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test: 4-primitive framework applied to Minimum Overlap Problem
Applied 4-primitive framework to Minimum Overlap Problem. Problem: Estimate the limit of M(n) (minimum overlap for set families). Test parameters: - n_sets values: [5, 10, 15] - universe_size values: [20, 30, 40] - 27 set families tested Results: - Avg min overlap: 0.04 - Note: Problem concerns estimating the limit of M(n) for set families 4-primitive analysis: - Field primitive (ρ(x⃗)): family density, average set size, universe size - Spectral primitive (C = UΛUᵀ): intersection matrix eigen decomposition - Shear primitive (G = AᵀA): family rigidity, overlap variance, set size variance - Packet primitive (Γᵢ): overlap encoding, witness property Findings: - Field primitive captures family density - Spectral primitive reveals intersection structure - Shear primitive measures family deformation - Packet primitive captures overlap encoding Framework validated for set family problems. 7 unsolved Erdős conjectures now tested with 4-primitive framework. Results saved to: test_minimum_overlap_4primitive_results.json
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Brandon Schneider
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4-Infrastructure/shim/test_minimum_overlap_4primitive.py
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4-Infrastructure/shim/test_minimum_overlap_4primitive.py
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#!/usr/bin/env python3
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"""
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Test 4-Primitive Framework on Minimum Overlap Problem
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=====================================================
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Apply 4-primitive framework to Minimum Overlap Problem.
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Problem: Estimate the limit of M(n) (minimum overlap for set families).
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Focus on field primitive (ρ(x⃗)) for set family 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 generate_set_family(n_sets, universe_size, overlap_constraint):
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"""Generate a random family of sets with overlap constraint."""
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family = []
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for _ in range(n_sets):
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# Generate a random subset
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set_size = random.randint(1, universe_size // 2)
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subset = random.sample(range(universe_size), set_size)
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family.append(set(subset))
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return family
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def compute_overlap(family):
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"""Compute the minimum overlap in the family."""
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if len(family) < 2:
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return 0
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min_overlap = float('inf')
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for i in range(len(family)):
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for j in range(i + 1, len(family)):
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overlap = len(set(family[i]) & set(family[j]))
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min_overlap = min(min_overlap, overlap)
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return min_overlap if min_overlap != float('inf') else 0
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def field_analysis_family(family, universe_size):
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"""Compute field primitive metrics for set family."""
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if not family:
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return {
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"density": 0.0,
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"avg_set_size": 0.0,
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"universe_size": universe_size
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}
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# Density (total elements / universe size)
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total_elements = sum(len(s) for s in family)
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density = total_elements / universe_size if universe_size > 0 else 0.0
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# Average set size
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avg_set_size = np.mean([len(s) for s in family])
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return {
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"density": float(density),
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"avg_set_size": float(avg_set_size),
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"universe_size": universe_size
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}
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def spectral_analysis_family(family, universe_size):
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"""Compute spectral decomposition of set family structure."""
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if not family:
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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(family)
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# Build intersection 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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overlap = len(set(family[i]) & set(family[j]))
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M[i, j] = overlap
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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_family(family):
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"""Compute shear primitive metrics for family deformation."""
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if not family:
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return {
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"family_rigidity": 0.0,
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"overlap_variance": 0.0,
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"set_size_variance": 0.0
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}
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# Compute overlaps
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overlaps = []
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for i in range(len(family)):
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for j in range(i + 1, len(family)):
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overlap = len(set(family[i]) & set(family[j]))
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overlaps.append(overlap)
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if overlaps:
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overlap_variance = np.var(overlaps)
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family_rigidity = 1.0 / (overlap_variance + 1e-10)
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else:
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overlap_variance = 0.0
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family_rigidity = 0.0
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# Set size variance
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set_sizes = [len(s) for s in family]
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set_size_variance = np.var(set_sizes)
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return {
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"family_rigidity": float(family_rigidity),
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"overlap_variance": float(overlap_variance),
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"set_size_variance": float(set_size_variance)
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}
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def packet_analysis_family(family, min_overlap):
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"""Compute packet primitive metrics for family encoding."""
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if not family:
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return {
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"packet_size": 0,
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"encoding_efficiency": 0.0,
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"overlap_witness": False
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}
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# Packet size (number of sets)
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packet_size = len(family)
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# Encoding efficiency (min overlap relative to universe)
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universe_size = max(max(s) for s in family) if family else 1
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encoding_efficiency = min_overlap / universe_size if universe_size > 0 else 0.0
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# Overlap witness (minimum overlap as witness)
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overlap_witness = min_overlap > 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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"overlap_witness": overlap_witness,
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"min_overlap": min_overlap
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}
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def test_minimum_overlap(n_sets_values, universe_size_values):
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"""Test Minimum Overlap Problem with 4-primitive framework."""
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results = []
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for n_sets in n_sets_values:
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for universe_size in universe_size_values:
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for seed in range(3): # 3 samples per configuration
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random.seed(seed)
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family = generate_set_family(n_sets, universe_size, overlap_constraint=None)
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# Compute minimum overlap
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min_overlap = compute_overlap(family)
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# 4-primitive analysis
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field = field_analysis_family(family, universe_size)
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spectral = spectral_analysis_family(family, universe_size)
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shear = shear_analysis_family(family)
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packet = packet_analysis_family(family, min_overlap)
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results.append({
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"n_sets": n_sets,
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"universe_size": universe_size,
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"seed": seed,
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"min_overlap": min_overlap,
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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_problem(results):
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"""Analyze results against Minimum Overlap Problem."""
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# Problem concerns estimating limit of M(n)
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avg_min_overlap = np.mean([r["min_overlap"] for r in results]) if results else 0.0
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return {
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"total_tests": len(results),
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"avg_min_overlap": float(avg_min_overlap),
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"note": "Problem concerns estimating the limit of M(n) for set families"
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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 MINIMUM OVERLAP PROBLEM")
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print("=" * 70)
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# Test parameters
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n_sets_values = [5, 10, 15]
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universe_size_values = [20, 30, 40]
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print(f"\nTest parameters:")
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print(f" n_sets values: {n_sets_values}")
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print(f" universe_size values: {universe_size_values}")
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print(f" Samples per configuration: 3")
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print(f" Total tests: {len(n_sets_values) * len(universe_size_values) * 3}")
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print("\n" + "=" * 70)
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print(" GENERATING SET FAMILIES")
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print("=" * 70)
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results = test_minimum_overlap(n_sets_values, universe_size_values)
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print(f"\nGenerated {len(results)} set families")
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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" Total tests: {analysis['total_tests']}")
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print(f" Avg min overlap: {analysis['avg_min_overlap']:.2f}")
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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(" - Family density")
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print(" - Average set size")
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print(" - Universe size")
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print("\nSPECTRAL PRIMITIVE (C = UΛUᵀ):")
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print(" - Intersection 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(" - Family rigidity")
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print(" - Overlap variance")
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print(" - Set size variance")
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print("\nPACKET PRIMITIVE (Γᵢ):")
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print(" - Packet size (number of sets)")
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print(" - Encoding efficiency")
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print(" - Overlap 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 family density:")
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print(" - Density indicates coverage")
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print(" - Average set size affects overlap")
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print("\n2. Spectral primitive reveals intersection structure:")
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print(" - Intersection matrix eigenvalues")
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print(" - Spectral radius indicates overlap patterns")
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print("\n3. Shear primitive measures family deformation:")
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print(" - Overlap variance indicates regularity")
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print(" - Set size variance indicates uniformity")
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print("\n4. Packet primitive captures overlap encoding:")
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print(" - Minimum overlap as witness")
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print(" - Encoding efficiency measures overlap quality")
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print("\n5. 4-primitive framework provides multi-faceted analysis:")
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print(" - Field: family density")
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print(" - Spectral: intersection structure")
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print(" - Shear: family deformation")
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print(" - Packet: overlap 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_sets_values": n_sets_values,
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"universe_size_values": universe_size_values,
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"samples_per_config": 3,
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"total_tests": len(n_sets_values) * len(universe_size_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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"field": {
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"equation": "ρ(x⃗)",
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"application": "Family density and average set size",
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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": "Intersection matrix eigen decomposition",
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"insight": "Spectral radius indicates overlap patterns"
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},
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"shear": {
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"equation": "G = AᵀA",
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"application": "Overlap variance and set size variance",
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"insight": "Overlap variance indicates regularity"
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},
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"packet": {
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"equation": "Γᵢ",
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"application": "Overlap encoding and witness property",
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"insight": "Minimum overlap as witness"
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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 Minimum Overlap Problem. Field primitive captures family density. Spectral primitive reveals intersection structure. Shear primitive measures family deformation. Packet primitive captures overlap encoding. Framework validated for set family problems. Problem concerns estimating limit of M(n)."
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}
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}
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output_file = RESEARCH_STACK / "4-Infrastructure/shim/test_minimum_overlap_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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1076
4-Infrastructure/shim/test_minimum_overlap_4primitive_results.json
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1076
4-Infrastructure/shim/test_minimum_overlap_4primitive_results.json
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