, vy₀): Initial velocity
• μ: Mass parameter
• t: Time

Output: x(t) (position at time t)

Characteristics:

• Some chaotic regions (near Lagrange points)
• No general analytical solution
• Famous for chaos (horseshoe orbits)
• Critical test: First potentially chaotic system

Expected: TRANSITION (max_corr ≈ 0.5-0.7)

Rationale: THIS IS THE KEY LEVEL - likely where cage begins to break

Level 4: Unrestricted 3-Body (18D)

Physics: Full 3-body problem, all masses free

Inputs: [m₁, m₂, m₃, x₁, y₁, x₂, y₂, x₃, y₃, vx₁, vy₁, vx₂, vy₂, vx₃, vy₃, G, t, target] (18 dimensions)

Output: x_target(t) (position of target body)

Characteristics:

• Fully chaotic
• No general analytical solution (only special cases)
• Sensitive to initial conditions
• High dimensionality (18D)

Expected: BROKEN (max_corr < 0.4)

Rationale: Confirms cage-breaking in intermediate high-D chaotic regime

Level 5: N-Body (42D)

Physics: N=7 gravitational bodies

Inputs: [m₁…m₇, x₁…x₇, y₁…y₇, vx₁…vx₇, vy₁…vy₇, G, t] (42 dimensions)

Output: Total energy E(t)

Characteristics:

• Strongly chaotic
• Known from Exp 10: At N=6 (36D), max_corr=0.13, R²=-0.17 (failure)
• Very high dimensionality

Expected: STRONGLY BROKEN (max_corr < 0.2)

Rationale: Confirms strong cage-breaking but potential performance degradation

Key Metrics

Primary: Cage Status

• max_corr < 0.5: BROKEN
• 0.5 ≤ max_corr < 0.7: TRANSITION
• max_corr ≥ 0.7: LOCKED

Performance

• R² (test): Must be >0.8 for reliable cage analysis
• R² (extrapolation): Tests generalization vs. memorization
• RMSE: Quantifies prediction error

Complexity Indicators

• Dimensionality: Input space size
• Lyapunov exponent: Quantifies chaos (not computed here, but implicit)
• Analytical solution: Yes/No

Success Criteria

Minimum Viable Success (MVS)

1. ✅ Clear monotonic trend: complexity ↑ → max_corr ↓
2. ✅ Levels 1-2 LOCKED (max_corr > 0.7)
3. ✅ Levels 4-5 BROKEN (max_corr < 0.5)
4. ✅ All levels R² > 0.7 (reliable results)

Strong Success

• MVS + Level 3 shows TRANSITION (0.5 < max_corr < 0.7)
• Transition occurs between Levels 2-4 (3D to 18D)
• Extrapolation R² > 0.7 for all levels

Breakthrough Success

• Clear phase transition with sharp boundary
• Quantitative model: max_corr = f(dimensionality, chaos_strength)
• Transfer to predicting cage status for new problems

Falsification Criteria

Experiment FAILS if:

1. No monotonic trend: max_corr doesn't decrease with complexity

   • Implication: Cage status is NOT complexity-dependent

2. All levels LOCKED: Even high-D systems reconstruct variables

   • Implication: Architecture too weak, or human representations more robust

3. Performance degradation: High-D systems have R² < 0.7

   • Implication: Dimensionality threshold exceeded, need stronger architecture

4. Inconsistent with previous experiments: Contradicts Exp 2, 3, 10 findings

   • Implication: Methodology issue, need to reconcile

All failure modes provide valuable information!

Implementation Details

Architecture

Optical Chaos Machine (from B1, validated):

• Random projection: 4096 optical features
• FFT-based interference
• Intensity detection: |FFT|²
• Nonlinear activation: tanh(brightness × intensity)
• Ridge readout: α=0.1

Hyperparameters:

• n_features: 4096
• brightness: 0.001 (tuned from B1)
• alpha: 0.1

Datasets

Training: 3000 samples per level Test: 500 samples per level Extrapolation: 500 samples with extended parameter ranges

Cage Analysis

For each input variable i:

1. Compute features = model.get_features(X)
2. Calculate corr_i = max(|corrcoef(X[:, i], features.T)|)
3. max_corr = max(corr_i for all i)
4. Status = BROKEN if max_corr < 0.5, else LOCKED

How to Run

Prerequisites

pip install numpy matplotlib scikit-learn scipy

Execution

python experiment_D1_complexity_ladder.py

Expected Runtime

• Total: ~15-20 minutes
• Per level: ~3-4 minutes

Outputs

Console: Detailed progress for each level Visualizations: results/level_*.png (5 files) Summary: results/D1_complete_results.json Phase Transition: results/D1_phase_transition_curve.png

Interpretation Guide

Reading Results

If max_corr decreases monotonically: ✅ SUCCESS - Cage status IS complexity-dependent

If transition occurs at Level 3 (6D): ✅ STRONG SUCCESS - Boundary identified precisely

If Levels 1-2 LOCKED, Levels 4-5 BROKEN: ✅ MVS ACHIEVED - Clear boundary exists

If all levels show similar max_corr: ❌ FAILURE - Cage status not complexity-dependent

If high-D levels have R² < 0.7: ⚠️ PARTIAL - Architecture capacity exceeded

Key Questions Answered

1. What is the dimensionality threshold?

   • Answer: The dimension at which max_corr drops below 0.5

2. Is chaos necessary for cage-breaking?

   • Compare Level 2 (integrable) vs. Level 3 (chaotic) at similar dimensions

3. Can the model handle high-D without performance loss?

   • Check if Level 5 maintains R² > 0.8

4. Is the transition sharp or gradual?

   • Examine slope of max_corr vs. dimensionality curve

Connection to Research Program

D1's Role

D1 is Phase 1 of the 4-phase Physics Discovery Engine:

D1 (Boundary Mapping)
  ↓ Identifies threshold τ
D2 (Forced Discovery)
  ↓ Uses τ to design problems
D3 (Law Extraction)
  ↓ Extracts equations from D2
D4 (Cross-Domain Transfer)
  ↓ Tests universality
Physics Discovery Engine

D1 provides the foundation - without knowing where the cage breaks, we cannot systematically force it (D2) or extract emergent laws (D3).

Expected Impact

If Successful:

• Quantitative threshold for cage-breaking
• Design principles for forcing emergent representations
• Validation that complexity alone can break the cage

Enables:

• D2: Design problems at threshold + 50% margin
• D3: Focus on cage-broken models from D1 Levels 4-5
• D4: Test if threshold transfers across domains

Scientific Significance

Immediate Contributions

1. Empirical boundary for cage-breaking in dynamical systems
2. Validation that dimensionality + chaos → cage-breaking
3. Quantitative model for predicting cage status

Future Applications

1. Problem design: Know when to expect emergent representations
2. Architecture design: Match capacity to target complexity
3. Interpretability: Understand when models use novel features

Broader Impact

This is the first systematic mapping of the "Darwin's Cage boundary" - the threshold where AI models transition from reconstructing human variables to discovering genuinely novel representations.

Potential breakthrough: If we can reliably predict and induce cage-breaking, we can systematically discover physics beyond human formulations.

Validation Checklist

Before trusting results:

Physics

• All simulators validated independently
• Energy/momentum conservation checked (where applicable)
• No NaN/Inf in generated data
• Output ranges span 2-3 orders of magnitude

Code Quality

• Fixed random seeds (seed=42)
• No data leakage (scaler fit on train only)
• All functions documented
• Edge cases handled (integration failures)

Consistency

• Level 2 reproduces Exp 10 2-body results (R² > 0.95, max_corr > 0.9)
• Level 5 similar to Exp 10 N-body (max_corr < 0.2)
• No contradictions with previous experiments

References

Theoretical Background

1. Poincaré, H. (1890). "Sur le problème des trois corps et les équations de la dynamique." (3-body problem)
2. Lorenz, E.N. (1963). "Deterministic Nonperiodic Flow." (Chaos theory)
3. Samid, G. (2024). "Darwin's Cage: The Trap of Human-Defined Variables in AI."

Previous Experiments

• Experiment 2 (Relativity): Best cage-breaking (max_corr=0.01)
• Experiment 10 (N-body): Dimensionality effect (36D → max_corr=0.13)
• Experiment B1 (Symmetry): 40D failure (threshold identification)

Next Steps

If D1 Succeeds

Immediate: Analyze phase transition curve

• Fit max_corr = f(dim, chaos)
• Identify optimal complexity for D2

Next Experiment: D2 (Forcing Emergent Representations)

• Use threshold + 50% margin
• Design "representation traps"

If D1 Partially Succeeds

Scenario: High-D levels fail (R² < 0.7)

• Reduce N-body from N=7 to N=5 (30D)
• Increase training data (3000 → 5000)
• Tune brightness parameter

If D1 Fails

Scenario: No monotonic trend

• Re-examine hypothesis
• Test alternative complexity measures (Lyapunov exponent)
• Consider architectural modifications

Conclusion

Experiment D1 is the critical first step in building a systematic Physics Discovery Engine. By empirically mapping the cage-breaking boundary, we establish:

1. When cage-breaking occurs (dimensionality threshold)
2. Why it occurs (complexity overwhelms reconstruction)
3. How to exploit it (design principles for D2)

This experiment transforms cage-breaking from a rare observation (3 cases in 11 experiments) to a systematic capability that can be predicted and induced.

Predicted probability of MVS: 85%

Predicted probability of strong success: 65%

Expected outcome: Clear phase transition between Levels 2-4, enabling systematic exploitation in D2-D4.

Last Updated: November 27, 2025 Authors: Francisco Angulo (Agnuxo1) & Claude Code Status: Ready for execution Expected Runtime: ~15-20 minutes Part of: Physics Discovery Engine Research Program (Phase 1/4)