# Payment Network Analysis UHG Theorem Mapping

## Core Network Structure Theorems

### Network Topology Fundamentals
1. **Join of points (1)** - Connect merchants to processors
2. **Meet of lines (2)** - Intersect transaction paths
3. **Collinear points (3)** - Identify linear transaction chains
4. **Concurrent lines (4)** - Detect converging transaction paths

### Network Hierarchy
5. **Line through null points (5)** - Model payment gateway connections
6. **Point on null lines (6)** - Represent terminal endpoints
7. **Perpendicular point (7)** - Model cross-network connections
8. **Perpendicular line (8)** - Represent inter-bank routes

### Network Relationships
9. **Opposite points (9)** - Model merchant-acquirer pairs
10. **Opposite lines (10)** - Represent dual transaction paths
11. **Altitude line (11)** - Model direct clearing paths
12. **Altitude point (12)** - Represent clearing nodes

## Transaction Path Analysis

### Path Parametrization
17. **Parametrizing a line (17)** - Model transaction sequences
18. **Parametrizing a point (18)** - Represent transaction states
19. **Parametrizing a join (19)** - Model merging transaction paths
20. **Parametrizing a meet (20)** - Represent splitting transactions

### Special Path Cases
21. **Parametrization of null points (21)** - Model failed transactions
22. **Parametrization of null lines (22)** - Represent blocked routes
23. **Join of null points (23)** - Connect failed transaction patterns
24. **Meet of null lines (24)** - Intersect blocked routes

## Transaction Metrics

### Distance Measures
32. **Quadrance (32)** - Measure transaction similarity
33. **Zero quadrance (33)** - Identify identical transactions
34. **Spread (34)** - Measure transaction dispersion
35. **Zero spread (35)** - Detect transaction convergence

### Transaction Relationships
37. **Quadrance cross ratio (37)** - Measure transaction path relationships
38. **Triple quad formula (38)** - Analyze three-way transaction patterns
39. **Triple spread formula (39)** - Measure transaction spread patterns

## Fraud Detection Patterns

### Pattern Analysis
47. **Quadrea (47)** - Measure transaction area patterns
48. **Quadreal (48)** - Analyze transaction reality checks
49. **Quadrea quadreal product (49)** - Combine pattern metrics
50. **Cross law (50)** - Analyze crossing transaction patterns

### Midpoint Analysis
54. **Midpoints (54)** - Identify transaction midpoints
55. **Midlines (55)** - Analyze transaction path centers
56. **Triple quad mid (56)** - Analyze three-way midpoints
57. **Pythagoras mid (57)** - Measure midpoint relationships

## Anomaly Detection

### Triangle Relations
75. **Triangle proportions (75)** - Analyze three-party relationships
76. **Menelaus (76)** - Detect collinear anomalies
77. **Menelaus' dual (77)** - Analyze dual anomaly patterns
78. **Ceva (78)** - Detect concurrent anomalies
79. **Ceva's dual (79)** - Analyze dual concurrency patterns

### Special Cases
80. **Nil cross law (80)** - Analyze null transaction crosses
81. **Doubly nil triangle (81)** - Detect double-null patterns
82. **Triply nil quadreal (82)** - Analyze triple-null patterns
83. **Triply nil balance (83)** - Check triple-null balance

## Implementation Priority

1. **Core Network Structure**
   - Join of points (1)
   - Meet of lines (2)
   - Collinear points (3)
   - Concurrent lines (4)

2. **Transaction Path Analysis**
   - Parametrizing a line (17)
   - Parametrizing a point (18)
   - Parametrizing a join (19)
   - Parametrizing a meet (20)

3. **Transaction Metrics**
   - Quadrance (32)
   - Spread (34)
   - Quadrance cross ratio (37)
   - Triple quad formula (38)

4. **Fraud Detection**
   - Quadrea (47)
   - Quadreal (48)
   - Cross law (50)
   - Midpoints (54)

5. **Anomaly Detection**
   - Triangle proportions (75)
   - Menelaus (76)
   - Ceva (78)
   - Nil cross law (80)

## Next Steps

1. **Core Implementation**
   - Implement basic point and line operations
   - Add network topology fundamentals
   - Create transaction path representations

2. **Metric Development**
   - Implement quadrance calculations
   - Add spread measurements
   - Create cross ratio analysis

3. **Pattern Analysis**
   - Implement pattern detection
   - Add anomaly detection
   - Create fraud detection metrics

4. **Integration**
   - Connect with Splunk SOAR
   - Add real-time processing
   - Create alert mechanisms 