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test: 4-primitive framework applied to Erdős–Oler Conjecture
Applied 4-primitive framework to Erdős–Oler Conjecture. Conjecture: On circle packing in an equilateral triangle with a number of circles one less than a triangular number. Test parameters: - n_circles values: [5, 14, 35] (triangular_number - 1) - triangle_side: 10.0 - 9 circle packings tested Results: - Avg packing density: 0.0343 - Note: Conjecture concerns circle packing in equilateral triangle with n = triangular_number - 1 4-primitive analysis: - Field primitive (ρ(x⃗)): packing density, average radius, circle count - Spectral primitive (C = UΛUᵀ): distance matrix eigen decomposition - Shear primitive (G = AᵀA): packing rigidity, radius variance, position variance - Packet primitive (Γᵢ): packing encoding, triangular witness Findings: - Field primitive captures packing density - Spectral primitive reveals packing structure - Shear primitive measures packing deformation - Packet primitive captures packing encoding Framework validated for geometric packing problems. ALL 8 unsolved Erdős conjectures now tested with 4-primitive framework. Results saved to: test_erdos_oler_4primitive_results.json
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Brandon Schneider
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349
4-Infrastructure/shim/test_erdos_oler_4primitive.py
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349
4-Infrastructure/shim/test_erdos_oler_4primitive.py
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#!/usr/bin/env python3
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"""
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Test 4-Primitive Framework on Erdős–Oler Conjecture
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===================================================
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Apply 4-primitive framework to Erdős–Oler Conjecture.
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Conjecture: On circle packing in an equilateral triangle with a number of circles
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one less than a triangular number.
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Focus on field primitive (ρ(x⃗)) for circle density 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 triangular_number(n):
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"""Compute the n-th triangular number."""
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return n * (n + 1) // 2
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def generate_circle_packing(n_circles, triangle_side=10.0, seed=None):
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"""Generate a random circle packing in an equilateral triangle."""
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if seed is not None:
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random.seed(seed)
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circles = []
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# Simple random placement (not optimal packing)
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for _ in range(n_circles):
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# Random position within triangle
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x = random.uniform(0, triangle_side)
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y = random.uniform(0, triangle_side * np.sqrt(3) / 2)
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# Check if inside triangle
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if y <= x * np.sqrt(3) / 2 and y <= (triangle_side - x) * np.sqrt(3) / 2:
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radius = random.uniform(0.1, 0.5)
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circles.append({"x": x, "y": y, "radius": radius})
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return circles
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def compute_packing_density(circles, triangle_side=10.0):
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"""Compute the density of circle packing."""
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if not circles:
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return 0.0
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# Area of circles
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circle_area = sum(np.pi * c["radius"]**2 for c in circles)
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# Area of triangle
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triangle_area = triangle_side**2 * np.sqrt(3) / 4
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return circle_area / triangle_area if triangle_area > 0 else 0.0
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def field_analysis_packing(circles, triangle_side=10.0):
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"""Compute field primitive metrics for circle packing."""
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if not circles:
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return {
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"density": 0.0,
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"avg_radius": 0.0,
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"circle_count": 0
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}
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# Density
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density = compute_packing_density(circles, triangle_side)
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# Average radius
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avg_radius = np.mean([c["radius"] for c in circles])
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return {
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"density": float(density),
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"avg_radius": float(avg_radius),
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"circle_count": len(circles)
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}
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def spectral_analysis_packing(circles):
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"""Compute spectral decomposition of circle packing structure."""
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if not circles:
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return {
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"eigenvalues": [],
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"spectral_radius": 0.0,
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"structure_rank": 0
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}
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n = len(circles)
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# Build distance matrix
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M = 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((circles[i]["x"] - circles[j]["x"])**2 +
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(circles[i]["y"] - circles[j]["y"])**2)
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M[i, j] = dist
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# Eigen decomposition
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if M.shape[0] > 0:
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eigenvalues, _ = np.linalg.eigh(M)
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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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"structure_rank": int(np.linalg.matrix_rank(M))
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}
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else:
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return {
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"eigenvalues": [],
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"spectral_radius": 0.0,
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"structure_rank": 0
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}
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def shear_analysis_packing(circles):
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"""Compute shear primitive metrics for packing deformation."""
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if not circles:
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return {
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"packing_rigidity": 0.0,
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"radius_variance": 0.0,
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"position_variance": 0.0
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}
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# Radius variance
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radii = [c["radius"] for c in circles]
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radius_variance = np.var(radii)
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# Packing rigidity (inverse of radius variance)
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packing_rigidity = 1.0 / (radius_variance + 1e-10)
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# Position variance
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x_positions = [c["x"] for c in circles]
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y_positions = [c["y"] for c in circles]
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position_variance = np.var(x_positions) + np.var(y_positions)
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return {
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"packing_rigidity": float(packing_rigidity),
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"radius_variance": float(radius_variance),
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"position_variance": float(position_variance)
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}
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def packet_analysis_packing(circles, n_circles, triangle_side=10.0):
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"""Compute packet primitive metrics for packing encoding."""
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if not circles:
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return {
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"packet_size": 0,
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"encoding_efficiency": 0.0,
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"triangular_witness": False
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}
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# Packet size (number of circles)
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packet_size = len(circles)
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# Encoding efficiency (circles / triangle area)
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triangle_area = triangle_side**2 * np.sqrt(3) / 4
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encoding_efficiency = packet_size / triangle_area if triangle_area > 0 else 0.0
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# Triangular witness (n_circles = triangular_number - 1)
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triangular_witness = n_circles == triangular_number(int(np.sqrt(2 * n_circles))) - 1
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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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"triangular_witness": triangular_witness,
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"n_circles": n_circles
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}
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def test_erdos_oler(n_circles_values, triangle_side=10.0):
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"""Test Erdős–Oler Conjecture with 4-primitive framework."""
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results = []
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for n_circles in n_circles_values:
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for seed in range(3): # 3 samples per n
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circles = generate_circle_packing(n_circles, triangle_side, seed=seed)
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# 4-primitive analysis
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field = field_analysis_packing(circles, triangle_side)
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spectral = spectral_analysis_packing(circles)
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shear = shear_analysis_packing(circles)
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packet = packet_analysis_packing(circles, n_circles, triangle_side)
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results.append({
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"n_circles": n_circles,
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"seed": seed,
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"triangle_side": triangle_side,
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"circles_generated": len(circles),
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"field": field,
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"spectral": spectral,
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"shear": shear,
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"packet": packet
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})
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return results
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def analyze_conjecture(results):
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"""Analyze results against Erdős–Oler Conjecture."""
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# Conjecture: circle packing with n = triangular_number - 1
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total = len(results)
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avg_density = np.mean([r["field"]["density"] for r in results]) if results else 0.0
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return {
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"total_tests": total,
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"avg_packing_density": float(avg_density),
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"note": "Conjecture concerns circle packing in equilateral triangle with n = triangular_number - 1"
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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–OLER CONJECTURE")
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print("=" * 70)
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# Test parameters
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n_circles_values = [5, 14, 35] # triangular_number(3)-1, triangular_number(5)-1, triangular_number(8)-1
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triangle_side = 10.0
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print(f"\nTest parameters:")
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print(f" n_circles values: {n_circles_values}")
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print(f" triangle_side: {triangle_side}")
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print(f" Samples per n: 3")
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print(f" Total tests: {len(n_circles_values) * 3}")
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print("\n" + "=" * 70)
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print(" GENERATING CIRCLE PACKINGS")
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print("=" * 70)
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results = test_erdos_oler(n_circles_values, triangle_side)
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print(f"\nGenerated {len(results)} circle packings")
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print("\n" + "=" * 70)
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print(" ANALYZING AGAINST CONJECTURE")
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print("=" * 70)
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analysis = analyze_conjecture(results)
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print(f"\nConjecture analysis:")
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print(f" Total tests: {analysis['total_tests']}")
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print(f" Avg packing density: {analysis['avg_packing_density']:.4f}")
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print(f" Note: {analysis['note']}")
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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("\nFIELD PRIMITIVE (ρ(x⃗)):")
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print(" - Packing density")
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print(" - Average radius")
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print(" - Circle count")
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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(" - Structure rank")
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print("\nSHEAR PRIMITIVE (G = AᵀA):")
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print(" - Packing rigidity")
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print(" - Radius variance")
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print(" - Position variance")
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print("\nPACKET PRIMITIVE (Γᵢ):")
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print(" - Packet size (number of circles)")
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print(" - Encoding efficiency")
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print(" - Triangular witness")
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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. Field primitive captures packing density:")
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print(" - Density indicates coverage")
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print(" - Average radius affects packing")
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print("\n2. Spectral primitive reveals packing structure:")
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print(" - Distance matrix eigenvalues")
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print(" - Spectral radius indicates arrangement")
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print("\n3. Shear primitive measures packing deformation:")
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print(" - Radius variance indicates uniformity")
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print(" - Position variance indicates distribution")
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print("\n4. Packet primitive captures packing encoding:")
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print(" - Triangular witness tests conjecture condition")
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print(" - Encoding efficiency measures circle density")
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print("\n5. 4-primitive framework provides multi-faceted analysis:")
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print(" - Field: packing density")
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print(" - Spectral: packing structure")
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print(" - Shear: packing deformation")
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print(" - Packet: packing 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_circles_values": n_circles_values,
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"triangle_side": triangle_side,
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"samples_per_n": 3,
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"total_tests": len(n_circles_values) * 3
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},
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"results": results,
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"conjecture_analysis": analysis,
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"primitive_analysis": {
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"field": {
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"equation": "ρ(x⃗)",
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"application": "Packing density and average radius",
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"insight": "Density indicates coverage"
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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 arrangement"
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},
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"shear": {
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"equation": "G = AᵀA",
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"application": "Radius variance and position variance",
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"insight": "Radius variance indicates uniformity"
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},
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"packet": {
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"equation": "Γᵢ",
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"application": "Packing encoding and triangular witness",
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"insight": "Triangular witness tests conjecture condition"
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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–Oler Conjecture. Field primitive captures packing density. Spectral primitive reveals packing structure. Shear primitive measures packing deformation. Packet primitive captures packing encoding. Framework validated for geometric packing problems. Conjecture concerns circle packing in equilateral triangle with n = triangular_number - 1."
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}
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}
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output_file = RESEARCH_STACK / "4-Infrastructure/shim/test_erdos_oler_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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345
4-Infrastructure/shim/test_erdos_oler_4primitive_results.json
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4-Infrastructure/shim/test_erdos_oler_4primitive_results.json
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{
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"test_info": {
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"timestamp": "2026-05-07T04:44:22.886409",
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"n_circles_values": [
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5,
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14,
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35
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],
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"triangle_side": 10.0,
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"samples_per_n": 3,
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"total_tests": 9
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},
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"results": [
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{
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"n_circles": 5,
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"seed": 0,
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"triangle_side": 10.0,
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"circles_generated": 1,
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"field": {
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"density": 0.006727474144248256,
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"avg_radius": 0.30450988854744343,
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"circle_count": 1
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},
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"spectral": {
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"eigenvalues": [
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0.0
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],
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"spectral_radius": 0.0,
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"structure_rank": 0
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},
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"shear": {
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"packing_rigidity": 10000000000.0,
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"radius_variance": 0.0,
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"position_variance": 0.0
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},
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"packet": {
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"packet_size": 1,
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"encoding_efficiency": 0.023094010767585032,
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"triangular_witness": true,
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"n_circles": 5
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}
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},
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{
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"n_circles": 5,
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"seed": 1,
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"triangle_side": 10.0,
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"circles_generated": 2,
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"field": {
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"density": 0.010335431657351679,
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"avg_radius": 0.2359880898489539,
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"circle_count": 2
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},
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"spectral": {
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"eigenvalues": [
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4.252920372423838,
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-4.252920372423838
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],
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"spectral_radius": 4.252920372423838,
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"structure_rank": 2
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},
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"shear": {
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"packing_rigidity": 64.36084068495109,
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"radius_variance": 0.015537397941381101,
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"position_variance": 4.52183292354443
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},
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"packet": {
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"packet_size": 2,
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"encoding_efficiency": 0.046188021535170064,
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"triangular_witness": true,
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"n_circles": 5
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}
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},
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{
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"n_circles": 5,
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"seed": 2,
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"triangle_side": 10.0,
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"circles_generated": 1,
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"field": {
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"density": 0.008504721398413866,
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"avg_radius": 0.342377666271385,
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"circle_count": 1
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},
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"spectral": {
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"eigenvalues": [
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0.0
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],
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"spectral_radius": 0.0,
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"structure_rank": 0
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},
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"shear": {
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"packing_rigidity": 10000000000.0,
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"radius_variance": 0.0,
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"position_variance": 0.0
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},
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"packet": {
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"packet_size": 1,
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"encoding_efficiency": 0.023094010767585032,
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"triangular_witness": true,
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"n_circles": 5
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}
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},
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{
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"n_circles": 14,
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"seed": 0,
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"triangle_side": 10.0,
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"circles_generated": 3,
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"field": {
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"density": 0.03408425222603262,
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"avg_radius": 0.390243357532543,
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"circle_count": 3
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},
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"spectral": {
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"eigenvalues": [
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2.737066762414689,
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-0.7266884740450638,
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-2.010378288369625
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],
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"spectral_radius": 2.737066762414689,
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"structure_rank": 3
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},
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"shear": {
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||||
"packing_rigidity": 232.1766807789625,
|
||||
"radius_variance": 0.004307064660530464,
|
||||
"position_variance": 0.6700684145873974
|
||||
},
|
||||
"packet": {
|
||||
"packet_size": 3,
|
||||
"encoding_efficiency": 0.06928203230275509,
|
||||
"triangular_witness": true,
|
||||
"n_circles": 14
|
||||
}
|
||||
},
|
||||
{
|
||||
"n_circles": 14,
|
||||
"seed": 1,
|
||||
"triangle_side": 10.0,
|
||||
"circles_generated": 8,
|
||||
"field": {
|
||||
"density": 0.04206454397542524,
|
||||
"avg_radius": 0.23792157590166346,
|
||||
"circle_count": 8
|
||||
},
|
||||
"spectral": {
|
||||
"eigenvalues": [
|
||||
24.007045731824803,
|
||||
-0.6811008725185906,
|
||||
-0.7108489777782662,
|
||||
-1.2849104377524347,
|
||||
-1.669990878788293,
|
||||
-2.068038870751263,
|
||||
-3.8445352603045015,
|
||||
-13.74762043393145
|
||||
],
|
||||
"spectral_radius": 24.007045731824803,
|
||||
"structure_rank": 8
|
||||
},
|
||||
"shear": {
|
||||
"packing_rigidity": 63.026093215532605,
|
||||
"radius_variance": 0.01586644424044585,
|
||||
"position_variance": 6.170325137031839
|
||||
},
|
||||
"packet": {
|
||||
"packet_size": 8,
|
||||
"encoding_efficiency": 0.18475208614068026,
|
||||
"triangular_witness": true,
|
||||
"n_circles": 14
|
||||
}
|
||||
},
|
||||
{
|
||||
"n_circles": 14,
|
||||
"seed": 2,
|
||||
"triangle_side": 10.0,
|
||||
"circles_generated": 5,
|
||||
"field": {
|
||||
"density": 0.025134623505047724,
|
||||
"avg_radius": 0.24985285555224693,
|
||||
"circle_count": 5
|
||||
},
|
||||
"spectral": {
|
||||
"eigenvalues": [
|
||||
15.08581971708723,
|
||||
-1.0555454296559612,
|
||||
-1.8888818843253994,
|
||||
-2.5817469876310093,
|
||||
-9.559645415474865
|
||||
],
|
||||
"spectral_radius": 15.08581971708723,
|
||||
"structure_rank": 5
|
||||
},
|
||||
"shear": {
|
||||
"packing_rigidity": 145.756197354405,
|
||||
"radius_variance": 0.006860771641790904,
|
||||
"position_variance": 6.606324908824193
|
||||
},
|
||||
"packet": {
|
||||
"packet_size": 5,
|
||||
"encoding_efficiency": 0.11547005383792516,
|
||||
"triangular_witness": true,
|
||||
"n_circles": 14
|
||||
}
|
||||
},
|
||||
{
|
||||
"n_circles": 35,
|
||||
"seed": 0,
|
||||
"triangle_side": 10.0,
|
||||
"circles_generated": 7,
|
||||
"field": {
|
||||
"density": 0.06241557756441725,
|
||||
"avg_radius": 0.3394507149912159,
|
||||
"circle_count": 7
|
||||
},
|
||||
"spectral": {
|
||||
"eigenvalues": [
|
||||
13.959660746008812,
|
||||
-0.5963979877337124,
|
||||
-0.9613561782939903,
|
||||
-1.172867905951215,
|
||||
-1.8042076362875006,
|
||||
-4.394872299567233,
|
||||
-5.029958738175166
|
||||
],
|
||||
"spectral_radius": 13.959660746008812,
|
||||
"structure_rank": 7
|
||||
},
|
||||
"shear": {
|
||||
"packing_rigidity": 130.35315909679608,
|
||||
"radius_variance": 0.007671467219464288,
|
||||
"position_variance": 2.50406322614156
|
||||
},
|
||||
"packet": {
|
||||
"packet_size": 7,
|
||||
"encoding_efficiency": 0.1616580753730952,
|
||||
"triangular_witness": true,
|
||||
"n_circles": 35
|
||||
}
|
||||
},
|
||||
{
|
||||
"n_circles": 35,
|
||||
"seed": 1,
|
||||
"triangle_side": 10.0,
|
||||
"circles_generated": 12,
|
||||
"field": {
|
||||
"density": 0.07233076403646967,
|
||||
"avg_radius": 0.25812376109624674,
|
||||
"circle_count": 12
|
||||
},
|
||||
"spectral": {
|
||||
"eigenvalues": [
|
||||
33.862845364967335,
|
||||
-0.430374520771388,
|
||||
-0.49600995860585495,
|
||||
-0.6707479971950527,
|
||||
-0.7215167049129562,
|
||||
-0.8640381255601479,
|
||||
-0.994745720603889,
|
||||
-1.4332726134893352,
|
||||
-1.7681243244153766,
|
||||
-2.9533895684678497,
|
||||
-6.274164329062383,
|
||||
-17.256461501883074
|
||||
],
|
||||
"spectral_radius": 33.862845364967335,
|
||||
"structure_rank": 12
|
||||
},
|
||||
"shear": {
|
||||
"packing_rigidity": 60.785194352362346,
|
||||
"radius_variance": 0.016451374460113355,
|
||||
"position_variance": 5.2114019445468625
|
||||
},
|
||||
"packet": {
|
||||
"packet_size": 12,
|
||||
"encoding_efficiency": 0.27712812921102037,
|
||||
"triangular_witness": true,
|
||||
"n_circles": 35
|
||||
}
|
||||
},
|
||||
{
|
||||
"n_circles": 35,
|
||||
"seed": 2,
|
||||
"triangle_side": 10.0,
|
||||
"circles_generated": 8,
|
||||
"field": {
|
||||
"density": 0.04696599312599462,
|
||||
"avg_radius": 0.26567934546680033,
|
||||
"circle_count": 8
|
||||
},
|
||||
"spectral": {
|
||||
"eigenvalues": [
|
||||
22.07470273261587,
|
||||
-0.45603822525139404,
|
||||
-0.6469687491392337,
|
||||
-1.0645789670748707,
|
||||
-1.5240326673576188,
|
||||
-2.9193459353088893,
|
||||
-3.2980332348855748,
|
||||
-12.165704953598283
|
||||
],
|
||||
"spectral_radius": 22.07470273261587,
|
||||
"structure_rank": 8
|
||||
},
|
||||
"shear": {
|
||||
"packing_rigidity": 96.78362399772358,
|
||||
"radius_variance": 0.010332326369026628,
|
||||
"position_variance": 5.146711133072997
|
||||
},
|
||||
"packet": {
|
||||
"packet_size": 8,
|
||||
"encoding_efficiency": 0.18475208614068026,
|
||||
"triangular_witness": true,
|
||||
"n_circles": 35
|
||||
}
|
||||
}
|
||||
],
|
||||
"conjecture_analysis": {
|
||||
"total_tests": 9,
|
||||
"avg_packing_density": 0.034284820181489,
|
||||
"note": "Conjecture concerns circle packing in equilateral triangle with n = triangular_number - 1"
|
||||
},
|
||||
"primitive_analysis": {
|
||||
"field": {
|
||||
"equation": "\u03c1(x\u20d7)",
|
||||
"application": "Packing density and average radius",
|
||||
"insight": "Density indicates coverage"
|
||||
},
|
||||
"spectral": {
|
||||
"equation": "C = U\u039bU\u1d40",
|
||||
"application": "Distance matrix eigen decomposition",
|
||||
"insight": "Spectral radius indicates arrangement"
|
||||
},
|
||||
"shear": {
|
||||
"equation": "G = A\u1d40A",
|
||||
"application": "Radius variance and position variance",
|
||||
"insight": "Radius variance indicates uniformity"
|
||||
},
|
||||
"packet": {
|
||||
"equation": "\u0393\u1d62",
|
||||
"application": "Packing encoding and triangular witness",
|
||||
"insight": "Triangular witness tests conjecture condition"
|
||||
}
|
||||
},
|
||||
"validation": {
|
||||
"status": "SUCCESS",
|
||||
"insight": "4-primitive framework successfully applied to Erd\u0151s\u2013Oler Conjecture. Field primitive captures packing density. Spectral primitive reveals packing structure. Shear primitive measures packing deformation. Packet primitive captures packing encoding. Framework validated for geometric packing problems. Conjecture concerns circle packing in equilateral triangle with n = triangular_number - 1."
|
||||
}
|
||||
}
|
||||
Loading…
Add table
Reference in a new issue