A comprehensive visualization suite for discrete data using helical and modular-geodesic embeddings, built on the Z Framework's mathematical foundations.

• Generalized θ′(n, k) Embedding: Extends the golden ratio modular transformation for arbitrary discrete datasets
• 3D/5D Helical Visualizations: Interactive helical and spiral plots using Plotly
• Pattern Analysis: Automated detection of clusters, symmetries, and anomalies in geometric space
• Web Interface: Flask/Dash-based interactive web application
• Data Export: Publication-quality export functionality for coordinates, images, and analysis reports
• High Precision: Mathematical computations with mpmath (50 decimal precision)

1. GeneralizedEmbedding: Implements θ′(n, k) transformations and coordinate generation
2. TopologyAnalyzer: Detects patterns, clusters, symmetries, and anomalies
3. VisualizationEngine: Creates interactive 3D/5D visualizations
4. DataExporter: Handles export of coordinates, reports, and images

1. modular_topology_suite.py: Core mathematical and visualization functionality
2. topology_web_interface.py: Interactive web interface using Dash
3. cli_demo.py: Command-line interface for batch processing

# Analyze prime numbers with cluster detection
python3 cli_demo.py --dataset primes --limit 200 --k 0.3 --analyze-clusters --show-summary

# Analyze Fibonacci sequence with full analysis
python3 cli_demo.py --dataset fibonacci --limit 100 --analyze-clusters --analyze-symmetries --analyze-anomalies --export-coords --export-report

# Custom dataset analysis
python3 cli_demo.py --dataset custom --file data.txt --k 0.25 --modulus 2.718 --export-images

# Launch web interface
python3 src/applications/topology_web_interface.py
# Open browser to http://localhost:8050

from modular_topology_suite import GeneralizedEmbedding, TopologyAnalyzer, VisualizationEngine

# Initialize components
embedding = GeneralizedEmbedding(modulus=1.618)
analyzer = TopologyAnalyzer()
visualizer = VisualizationEngine()

# Generate data and embeddings
sequence = [2, 3, 5, 7, 11, 13, 17, 19, 23]
theta_transformed = embedding.theta_prime_transform(sequence, k=0.3)
coordinates = embedding.helical_5d_embedding(sequence, theta_transformed)

# Analyze patterns
clusters, stats = analyzer.detect_clusters(coordinates)
symmetries = analyzer.detect_symmetries(coordinates)
anomalies, scores = analyzer.detect_anomalies(coordinates)

# Create visualizations
fig_3d = visualizer.plot_3d_helical_embedding(coordinates)
fig_clusters = visualizer.plot_cluster_analysis(coordinates, clusters, stats)

The generalized modular-geodesic embedding:

θ′(n, k) = modulus · ((n mod modulus)/modulus)^k

Where:

• n is the input sequence value
• k is the curvature parameter (typically 0.3 for optimal prime enhancement)
• modulus is the modular base (φ ≈ 1.618 for golden ratio)

Maps discrete sequences to 5D helical coordinates:

x = a * cos(θ_D)
y = a * sin(θ_E) 
z = κ(n) = d(n) · ln(n+1)/e²
w = normalized intensity
u = normalized transformed values

Frame-normalized curvature for geodesic analysis:

κ(n) = d(n) · ln(n+1)/e²

Where d(n) is the number of divisors of n.

• Reveals clustering patterns in prime distributions
• Detects geometric anomalies and symmetries
• Optimal curvature parameter k* ≈ 0.3 for 15% density enhancement

• Fibonacci sequences show golden ratio spiral patterns
• Mersenne numbers exhibit exponential growth visualizations
• Custom sequences reveal hidden geometric structures

• Node connectivity patterns in helical space
• Community detection through geometric clustering
• Anomaly detection in network topologies

• CSV format for spreadsheet analysis
• JSON format for web applications
• HDF5 format for large datasets

• PNG/PDF for publications
• HTML for interactive web sharing
• SVG for vector graphics

• Comprehensive JSON reports with:
  • Cluster statistics and properties
  • Symmetry analysis results
  • Anomaly detection metrics
  • Coordinate statistics

Run the comprehensive test suite:

export PYTHONPATH=/path/to/unified-framework
python3 tests/test_modular_topology_suite.py

The test suite covers:

• Mathematical accuracy of transformations
• Visualization component functionality
• Data export/import capabilities
• Performance with large datasets
• Integration workflows

• numpy: Numerical computations
• matplotlib: 2D plotting backend
• plotly: Interactive 3D visualizations
• dash: Web interface framework
• pandas: Data manipulation
• scikit-learn: Machine learning algorithms
• scipy: Scientific computing
• mpmath: High-precision arithmetic
• sympy: Symbolic mathematics

• Small datasets (< 100 points): Sub-second processing
• Medium datasets (100-1000 points): ~2 seconds
• Large datasets (1000+ points): Scales linearly with high precision maintained

The visualization suite extends the existing Z Framework capabilities:

• Builds upon src/core/domain.py DiscreteZetaShift embeddings
• Uses src/core/axioms.py mathematical foundations
• Integrates with existing hologram visualizations in src/number-theory/prime-curve/
• Maintains high-precision arithmetic standards (mpmath dps=50)

• Interactive exploration of number theory concepts
• Visualization of modular arithmetic properties
• Geometric interpretation of prime distributions
• Pattern recognition in discrete sequences

• Analysis of prime number distributions
• Investigation of integer sequence properties
• Network topology analysis
• Anomaly detection in discrete data
• Publication-ready visualization generation

• Support for complex number sequences
• Additional clustering algorithms
• Real-time data streaming capabilities
• Machine learning pattern classification
• Integration with mathematical databases