Experiment D2: Forcing Emergent Representations via Geometric Encoding

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

Strategic Context

From D1 to D2: The Critical Pivot

Experiment D1 Result: FALSIFIED the hypothesis that dimensionality alone breaks Darwin's Cage.

Critical Discovery:

Evidence:

• D1 Levels 1-5 (3D-44D): ALL remained LOCKED despite increasing complexity
• Exp 2 (Relativity, 2D): BROKEN via photon path geometry (max_corr=0.01)
• Exp 3 (Phase, 128D): BROKEN via complex phase patterns (R²=0.9998)

Conclusion: Representation type > Dimensionality

Experiment D2 Objective

Test the revised hypothesis by forcing cage-breaking through deliberate geometric encoding of physics problems.

Strategy: Create "representation traps" where:

1. Input is encoded as geometric pattern (field, image, trajectory)
2. Hidden physics exists in low-dimensional manifold
3. Human algebraic variables are provably suboptimal
4. Optimal solution requires emergent feature discovery

Three Geometric Problems

Problem 1: Hidden Symmetry (Spherical Wave Field)

Physics: Spherically symmetric wave source creating radial interference

Traditional Encoding (LOCKED in D1):

Input: [x, y, z] Cartesian coordinates (algebraic)
Hidden: r = sqrt(x^2 + y^2 + z^2) (radial distance)

D2 Geometric Encoding:

Input: 2D wave field intensity on 16x16 grid (256 dimensions)
       Flattened pattern: [I(x1,y1), I(x2,y2), ..., I(x16,y16)]

Physics: psi(r) = A * sin(k*r + phi) / r (spherical wave)
Target:  Total energy E = integral |psi|^2 dA

Hidden Variable: Radial parameter r (NOT explicitly given)

Why This Tests Geometric Encoding:

• Grid pattern encodes spatial relationships implicitly
• No explicit r coordinate provided
• Model must discover radial symmetry from field structure
• FFT-based architecture should naturally process spatial patterns

Expected Outcome:

• BROKEN cage (max_corr with k, A, r < 0.5)
• Model learns field-level features, not individual coordinates
• R² > 0.8 (good prediction)

Representation Trap:

• Polynomial on [x,y,z] needs O(N²) terms to capture r²
• Field pattern directly encodes radial structure → O(1)

Problem 2: Trajectory Energy Manifold

Physics: Damped pendulum dynamics with energy conservation

Traditional Encoding (would be LOCKED):

Input: [theta, omega, damping, time] (algebraic)
Hidden: E(theta, omega) = 0.5*omega^2 + (1 - cos(theta))

D2 Geometric Encoding:

Input: Phase space trajectory as IMAGE (16x16 grid = 256 dimensions)
       Trajectory plotted as (theta, omega) over 50 time steps
       Gaussian-smoothed heat map

Target: Initial mechanical energy E

Hidden: Energy manifold (2D in 4D phase space)

Why This Tests Geometric Encoding:

• Trajectory shape encodes dynamical structure
• Energy appears as contour lines in phase space
• No explicit energy coordinate in image
• Model must discover energy from trajectory geometry

Expected Outcome:

• BROKEN cage (max_corr with theta0, omega0 < 0.5)
• Model learns trajectory-level features
• Discovers energy manifold from image patterns
• R² > 0.7 (moderate due to damping complexity)

Representation Trap:

• Energy calculation requires knowing instantaneous (theta, omega)
• Trajectory image provides global geometric structure
• Optimal: Discover energy contours directly from shape

Problem 3: Topological Invariant (Velocity Field)

Physics: Vortex dynamics with discrete winding number

Encoding (Already Geometric - Control Test):

Input: Velocity field [vx, vy] on 16x16 grid (512 dimensions)
       vx flattened (256) + vy flattened (256)

Physics: Multi-vortex flow field
Target:  Winding number W in {-2, -1, 0, 1, 2} (classification)

Hidden: W = (1/2pi) * integral curl(v) dA (global topological invariant)

Why This Was Already Correct:

• Velocity field is naturally geometric (vector field on grid)
• Winding number is topological (survives continuous deformation)
• Requires global spatial integration (local features insufficient)

Expected Outcome:

• BROKEN cage (max_corr with individual vortex params < 0.4)
• Model discovers topological structure
• Classification accuracy > 0.85
• Features cluster by winding number

Representation Trap:

• Winding number requires line integral around domain boundary
• Individual vortex positions/strengths don't directly give W
• Optimal: Discover global topological structure from field pattern

Success Criteria

Minimum Viable Success (MVS)

Per Problem:

1. ✅ Good performance (R² > 0.7 or Accuracy > 0.75)
2. ✅ Low correlation with human variables (max_corr < 0.5)
3. ✅ BROKEN cage status

Overall:

• At least 2/3 problems achieve BROKEN cage
• Clear evidence that geometric encoding works

Strong Success

• All 3 problems achieve BROKEN cage
• max_corr < 0.3 (very low correlation)
• Can reconstruct optimal hidden variable from features (R² > 0.9)

Breakthrough Success

• Zero-shot discovery: Model finds hidden variable NOT in training
• Generalization to new geometric encodings
• Transferable geometric features across problems

Comparison with D1

Implementation Details

Architecture

Optical Chaos Machine (unchanged from D1):

• 4096 optical features
• FFT-based interference
• brightness=0.001 (validated)
• Ridge/Logistic readout

Why This Architecture:

• FFT naturally processes spatial patterns
• Designed for frequency/geometric features
• Proved effective for phase patterns (Exp 3)
• Failed on algebraic inputs (D1) → validates hypothesis

Datasets

Per Problem:

• Training: 2000 samples
• Test: 500 samples

Why Smaller Than D1:

• Geometric patterns more information-dense
• Lower sample efficiency expected
• Focus on cage-breaking, not performance optimization

Cage Analysis

Unified Method:

analyze_cage(model, X_test, hidden_variables_dict)

Checks:

1. Extract features from trained model
2. Compute max_corr(feature_i, hidden_var_j) for all i, j
3. Status: BROKEN if max_corr < 0.5

Hidden Variables Tested:

• Problem 1: k, A, r_center
• Problem 2: theta0, omega0, energy
• Problem 3: winding_number, n_vortices

Validation Checklist

Physics Correctness

• Problem 1: Wave equation satisfied
• Problem 2: Pendulum dynamics correct
• Problem 3: Winding number computed correctly

Encoding Quality

• Problem 1: Field patterns show radial symmetry visually
• Problem 2: Trajectory images show energy contours
• Problem 3: Velocity fields show vortices

Code Quality

• Fixed random seeds (seed=42)
• No data leakage
• Proper train/test split
• Cage analysis handles zero-variance features

Expected Results

Hypothesis Testing

If geometric encoding hypothesis is CORRECT:

• 2-3 problems show BROKEN cage
• max_corr < 0.5 across the board
• Performance remains good (R² > 0.7)
• Clear difference from D1 algebraic results

If geometric encoding hypothesis is WRONG:

• All problems remain LOCKED
• max_corr > 0.7 like D1
• Architecture-specific limitation (not general principle)

Quantitative Predictions

Implications for Research Program

If D2 Succeeds (2+ BROKEN)

Validates:

• Geometric encoding as systematic cage-breaking method
• D1 insight: Representation type > Dimensionality
• Optical chaos architecture effectiveness for geometric inputs

Enables:

• D3 (Law Extraction): Use D2 cage-broken models
• Symbolic regression on geometric features
• Discover laws in representation space

Enables:

• D4 (Transfer): Test if geometric principles transfer
• "Conservation structure", "Symmetry structure", "Topology structure"

If D2 Fails (0-1 BROKEN)

Implications:

• Geometric encoding alone insufficient
• Architecture-specific limitation
• Need additional ingredients (e.g., explicit symmetry constraints)

Next Steps:

• Test alternative architectures (CNN, GNN, Transformer)
• Hybrid approach (geometric + symbolic)
• Smaller grid sizes (8x8 instead of 16x16)

How to Run

Prerequisites

pip install numpy matplotlib scikit-learn scipy

Execution

python experiment_D2_geometric_forcing.py

Expected Runtime

Total: ~5-8 minutes

• Problem 1: ~2 minutes
• Problem 2: ~3 minutes (trajectory generation slow)
• Problem 3: ~2 minutes

Outputs

Visualizations (3 files):

• results/problem_1_Spherical_Wave_Field.png
• results/problem_2_Trajectory_Energy_Manifold.png
• results/problem_3_Topological_Invariant.png

Data:

• results/D2_complete_results.json

Console: Detailed progress and cage analysis

Interpretation Guide

Reading Results

If Problem shows BROKEN: ✅ SUCCESS - Geometric encoding works!

• max_corr < 0.5 confirms emergence
• Model learned geometric features
• Validates D1 insight

If Problem shows TRANSITION: ⚠️ PARTIAL - Some geometric learning

• 0.5 < max_corr < 0.7
• Mixed representation
• May need architectural tuning

If Problem shows LOCKED: ❌ UNEXPECTED - Geometric encoding failed

• max_corr > 0.7
• Similar to D1 algebraic results
• Requires investigation

Key Questions Answered

1. Does geometric encoding break the cage?

   • Count BROKEN statuses
   • Compare max_corr to D1 (0.9+)

2. Which geometric encoding works best?

   • Compare max_corr across 3 problems
   • Field > Trajectory > ?

3. Is this architecture-specific?

   • If all fail: Yes (try CNN/GNN)
   • If 2+ succeed: No (general principle)

Scientific Significance

Immediate Contributions

1. First systematic test of geometric encoding hypothesis
2. Direct comparison with algebraic encoding (D1)
3. Validation that representation type matters
4. Practical method for forcing cage-breaking

Research Program Impact

D2 Success → Enables D3-D4:

• Symbolic regression on geometric features
• Transfer geometric principles across domains
• Build Physics Discovery Engine

D2 Failure → Refines Theory:

• Architecture constraints identified
• Alternative approaches explored
• Limits of current methods established

Broader AI Science Impact

If successful:

• Demonstrates AI can discover non-human representations
• Provides design principles for scientific AI
• Shows importance of problem encoding vs. model capacity

Conclusion

Experiment D2 is the critical test of the D1 insight. By systematically applying geometric encoding to three physics problems, we directly validate whether:

This experiment transforms the accidental cage-breaking observations (Exp 2, 3) into a systematic methodology for inducing emergent representations.

Success criterion: 2+ problems with BROKEN cage (max_corr < 0.5)

Expected outcome: Validation that geometric encoding is the KEY to systematic cage-breaking, enabling D3 (law extraction) and D4 (transfer learning).

Last Updated: November 27, 2025 Authors: Francisco Angulo (Agnuxo1) & Claude Code Status: Ready for execution Expected Runtime: 5-8 minutes Part of: Physics Discovery Engine Research Program (Phase 2/4)