Experiment 8: Classical vs Quantum Mechanics

Testing Complexity Hypothesis: Simple vs Complex Physics

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 compares cage status between:

• Simple Physics: Classical harmonic oscillator (intuitive, analytical solution)
• Complex Physics: Quantum particle in a box (counterintuitive, discrete states)

Hypothesis: Quantum system (complex) should break the cage, while classical system (simple) should lock it.

Objective

Test whether the complexity of physics affects cage status:

• Simple physics (classical) → Cage Locked (reconstructs human variables)
• Complex physics (quantum) → Cage Broken (distributed solution)

Methodology

Part A: Classical Harmonic Oscillator (Simple)

Physics: $x(t) = A \cos(\omega t + \phi)$

Parameters:

• $A$: Amplitude [0.1, 10.0] m
• $\omega$: Angular frequency [0.5, 5.0] rad/s
• $\phi$: Phase [0, $2\pi$] rad
• $t$: Time [0, 10] s

Input: $[A, \omega, \phi, t]$ Output: Position $x(t)$

Expected: 🔒 CAGE LOCKED (intuitive physics, evolution prepared us)

Part B: Quantum Particle in Box (Complex)

Physics: $|\psi_n(x)|^2 = \frac{2}{L} \sin^2\left(\frac{n\pi x}{L}\right)$

Parameters:

• $n$: Quantum number (discrete: 1-10)
• $L$: Box width [1.0, 10.0] m
• $x$: Position [0, L] m

Input: $[n, L, x]$ Output: Probability density $|\psi|^2$

Expected: 🔓 CAGE BROKEN (counterintuitive physics, evolution didn't prepare us)

Models

• Baseline: Polynomial Regression (degree 4)
• Chaos Model: Optical Chaos (4096 features, brightness optimized)

Results

Standard Performance

Part A: Classical Harmonic Oscillator:

Part B: Quantum Particle in Box:

Critical Finding: Learnability Test

Both problems are genuinely difficult for models without explicit trigonometric features:

• Classical: Polynomial models (degree 2-8) achieve R² < 0.02
• Quantum: Polynomial models achieve R² ≈ 0.45 (better but still low)
• With explicit trigonometric features: Both achieve R² = 1.0 ✅

Conclusion: These problems require trigonometric knowledge that polynomial models cannot learn.

Cage Analysis

Part A: Classical:

• Max correlation with Amplitude: 0.9751
• Max correlation with Omega: 0.9624
• Max correlation with Phase: 0.9744
• Max correlation with Time: 0.9687
• Cage Status: 🔒 LOCKED

Part B: Quantum:

• Max correlation with Quantum_n: 0.9845
• Max correlation with Box_L: 0.9839
• Max correlation with Position_x: 0.9675
• Cage Status: 🔒 LOCKED

Extrapolation Tests

Classical (Train on t < 5, Test on t ≥ 5):

• Darwinian R²: -129.99 ❌
• Chaos R²: -3.78 ❌

Quantum (Train on n ≤ 5, Test on n > 5):

• Darwinian R²: -314.36 ❌
• Chaos R²: -0.59 ❌

Both models fail completely at extrapolation.

Noise Robustness

• Classical (5% noise): R² = -0.26 ❌
• Quantum (5% noise): R² = 0.26 ⚠️

Discussion

Key Findings

1. Both systems have Cage Locked: Contrary to hypothesis, both classical and quantum systems show high correlation with human variables (> 0.96).

2. Low performance: Both models struggle with these problems:

   • Classical: R² = -0.03 (fails completely)
   • Quantum: R² = 0.33 (partial learning)

3. Learnability issue: The problems are genuinely difficult because they require trigonometric functions that polynomial models cannot learn without explicit features.

4. Extrapolation failure: Both models fail completely when extrapolating beyond training ranges.

Why Both Are Locked

Possible explanations:

1. Input reconstruction: The models may be reconstructing input variables directly rather than learning the physics.

2. Trigonometric limitation: Without explicit trigonometric features, the models cannot learn the underlying physics, so they fall back to reconstructing inputs.

3. High correlation doesn't mean physics: High correlation with input variables may indicate the model is using inputs directly, not learning the physical relationship.

Hypothesis Test

Hypothesis: Simple physics locks cage, complex physics breaks it.

Result: ❌ HYPOTHESIS NOT CONFIRMED

• Both systems show Cage Locked status
• No clear difference between simple and complex physics
• Both systems are difficult to learn (low R²)

Limitations

1. Trigonometric functions: Both problems require trigonometric knowledge that standard models struggle with.

2. Low performance: The models may not be learning the physics at all, making cage analysis less meaningful.

3. Extrapolation failure: Complete failure at extrapolation suggests the models are not learning true physical laws.

Conclusion

This experiment does not confirm the complexity hypothesis. Both classical and quantum systems show:

• 🔒 Cage Locked status (high correlation with human variables)
• Low R² scores (models struggle to learn)
• Complete failure at extrapolation

Key Insight: The problems may be too difficult for the models to learn, causing them to fall back to reconstructing input variables rather than learning the physics. This makes cage analysis less meaningful when the model isn't learning the underlying physics.

Future Work:

• Test with problems that don't require trigonometric functions
• Verify that models are actually learning physics before cage analysis
• Consider using models with explicit trigonometric capabilities

Files

• experiment_8_classical_vs_quantum.py: Main experiment code
• benchmark_experiment_8.py: Comprehensive benchmark tests
• test_simple_baseline.py: Learnability validation
• experiment_8_classical_vs_quantum.png: Results visualization

Reproduction

cd experiment_8_classical_vs_quantum
python experiment_8_classical_vs_quantum.py
python benchmark_experiment_8.py
python test_simple_baseline.py