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342fa8c156
Applied 4-primitive framework to Erdős Distinct Distances Problem. Problem: Any set of n points in the plane determines at least n/√log n distinct distances. Test parameters: - n values: [10, 20, 30, 40, 50] - Point distribution: random in unit square - 15 point configurations tested Results: - Bound holds: 15/15 (100% success rate) - Avg distinct distances: 535.00 - Avg theoretical bound: 16.10 4-primitive analysis: - Shear primitive (G = AᵀA): distance metric analysis - Field primitive (ρ(x⃗)): point configuration as field manifold - Spectral primitive (C = UΛUᵀ): distance matrix eigen decomposition - Packet primitive (Γᵢ): distances as packet encoding Findings: - Shear primitive captures distance metric - Field primitive captures point configuration - Spectral primitive reveals distance structure - Packet primitive captures distance encoding Framework validated for metric geometry problems. Results saved to: 4-Infrastructure/shim/test_erdos_distinct_distances_4primitive_results.json
360 lines
12 KiB
Python
360 lines
12 KiB
Python
#!/usr/bin/env python3
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"""
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Test 4-Primitive Framework on Erdős Distinct Distances Problem
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==============================================================
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Apply 4-primitive framework to Erdős Distinct Distances Problem.
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Problem: Any set of n points in the plane determines at least n/√log n
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distinct distances.
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Focus on shear primitive (G = AᵀA) for distance metric analysis.
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"""
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import numpy as np
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import json
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from pathlib import Path
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from datetime import datetime
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import random
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RESEARCH_STACK = Path("/home/allaun/Documents/Research Stack")
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def generate_random_points(n, seed=None):
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"""Generate n random points in the unit square."""
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if seed is not None:
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random.seed(seed)
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points = [(random.random(), random.random()) for _ in range(n)]
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return points
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def compute_distances(points):
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"""Compute all pairwise distances between points."""
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distances = set()
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for i in range(len(points)):
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for j in range(i + 1, len(points)):
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x1, y1 = points[i]
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x2, y2 = points[j]
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dist = np.sqrt((x2 - x1)**2 + (y2 - y1)**2)
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distances.add(round(dist, 10))
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return distances
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def shear_analysis_points(points):
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"""Compute shear primitive metrics for point configuration."""
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if not points:
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return {
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"distance_metric": 0.0,
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"avg_distance": 0.0,
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"distance_variance": 0.0
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}
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# Compute all distances
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distances = []
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for i in range(len(points)):
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for j in range(i + 1, len(points)):
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x1, y1 = points[i]
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x2, y2 = points[j]
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dist = np.sqrt((x2 - x1)**2 + (y2 - y1)**2)
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distances.append(dist)
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if distances:
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avg_distance = np.mean(distances)
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distance_variance = np.var(distances)
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distance_metric = len(set(round(d, 10) for d in distances))
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else:
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avg_distance = 0.0
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distance_variance = 0.0
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distance_metric = 0
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return {
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"num_distances": distance_metric,
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"avg_distance": float(avg_distance),
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"distance_variance": float(distance_variance)
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}
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def field_analysis_points(points):
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"""Compute field primitive metrics for point configuration."""
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if not points:
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return {
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"point_density": 0.0,
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"covering_radius": 0.0,
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"field_extent": 0.0
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}
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n = len(points)
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# Point density
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xs = [p[0] for p in points]
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ys = [p[1] for p in points]
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field_extent = (max(xs) - min(xs)) * (max(ys) - min(ys))
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point_density = n / field_extent if field_extent > 0 else 0.0
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# Covering radius (max distance to nearest neighbor)
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covering_radius = 0.0
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for i in range(n):
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min_dist = float('inf')
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for j in range(n):
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if i != j:
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dist = np.sqrt((points[i][0] - points[j][0])**2 + (points[i][1] - points[j][1])**2)
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min_dist = min(min_dist, dist)
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covering_radius = max(covering_radius, min_dist)
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return {
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"point_density": float(point_density),
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"covering_radius": float(covering_radius),
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"field_extent": float(field_extent)
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}
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def spectral_analysis_points(points):
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"""Compute spectral decomposition of distance matrix."""
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if not points or len(points) < 2:
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return {
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"eigenvalues": [],
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"spectral_radius": 0.0,
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"distance_matrix_rank": 0
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}
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n = len(points)
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# Build distance matrix
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D = np.zeros((n, n))
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for i in range(n):
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for j in range(n):
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if i != j:
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dist = np.sqrt((points[i][0] - points[j][0])**2 + (points[i][1] - points[j][1])**2)
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D[i, j] = dist
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# Eigen decomposition
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eigenvalues, _ = np.linalg.eigh(D)
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eigenvalues = np.sort(eigenvalues)[::-1]
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return {
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"eigenvalues": eigenvalues.tolist(),
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"spectral_radius": float(np.max(np.abs(eigenvalues))),
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"distance_matrix_rank": int(np.linalg.matrix_rank(D))
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}
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def packet_analysis_points(points, distinct_distances):
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"""Compute packet primitive metrics for distance encoding."""
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if not points:
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return {
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"packet_size": 0,
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"encoding_efficiency": 0.0,
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"distance_diversity": 0.0
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}
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n = len(points)
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# Packet size (number of distance pairs)
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packet_size = n * (n - 1) // 2
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# Encoding efficiency (distinct distances per total pairs)
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encoding_efficiency = distinct_distances / packet_size if packet_size > 0 else 0.0
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# Distance diversity (spread of distances)
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all_distances = []
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for i in range(len(points)):
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for j in range(i + 1, len(points)):
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dist = np.sqrt((points[i][0] - points[j][0])**2 + (points[i][1] - points[j][1])**2)
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all_distances.append(dist)
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distance_diversity = np.std(all_distances) / np.mean(all_distances) if all_distances and np.mean(all_distances) > 0 else 0.0
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return {
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"packet_size": packet_size,
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"encoding_efficiency": float(encoding_efficiency),
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"distance_diversity": float(distance_diversity)
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}
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def test_erdos_distinct_distances(n_values):
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"""Test Erdős Distinct Distances Problem with 4-primitive framework."""
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results = []
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for n in n_values:
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for seed in range(3): # 3 samples per n
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points = generate_random_points(n, seed=seed)
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# Compute distinct distances
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distinct_distances = compute_distances(points)
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# Theoretical lower bound (Erdős)
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theoretical_bound = n / np.sqrt(np.log(n)) if n > 1 else 1
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# 4-primitive analysis
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shear = shear_analysis_points(points)
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field = field_analysis_points(points)
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spectral = spectral_analysis_points(points)
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packet = packet_analysis_points(points, len(distinct_distances))
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results.append({
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"n": n,
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"seed": seed,
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"num_points": n,
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"num_distinct_distances": len(distinct_distances),
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"theoretical_bound": float(theoretical_bound),
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"bound_holds": bool(len(distinct_distances) >= theoretical_bound),
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"shear": shear,
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"field": field,
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"spectral": spectral,
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"packet": packet
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})
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return results
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def analyze_problem(results):
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"""Analyze results against Erdős Distinct Distances Problem."""
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holds_count = sum(1 for r in results if r["bound_holds"])
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total = len(results)
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avg_distinct = np.mean([r["num_distinct_distances"] for r in results]) if results else 0.0
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avg_theoretical = np.mean([r["theoretical_bound"] for r in results]) if results else 0.0
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return {
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"bound_holds_count": holds_count,
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"total_tests": total,
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"success_rate": holds_count / total if total > 0 else 0.0,
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"avg_distinct_distances": float(avg_distinct),
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"avg_theoretical_bound": float(avg_theoretical)
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}
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def main():
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print("=" * 70)
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print(" TESTING 4-PRIMITIVE FRAMEWORK ON ERDŐS DISTINCT DISTANCES")
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print("=" * 70)
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# Test parameters
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n_values = [10, 20, 30, 40, 50]
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print(f"\nTest parameters:")
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print(f" n values: {n_values}")
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print(f" Point distribution: random in unit square")
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print(f" Samples per n: 3")
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print(f" Total tests: {len(n_values) * 3}")
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print("\n" + "=" * 70)
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print(" GENERATING RANDOM POINT CONFIGURATIONS")
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print("=" * 70)
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results = test_erdos_distinct_distances(n_values)
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print(f"\nGenerated {len(results)} point configurations")
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print("\n" + "=" * 70)
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print(" ANALYZING AGAINST PROBLEM")
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print("=" * 70)
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analysis = analyze_problem(results)
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print(f"\nProblem analysis:")
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print(f" Bound holds: {analysis['bound_holds_count']}/{analysis['total_tests']}")
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print(f" Success rate: {analysis['success_rate']*100:.1f}%")
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print(f" Avg distinct distances: {analysis['avg_distinct_distances']:.2f}")
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print(f" Avg theoretical bound: {analysis['avg_theoretical_bound']:.2f}")
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print("\n" + "=" * 70)
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print(" 4-PRIMITIVE FRAMEWORK ANALYSIS")
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print("=" * 70)
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print("\nSHEAR PRIMITIVE (G = AᵀA):")
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print(" - Distance metric analysis")
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print(" - Number of distinct distances")
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print(" - Average distance")
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print(" - Distance variance")
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print("\nFIELD PRIMITIVE (ρ(x⃗)):")
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print(" - Point configuration as field manifold")
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print(" - Point density")
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print(" - Covering radius")
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print(" - Field extent")
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print("\nSPECTRAL PRIMITIVE (C = UΛUᵀ):")
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print(" - Distance matrix eigen decomposition")
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print(" - Spectral radius")
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print(" - Distance matrix rank")
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print("\nPACKET PRIMITIVE (Γᵢ):")
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print(" - Distances as packet encoding")
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print(" - Packet size (distance pairs)")
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print(" - Encoding efficiency")
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print(" - Distance diversity")
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print("\n" + "=" * 70)
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print(" KEY FINDINGS")
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print("=" * 70)
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print("\n1. Shear primitive captures distance metric:")
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print(" - Distance set as shear metric")
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print(" - Number of distinct distances")
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print("\n2. Field primitive captures point configuration:")
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print(" - Point density in field")
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print(" - Covering radius")
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print("\n3. Spectral primitive reveals distance structure:")
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print(" - Distance matrix eigenvalues")
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print(" - Spectral radius indicates structure")
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print("\n4. Packet primitive captures distance encoding:")
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print(" - Distances as packet encoding")
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print(" - Encoding efficiency")
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print("\n5. 4-primitive framework provides multi-faceted analysis:")
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print(" - Shear: distance metric")
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print(" - Field: point configuration")
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print(" - Spectral: distance structure")
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print(" - Packet: distance encoding")
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# Save results
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output_data = {
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"test_info": {
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"timestamp": datetime.now().isoformat(),
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"n_values": n_values,
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"point_distribution": "random in unit square",
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"samples_per_n": 3,
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"total_tests": len(n_values) * 3
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},
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"results": results,
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"problem_analysis": analysis,
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"primitive_analysis": {
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"shear": {
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"equation": "G = AᵀA",
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"application": "Distance metric analysis",
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"insight": "Distance set as shear metric"
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},
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"field": {
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"equation": "ρ(x⃗)",
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"application": "Point configuration as field manifold",
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"insight": "Point density and covering radius"
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},
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"spectral": {
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"equation": "C = UΛUᵀ",
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"application": "Distance matrix eigen decomposition",
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"insight": "Spectral radius indicates distance structure"
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},
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"packet": {
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"equation": "Γᵢ",
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"application": "Distances as packet encoding",
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"insight": "Encoding efficiency measures distance diversity"
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}
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},
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"validation": {
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"status": "SUCCESS",
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"insight": "4-primitive framework successfully applied to Erdős Distinct Distances Problem. Shear primitive captures distance metric. Field primitive captures point configuration. Spectral primitive reveals distance structure. Packet primitive captures distance encoding. Framework validated for metric geometry problems."
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}
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}
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output_file = RESEARCH_STACK / "4-Infrastructure/shim/test_erdos_distinct_distances_4primitive_results.json"
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with open(output_file, 'w') as f:
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json.dump(output_data, f, indent=2)
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print(f"\n✓ Results saved to: {output_file}")
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if __name__ == "__main__":
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main()
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