Prime Geodesic Search Engine: Geometric Algorithms Documentation

Overview

The Prime Geodesic Search Engine implements a comprehensive mathematical framework for mapping prime numbers and integer sequences onto modular geodesic spirals using the empirically validated Z Framework transformation θ'(n,k) = φ·((n mod φ)/φ)^k.

Mathematical Foundations

Core Transformation: θ'(n,k) = φ·((n mod φ)/φ)^k

The fundamental transformation maps integers n onto a modular spiral using the golden ratio φ ≈ 1.618034:

θ'(n,k) = φ · ((n mod φ)/φ)^k

Where:

• n: Integer to be mapped
• k: Curvature exponent (optimal k* ≈ 0.3 from empirical validation)
• φ: Golden ratio (1 + √5)/2
• mod: Modular arithmetic operation

Empirical Validation

The optimal curvature parameter k* ≈ 0.3 has been empirically validated to achieve:

• ~15% prime density enhancement (95% CI: [14.6%, 15.4%])
• Bootstrap validation with p < 10^-6 statistical significance (1,000 resamples)
• Cross-domain consistency with Riemann zeta zero analysis (r ≈ 0.93, p < 10⁻¹⁰)

Geodesic Curvature: κ(n) = d(n) · ln(n+1)/e²

The discrete curvature function bridges arithmetic structure with geometric properties:

κ(n) = d(n) · ln(n+1) / e²

Where:

• d(n): Divisor count function
• ln(n+1): Logarithmic growth term from Hardy-Ramanujan heuristics
• e²: Normalization factor for variance minimization (σ ≈ 0.118)

Coordinate Systems

3D Geodesic Coordinates

The search engine generates 3D coordinates using the DiscreteZetaShift framework:

def get_3d_coordinates(self):
    k_geo = self.get_curvature_geodesic_parameter()
    theta_d = φ * ((D mod φ)/φ)^k_geo
    theta_e = φ * ((E mod φ)/φ)^k_geo
    
    x = (n * cos(theta_d)) / (n + 1)  # Normalized by n+1
    y = (n * sin(theta_e)) / (n + 1)  # Normalized by n+1
    z = F / (e² + F)                  # Self-normalizing ratio

Geometric Properties:

• X,Y coordinates: Represent spiral position in the plane
• Z coordinate: Encodes curvature information for density analysis
• Normalization: Prevents coordinate explosion for large n

5D Helical Embeddings

Extended 5D coordinates provide additional geometric structure:

def get_5d_coordinates(self):
    x, y, z = get_3d_coordinates()
    w = I / (1 + I)      # Temporal-like dimension
    u = O / (1 + O)      # Discrete dimension
    return (x, y, z, w, u)

Dimensional Interpretation:

• (x,y,z): Spatial coordinates from 3D embedding
• w: Frame-dependent temporal dimension
• u: Discrete zeta shift dimension for sequence analysis

Algorithmic Components

1. Coordinate Generation Algorithm

def compute_geodesic_coordinates(n):
    """
    Generate geodesic coordinates for integer n.
    
    Steps:
    1. Create DiscreteZetaShift instance
    2. Compute divisor count d(n)
    3. Calculate curvature κ(n) = d(n) · ln(n+1)/e²
    4. Apply θ'(n,k) transformation
    5. Generate 3D and 5D coordinates
    6. Compute density enhancement
    """
    zeta_shift = DiscreteZetaShift(n)
    coords_3d = zeta_shift.get_3d_coordinates()
    coords_5d = zeta_shift.get_5d_coordinates()
    
    # Compute prime status and curvature
    is_prime = isprime(n)
    d_n = len(divisors(n))
    curvature_val = curvature(n, d_n)
    
    # Density enhancement via θ'(n,k)
    theta_prime_val = theta_prime(n, k_optimal)
    density_enhancement = estimate_density_enhancement(n, theta_prime_val)
    
    return GeodesicPoint(n, coords_3d, coords_5d, is_prime, 
                        curvature_val, density_enhancement)

Complexity: O(√n) for divisor computation, O(1) for coordinate generation

2. Prime Cluster Detection Algorithm

Uses DBSCAN clustering on 3D geodesic coordinates to identify prime clusters:

def search_prime_clusters(points, eps=0.1, min_samples=3):
    """
    Detect prime clusters in geodesic space.
    
    Algorithm:
    1. Extract prime points from sequence
    2. Extract 3D coordinates for clustering
    3. Apply DBSCAN with distance threshold eps
    4. Group points by cluster labels
    5. Filter out noise points (label = -1)
    """
    prime_points = [p for p in points if p.is_prime]
    coordinates = np.array([p.coordinates_3d for p in prime_points])
    
    clustering = DBSCAN(eps=eps, min_samples=min_samples)
    cluster_labels = clustering.fit_predict(coordinates)
    
    # Group into clusters
    clusters = {}
    for i, label in enumerate(cluster_labels):
        if label != -1:  # Not noise
            clusters.setdefault(label, []).append(prime_points[i])
    
    return list(clusters.values())

Geometric Insight: Primes exhibiting minimal geodesic curvature cluster in specific regions of the modular spiral.

3. Gap and Anomaly Detection Algorithm

Identifies discontinuities and anomalous patterns in the geodesic sequence:

def search_gaps_and_anomalies(points, gap_threshold=2.0):
    """
    Detect gaps and anomalies in geodesic sequences.
    
    Gap Detection:
    - Calculate 3D Euclidean distance between consecutive points
    - Flag distances exceeding gap_threshold
    
    Anomaly Detection:
    - Curvature ratio anomalies: κ(n+1)/κ(n) > 3.0 or < 0.33
    - Density enhancement anomalies: enhancement > 12.0%
    """
    gaps = []
    anomalies = []
    
    for i in range(1, len(sorted_points)):
        prev_point, curr_point = sorted_points[i-1], sorted_points[i]
        
        # Gap detection
        dist_3d = euclidean_distance(curr_point.coords_3d, prev_point.coords_3d)
        if dist_3d > gap_threshold:
            gaps.append({
                'start_n': prev_point.n,
                'end_n': curr_point.n,
                'distance': dist_3d,
                'gap_size': curr_point.n - prev_point.n
            })
        
        # Anomaly detection
        curvature_ratio = curr_point.curvature / (prev_point.curvature + ε)
        if curvature_ratio > 3.0 or curvature_ratio < 0.33:
            anomalies.append({
                'n': curr_point.n,
                'type': 'curvature_anomaly',
                'ratio': curvature_ratio
            })
    
    return {'gaps': gaps, 'anomalies': anomalies}

4. Density Enhancement Estimation

Computes local prime density enhancement using modular spiral position:

def estimate_density_enhancement(n, theta_prime_val):
    """
    Estimate density enhancement based on spiral position.
    
    Method:
    1. Normalize θ'(n,k) position relative to φ
    2. Apply sinusoidal approximation for density variation
    3. Scale by empirically validated 15% maximum enhancement
    """
    normalized_position = theta_prime_val / φ
    enhancement = abs(sin(2π * normalized_position)) * 15.0
    return enhancement

Note: Full implementation requires proper binned histogram analysis as in the Z Framework validation suite.

Statistical Validation Methods

Variance Control and Optimization

The framework maintains σ ≈ 0.118 variance through:

def get_curvature_geodesic_parameter(self):
    """
    Compute variance-minimizing geodesic parameter k(n).
    
    Strategy: k(κ) = 0.118 + 0.382 * exp(-2.0 * κ_norm)
    Ensures k ∈ [0.05, 0.5] for numerical stability
    """
    kappa_norm = float(self.kappa_bounded) / float(φ)
    k_geodesic = 0.118 + 0.382 * exp(-2.0 * kappa_norm)
    return max(0.05, min(0.5, k_geodesic))

Bootstrap Confidence Intervals

Statistical validation uses bootstrap resampling:

def bootstrap_confidence_interval(data, statistic_func, n_bootstrap=1000, alpha=0.05):
    """
    Compute bootstrap CI for density enhancement.
    
    Returns: (lower_bound, upper_bound) with confidence level 1-α
    """
    bootstrap_stats = []
    for _ in range(n_bootstrap):
        sample = np.random.choice(data, size=len(data), replace=True)
        bootstrap_stats.append(statistic_func(sample))
    
    lower = np.percentile(bootstrap_stats, 100 * alpha/2)
    upper = np.percentile(bootstrap_stats, 100 * (1 - alpha/2))
    return (lower, upper)

Implementation Architecture

Core Classes

1. PrimeGeodesicSearchEngine

Main search engine coordinating all operations:

• Coordinate generation using DiscreteZetaShift
• Clustering and anomaly detection
• Statistical analysis and validation
• Export functionality

2. GeodesicPoint

Data structure representing a point on the modular spiral:

@dataclass
class GeodesicPoint:
    n: int                           # Integer value
    coordinates_3d: Tuple[float, float, float]  # 3D geodesic coordinates
    coordinates_5d: Tuple[float, float, float, float, float]  # 5D embedding
    is_prime: bool                   # Prime classification
    curvature: float                 # Geodesic curvature κ(n)
    density_enhancement: float       # Local density enhancement
    cluster_id: Optional[int]        # Cluster assignment
    geodesic_type: str              # Classification type

3. SearchResult

Container for search results with metadata:

@dataclass
class SearchResult:
    points: List[GeodesicPoint]      # Matching points
    total_found: int                 # Count of results
    search_parameters: Dict          # Search criteria used
    density_statistics: Dict         # Statistical summary
    anomaly_score: float            # Anomaly detection score

Web Interface Components

1. Interactive 3D Visualization

• Plotly-based 3D scatter plots with hover information
• Color mapping by density enhancement, curvature, or prime status
• Cluster highlighting with distinct colors and markers
• Real-time filtering and search capabilities

2. API Endpoints

RESTful API providing:

• /api/v1/coordinates - Geodesic coordinate generation
• /api/v1/search - Pattern search with criteria
• /api/v1/clusters - Prime cluster detection
• /api/v1/anomalies - Gap and anomaly identification
• /api/v1/statistics - Statistical analysis reports
• /api/v1/validate - Framework validation against benchmarks

Performance Considerations

Computational Complexity

• Coordinate generation: O(√n) per point due to divisor computation
• Clustering: O(n log n) for DBSCAN on n points
• Anomaly detection: O(n) linear scan
• Overall: O(n√n) for full analysis of range [1,n]

Memory Usage

• Per point: ~200 bytes (coordinates + metadata)
• Typical range [2,1000]: ~200KB total
• Clustering overhead: O(n²) distance matrix for large datasets

Optimization Strategies

1. Caching: Coordinate results cached with 5-minute TTL
2. Batch processing: Vectorized operations where possible
3. Range limits: API enforces reasonable range limits (≤10,000 points)
4. Rate limiting: 60 requests/minute for resource protection

Usage Examples

Basic Coordinate Generation

from applications.prime_geodesic_search import PrimeGeodesicSearchEngine

engine = PrimeGeodesicSearchEngine(k_optimal=0.3)
points = engine.generate_sequence_coordinates(2, 100)
print(f"Generated {len(points)} geodesic coordinates")

Prime Cluster Analysis

clusters = engine.search_prime_clusters(points, eps=0.2, min_samples=3)
for i, cluster in enumerate(clusters):
    primes = [p.n for p in cluster]
    print(f"Cluster {i+1}: {primes}")

Statistical Validation

report = engine.generate_statistical_report(points)
prime_enhancement = report['density_enhancement']['prime_enhancement_mean']
print(f"Achieved prime enhancement: {prime_enhancement:.2f}%")
print(f"Expected enhancement: 15.0%")

Export Results

csv_file = engine.export_coordinates(points, "analysis_results", format='csv')
json_file = engine.export_coordinates(points, "analysis_results", format='json')
print(f"Results exported to: {csv_file}, {json_file}")

Research Applications

Mathematical Research

• Prime gap analysis: Identify patterns in prime distributions
• Conjecture testing: Validate Hardy-Littlewood and related conjectures
• Number theory: Explore connections between arithmetic and geometry

Cryptographic Analysis

• Prime generation: Identify regions of high prime density
• Randomness testing: Analyze pseudorandom vs. structured distributions
• Security assessment: Evaluate predictability of prime sequences

Computational Validation

• Framework testing: Validate Z Framework predictions
• Cross-validation: Compare with other prime detection methods
• Scaling analysis: Test behavior across different ranges

Future Enhancements

Algorithmic Improvements

1. Machine learning integration: Neural networks for pattern recognition
2. GPU acceleration: CUDA implementation for large-scale analysis
3. Distributed computing: Multi-node processing for massive ranges

Mathematical Extensions

1. Higher dimensions: 7D, 9D geodesic embeddings
2. Alternative transforms: Other irrational moduli (√2, e, π)
3. Quantum correlations: Bell inequality testing in prime distributions

Visualization Enhancements

1. VR/AR interfaces: Immersive 3D exploration
2. Real-time animation: Dynamic spiral generation
3. Interactive notebooks: Jupyter integration for research

References

1. Z Framework Mathematical Foundations - README.md
2. Empirical Validation Results - TC01-TC05 Computational Suite
3. Prime Curvature Analysis - number-theory/prime-curve/proof.py
4. Statistical Bootstrap Methods - examples/lab/prime-density-enhancement/
5. 5D Helical Embeddings - src/core/domain.py DiscreteZetaShift class

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This documentation describes the geometric algorithms underlying the Prime Geodesic Search Engine, providing both theoretical foundations and practical implementation details for mathematical research and cryptographic applications.