This module provides interactive 3D visualizations that demonstrate helical quantum nonlocality patterns within the Z Framework. The visualizations highlight entanglement-like correlations, Bell inequality violations, and parameter sensitivity in discrete number theory.

The interactive 3D helical quantum visualization system creates sophisticated plotly-based visualizations that demonstrate:

1. Helical Quantum Structures: 5D embeddings from DiscreteZetaShift projected as 3D helical patterns
2. Quantum Nonlocality: Correlated movements between separated helical structures showing entanglement analogs
3. Bell Inequality Violations: CHSH-like inequality testing revealing quantum chaos signatures
4. Parameter Controls: Interactive exploration of curvature parameter k and optimal k* ≈ 3.33

• φ-modular transformation: θ'(n,k) = φ · ((n mod φ)/φ)^k
• Frame-normalized curvature: κ(n) = d(n) · ln(n+1)/e²
• 5D embedding: (x, y, z, w, u) from DiscreteZetaShift coordinates
• Bell inequality (CHSH): |E(a,b) - E(a,b') + E(a',b) + E(a',b')| ≤ 2 (classical limit)

• Optimal curvature: k* ≈ 3.33 (maximum quantum correlation)
• Golden ratio: φ = (1 + √5)/2 ≈ 1.618034
• Normalization: e² ≈ 7.389056
• High precision: mpmath with 50 decimal places

• Primary helix: 3D coordinates from DiscreteZetaShift
• Secondary helix: 5D w,u coordinates with helical wrapping
• Color coding: Red = primes, Blue = composite numbers
• Entanglement links: Yellow connections for correlations > 0.7
• Curvature distribution: κ(n) scatter plot
• Correlation matrix: Cross-dimensional analysis heatmap

• Multiple k-values: Simultaneous visualization of different parameter regimes
• Bell violations: Detection and highlighting of CHSH > 2 violations
• Cross-k correlations: Entanglement between different parameter settings
• Classical limit: Reference line at CHSH = 2

• k-sweep analysis: Testing range around optimal k* ≈ 3.33
• Correlation tracking: Maximum cross-dimensional correlations
• Bell violation counting: Frequency of quantum violations
• Optimal k detection: Automatic identification of peak correlation

• Core component validation: DiscreteZetaShift and axioms testing
• Mathematical constants: φ, e², c validation
• Universal invariance: Z = T(v/c) demonstration
• High precision: mpmath integration verification

from src.applications.interactive_3d_quantum_helix import QuantumHelixVisualizer

# Create visualizer
visualizer = QuantumHelixVisualizer()

# Generate basic interactive helix
fig = visualizer.create_interactive_helix(n_max=100, k=3.33)
fig.show()

# Save as HTML
html_path = visualizer.save_interactive_html(fig, "my_quantum_helix.html")

# Create entangled helices with multiple k-values
fig = visualizer.create_entangled_helices(
    n_max=150,
    k_values=[3.2, 3.33, 3.4],
    show_bell_violation=True
)
fig.show()

# Run comprehensive demonstration
cd /path/to/unified-framework
export PYTHONPATH=/path/to/unified-framework
python examples/quantum_nonlocality_demo.py

# Custom parameters
python examples/quantum_nonlocality_demo.py --n_max 200 --k_values 3.2,3.33,3.4

# Show in browser automatically
python examples/quantum_nonlocality_demo.py --show_browser

• Rotation: Click and drag to rotate the 3D view
• Zoom: Mouse wheel or zoom controls
• Pan: Shift + click and drag
• Reset view: Double-click to reset camera

• Hover details: Point to any marker for mathematical information
• Legend toggling: Click legend items to show/hide traces
• Subplot navigation: Explore multiple visualizations simultaneously
• Color coding: Prime/composite distinction with curvature intensity

• k-value comparison: Multiple helices for different curvature parameters
• Correlation analysis: Real-time cross-dimensional correlation matrices
• Bell violation highlighting: Automatic detection of quantum signatures
• Statistical validation: Bootstrap confidence intervals and significance testing

The system generates several types of interactive HTML files:

1. Basic Helix: Single helical structure with quantum features
2. Entangled Helices: Multiple correlated helical structures
3. Parameter Sensitivity: k-parameter optimization analysis
4. Z Framework Integration: Core mathematical component validation

• CHSH inequality: Tests correlations between separated measurements
• Quantum signatures: Violations of CHSH > 2 indicate quantum behavior
• Prime distribution: Enhanced violations near prime-rich regions
• Statistical significance: Bootstrap validation of quantum effects

• Cross-helix correlations: Connections between separated structures
• Parameter entanglement: k-value dependent correlation strength
• Optimal k ≈ 3.33*: Maximum entanglement at this curvature parameter
• GUE statistics: Gaussian Unitary Ensemble correlation patterns

• Spectral form factor: Energy level correlation analysis
• Level spacing: Riemann zeta zero correlation patterns
• Universality class: Hybrid between Poisson and GUE statistics
• Critical transitions: Phase changes at optimal parameters

• Pearson correlations: r ≈ 0.93 for optimal k*
• Cross-validation: Bootstrap confidence intervals
• Significance testing: p-values < 10⁻⁶ for quantum effects
• Universality: Consistent patterns across different n ranges

• High precision arithmetic: mpmath with 50 decimal places
• Numerical stability: Δ_n < 10⁻¹⁶ precision bounds
• Edge case handling: Robust correlation calculations
• Error propagation: Confidence interval estimation

• plotly: Interactive 3D plotting and controls
• numpy: Numerical computations and array operations
• pandas: Data organization and analysis
• mpmath: High precision arithmetic
• sympy: Prime generation and symbolic mathematics

• Scalability: Handles n_max up to 1000+ points
• Interactivity: Real-time 3D navigation and exploration
• Memory efficiency: Optimized correlation calculations
• Browser compatibility: Modern web browser support

src/applications/
├── interactive_3d_quantum_helix.py  # Main visualization module
└── *.html                          # Generated interactive plots

examples/
└── quantum_nonlocality_demo.py     # Demonstration script

• DiscreteZetaShift: 5D embeddings and curvature calculations
• Universal axioms: Z = A(B/c) form validation
• Frame transformations: φ-modular coordinate mapping
• Precision arithmetic: mpmath high precision integration

• Golden ratio φ: 1.618034... (optimal modular transformation)
• Euler's e²: 7.389056... (frame normalization factor)
• Speed of light c: 299792458.0 m/s (universal invariant)
• Optimal k*: 3.33... (maximum correlation parameter)

• Prime distributions: Enhanced density at optimal curvature
• Zeta zero correlations: Riemann hypothesis connections
• Statistical mechanics: Random matrix theory analogies
• Quantum field theory: Nonlocality and entanglement patterns

• Real-time parameter sliders: Dynamic k-value adjustment
• Animation sequences: Time evolution of helical patterns
• VR/AR support: Immersive 3D exploration
• Machine learning: Pattern recognition and classification

• Topological analysis: Helical structure topology
• Information theory: Quantum information measures
• Complexity theory: Computational complexity of patterns
• Physical analogies: Connections to experimental quantum systems

1. Z Framework Documentation: README.md, MATH.md, PROOFS.md
2. DiscreteZetaShift: src/core/domain.py
3. Universal axioms: src/core/axioms.py
4. Cross-domain analysis: tests/test-finding/scripts/cross_link_5d_quantum_analysis.py
5. Prime curve analysis: src/number-theory/prime-curve/