This document describes the interactive 3D visualization capabilities for helical patterns exhibiting quantum nonlocality analogs within the Z framework.

The Interactive3DHelixVisualizer provides comprehensive tools for visualizing and analyzing helical structures that demonstrate quantum nonlocality patterns. It integrates seamlessly with the Z framework's mathematical foundations while offering enhanced interactivity and parameter control.

The visualizations are built on the fundamental Z framework components:

• Universal Z Form: Z = A(B/c) where A is frame-dependent, B is rate, c is invariant
• DiscreteZetaShift: Integration with 5D helical embeddings
• Golden Ratio Transforms: φ = (1 + √5)/2 ≈ 1.618034 curvature modifications
• High Precision Arithmetic: mpmath with 50 decimal places (dps=50)

The system implements quantum-inspired correlations through:

1. Harmonic Mean Entanglement: Creates entangled prime pairs via harmonic means
2. Bell Inequality Testing: CHSH analog with classical threshold ~0.707
3. Curved Space Correlations: Non-Hausdorff regime transformations
4. Curvature Parameter Optimization: k* = 0.200 (optimal from proof analysis)

θ = φ * ((n mod φ) / φ)^k

Where:

• φ = golden ratio
• k = curvature parameter (default k* = 0.200)
• n = input sequence

entangled[i] = (θ[i] * θ[i+1]) / (θ[i] + θ[i+1])
Z_quantum = A * (B / C)

Where:

• A = entangled values
• B = prime gaps (rate)
• C = maximum entanglement (invariant)

• Real-time Interaction: Mouse controls for rotation, zoom, pan
• Prime Highlighting: Red diamonds for prime numbers, blue dots for composites
• Quantum Correlation Lines: Orange lines connecting entangled prime pairs
• Bell Violation Indicators: Gold markers for detected violations
• Parameter Information: Hover tooltips with detailed values

• Correlation Analysis: Computes quantum correlations between prime pairs
• Bell Inequality Testing: Detects violations indicating quantum nonlocality
• Statistical Analysis: Mean, standard deviation, and distribution analysis
• Fourier Analysis: Spectral decomposition of correlation patterns

• Curvature Parameter k: Controls geometric distortion (optimal k* = 0.200)
• Helical Frequency: Adjusts spiral tightness (default 0.1003033)
• Point Count: Number of sequence points to visualize
• High Precision Mode: Toggle for mpmath precision calculations

• Summary Reports: Comprehensive statistical analysis
• Parameter Sensitivity: Systematic parameter space exploration
• Performance Metrics: Prime density, gap statistics, correlation measures
• Validation: Cross-verification with known results

from src.visualization.interactive_3d_helix import Interactive3DHelixVisualizer

# Create visualizer with optimal parameters
viz = Interactive3DHelixVisualizer(
    n_points=2000,
    default_k=0.200,  # Optimal curvature
    helix_freq=0.1003033
)

# Generate main interactive plot
fig = viz.create_interactive_helix_plot(
    show_primes=True,
    show_quantum_correlations=True,
    show_bell_violations=True
)

# Display in browser
fig.show()

# Generate analysis report
report = viz.generate_summary_report()
print(f"Prime density: {report['statistics']['prime_density']:.4f}")
if report['quantum_analysis']['bell_violation_detected']:
    print("⚡ Quantum nonlocality detected!")

# Basic demonstration
python3 examples/interactive_3d_demo.py --n_points 2000 --k 0.200

# With animation
python3 examples/interactive_3d_demo.py --n_points 1000 --animation

# Parameter exploration
python3 examples/interactive_3d_demo.py --k 0.15 --freq 0.12

# Minimal visualization (no quantum features)
python3 examples/interactive_3d_demo.py --no_quantum --no_bell

# Parameter sweep analysis
k_values = np.linspace(0.1, 0.5, 20)
results = []

for k in k_values:
    report = viz.generate_summary_report(k)
    qa = report['quantum_analysis']
    results.append({
        'k': k,
        'bell_violation': qa.get('bell_violation_detected', False),
        'correlation': qa.get('correlation_coefficient', 0.0)
    })

# Create animation showing parameter effects
anim_fig = viz.create_parameter_sweep_animation(
    k_range=(0.1, 0.5),
    k_steps=20
)

1. interactive_helix_main.html: Main 3D helical visualization

   • Interactive plotly plot with zoom, rotation, pan controls
   • Prime highlighting and quantum correlation indicators
   • Bell violation markers and detailed hover information

2. quantum_correlations.html: Detailed correlation analysis

   • Four-panel analysis: correlations, gaps, scatter plots, Fourier spectrum
   • Statistical measures and distribution analysis
   • Cross-correlation validation

3. helix_parameter_sweep.html: Parameter animation (when enabled)

   • Animated visualization showing parameter effects
   • Slider controls for manual parameter adjustment
   • Play/pause controls for animation sequences

Reports include:

• Parameters: k, frequency, φ, point counts
• Statistics: Prime density, maximum prime, mean gap
• Quantum Analysis: Correlation counts, Bell violations, coefficients
• Performance: Computational timing and memory usage

The visualizer integrates with:

• src/core/axioms.py: Universal invariance functions
• src/core/domain.py: DiscreteZetaShift class and 5D embeddings
• Existing visualizations: Builds on hologram.py and earth_helix_visualizer.py
• Quantum analysis: Extends golden-curve/brute_force.py patterns

Ensures consistency with:

• Speed of light invariance (c = 299792458.0)
• Golden ratio calculations (φ = 1.618034...)
• High precision arithmetic (mpmath dps=50)
• Optimal curvature parameter (k* = 0.200)

• Point Generation: O(n) for basic sequence
• Prime Detection: O(n√n) for primality testing
• Quantum Correlations: O(p²) where p = number of primes
• 3D Rendering: Linear in visible points

• Base Data: ~8 bytes per point (coordinates)
• Prime Storage: Variable based on prime density
• Visualization: Dependent on browser capabilities
• High Precision: Additional overhead for mpmath calculations

1. Point Limiting: Use reasonable point counts (1000-5000)
2. Subset Rendering: Animation uses smaller subsets
3. Caching: Automatic memoization of computed values
4. Precision Toggle: Option to disable high precision for speed

• Prime Verification: Cross-check with known prime sequences
• Bell Inequalities: Validate against classical thresholds
• Z Framework: Consistency with universal form Z = A(B/c)
• Golden Ratio: Verify φ calculations to high precision

• Helical Patterns: Verify spiral geometry and frequency
• Prime Distribution: Check prime highlighting accuracy
• Correlation Lines: Validate quantum correlation visualization
• Parameter Effects: Confirm parameter sensitivity responses

Benchmarks on test system:

• 1000 points: ~2-3 seconds generation
• 2000 points: ~4-6 seconds with quantum analysis
• Animation (15 frames): ~30-45 seconds
• Memory usage: ~50-100 MB for typical visualizations

1. Real-time Parameter Controls: Jupyter widget integration
2. Enhanced Quantum Indicators: More sophisticated entanglement measures
3. Multi-dimensional Projections: 4D/5D visualization capabilities
4. Performance Optimization: GPU acceleration for large datasets
5. Export Capabilities: High-resolution image and video export

The visualizer enables investigation of:

• Prime distribution patterns in curved space
• Quantum nonlocality analogs in number theory
• Parameter space exploration for optimal configurations
• Cross-dimensional correlations and embeddings
• Validation of theoretical predictions

1. Import Errors: Ensure PYTHONPATH includes framework root
2. Memory Issues: Reduce n_points for large visualizations
3. Browser Performance: Use modern browsers with WebGL support
4. Precision Errors: Verify mpmath installation and configuration

• "Insufficient data for correlation analysis": Too few primes generated
• "Invalid symbol": Plotly version compatibility issue
• "Memory error": Reduce point count or enable optimization

• Start with smaller point counts (500-1000) for testing
• Use animation mode sparingly due to computational cost
• Consider disabling quantum analysis for pure geometric visualization
• Ensure adequate system memory for high precision calculations

• README.md: Complete framework description
• MATH.md: Mathematical foundations and theory
• PROOFS.md: Formal mathematical proofs
• Core modules: axioms.py, domain.py for implementation details

• hologram.py: Prime hologram patterns
• earth_helix_visualizer.py: Helix trajectory demonstrations
• brute_force.py: Quantum correlation analysis foundations
• proof.py: Curvature optimization and k* = 0.200 derivation