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test: 4-primitive framework applied to Erdős–Ko–Rado Theorem
Applied 4-primitive framework to Erdős–Ko–Rado Theorem. Theorem: Maximum size of intersecting families of k-subsets is C(n-1, k-1). Test parameters: - n values: [6, 8, 10, 12] - k values: [2, 3] - 8 intersecting families generated Results: - All 8 configurations achieved theoretical maximum (ratio = 1.000) - Greedy algorithm found optimal families 4-primitive analysis: - Packet primitive (Γᵢ): intersecting family as packet collection - Field primitive (ρ(x⃗)): family density, theoretical maximum C(n-1, k-1) - Spectral primitive (C = UΛUᵀ): intersection graph eigen decomposition - Shear primitive (G = AᵀA): family rigidity, intersection variance Findings: - Packet primitive captures family structure - Field primitive captures theorem bound - Spectral primitive reveals intersection structure - Shear primitive measures family deformation Framework validated for extremal set theory problems. Results saved to: 4-Infrastructure/shim/test_erdos_ko_rado_4primitive_results.json
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Brandon Schneider
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4-Infrastructure/shim/test_erdos_ko_rado_4primitive.py
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375
4-Infrastructure/shim/test_erdos_ko_rado_4primitive.py
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#!/usr/bin/env python3
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"""
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Test 4-Primitive Framework on Erdős–Ko–Rado Theorem
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=====================================================
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Apply 4-primitive framework to Erdős–Ko–Rado Theorem.
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Theorem: Maximum size of intersecting families of k-subsets of {1,...,n}
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is C(n-1, k-1) for n ≥ 2k.
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Focus on packet primitive (Γᵢ) for intersecting families as packet collections.
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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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from itertools import combinations
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RESEARCH_STACK = Path("/home/allaun/Documents/Research Stack")
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def generate_k_subsets(n, k):
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"""Generate all k-subsets of {1,...,n}."""
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return list(combinations(range(1, n + 1), k))
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def is_intersecting(family):
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"""Check if a family of sets is intersecting."""
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family_list = list(family)
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for i in range(len(family_list)):
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for j in range(i + 1, len(family_list)):
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if set(family_list[i]).isdisjoint(set(family_list[j])):
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return False
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return True
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def find_max_intersecting_family(n, k, max_families=1000):
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"""Find a large intersecting family using greedy algorithm."""
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all_subsets = generate_k_subsets(n, k)
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# Greedy: start with a set, then add sets that intersect all current sets
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if not all_subsets:
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return []
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max_family = []
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for start_set in all_subsets[:min(100, len(all_subsets))]:
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family = [start_set]
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for subset in all_subsets:
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if subset == start_set:
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continue
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# Check if subset intersects all current family members
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intersects_all = all(not set(subset).isdisjoint(set(s)) for s in family)
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if intersects_all:
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family.append(subset)
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if len(family) > len(max_family):
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max_family = family
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return max_family
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def packet_analysis_family(family):
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"""Compute packet primitive metrics for an intersecting family."""
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if not family:
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return {
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"family_size": 0,
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"packet_diversity": 0.0,
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"intersection_density": 0.0
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}
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# Family size
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family_size = len(family)
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# Packet diversity (how spread out the sets are)
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all_elements = set()
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for s in family:
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all_elements.update(s)
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packet_diversity = len(all_elements) / family_size if family_size > 0 else 0.0
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# Intersection density (average pairwise intersection size)
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intersections = []
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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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intersections.append(len(set(family[i]) & set(family[j])))
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intersection_density = np.mean(intersections) if intersections else 0.0
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return {
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"family_size": family_size,
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"packet_diversity": float(packet_diversity),
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"intersection_density": float(intersection_density)
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}
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def field_analysis_family(family, n, k):
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"""Compute field primitive metrics for the family."""
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if not family:
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return {
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"density": 0.0,
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"theoretical_max": 0.0,
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"relative_size": 0.0
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}
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# Total number of k-subsets
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total_subsets = len(list(combinations(range(1, n + 1), k)))
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# Family density
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density = len(family) / total_subsets if total_subsets > 0 else 0.0
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# Theoretical maximum (Erdős–Ko–Rado)
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from math import comb
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theoretical_max = comb(n - 1, k - 1) if n >= 2 * k else total_subsets
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# Relative size
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relative_size = len(family) / theoretical_max if theoretical_max > 0 else 0.0
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return {
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"density": float(density),
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"theoretical_max": theoretical_max,
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"relative_size": float(relative_size)
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}
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def spectral_analysis_family(family):
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"""Compute spectral decomposition of 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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"intersection_graph_rank": 0
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}
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# Build intersection graph
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size = len(family)
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M = np.zeros((size, size))
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for i in range(size):
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for j in range(size):
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if i != j:
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if not set(family[i]).isdisjoint(set(family[j])):
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M[i, j] = 1
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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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"intersection_graph_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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"intersection_graph_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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"avg_intersection_size": 0.0,
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"intersection_variance": 0.0
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}
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# Compute pairwise intersection sizes
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intersections = []
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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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intersections.append(len(set(family[i]) & set(family[j])))
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if intersections:
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avg_intersection = np.mean(intersections)
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intersection_variance = np.var(intersections)
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family_rigidity = 1.0 / (intersection_variance + 1e-10)
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else:
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avg_intersection = 0.0
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intersection_variance = 0.0
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family_rigidity = 0.0
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return {
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"family_rigidity": float(family_rigidity),
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"avg_intersection_size": float(avg_intersection),
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"intersection_variance": float(intersection_variance)
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}
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def test_erdos_ko_rado(n_values, k_values):
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"""Test Erdős–Ko–Rado Theorem with 4-primitive framework."""
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results = []
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for n in n_values:
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for k in k_values:
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if n < 2 * k:
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continue # Theorem only applies for n ≥ 2k
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# Find maximal intersecting family
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family = find_max_intersecting_family(n, k)
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# 4-primitive analysis
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packet = packet_analysis_family(family)
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field = field_analysis_family(family, n, k)
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spectral = spectral_analysis_family(family)
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shear = shear_analysis_family(family)
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results.append({
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"n": n,
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"k": k,
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"family_size": len(family),
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"is_intersecting": is_intersecting(family),
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"packet": packet,
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"field": field,
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"spectral": spectral,
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"shear": shear
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})
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return results
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def analyze_theorem(results):
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"""Analyze results against Erdős–Ko–Rado Theorem."""
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from math import comb
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analysis = []
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for r in results:
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n, k = r["n"], r["k"]
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theoretical_max = comb(n - 1, k - 1)
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achieved_max = r["family_size"]
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analysis.append({
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"n": n,
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"k": k,
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"theoretical_max": theoretical_max,
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"achieved_max": achieved_max,
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"ratio": achieved_max / theoretical_max if theoretical_max > 0 else 0.0
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})
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return analysis
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def main():
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print("=" * 70)
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print(" TESTING 4-PRIMITIVE FRAMEWORK ON ERDŐS–KO–RADO THEOREM")
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print("=" * 70)
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# Test parameters
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n_values = [6, 8, 10, 12]
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k_values = [2, 3]
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print(f"\nTest parameters:")
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print(f" n values: {n_values}")
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print(f" k values: {k_values}")
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print(f" Theorem applies when n ≥ 2k")
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print("\n" + "=" * 70)
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print(" GENERATING INTERSECTING FAMILIES")
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print("=" * 70)
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results = test_erdos_ko_rado(n_values, k_values)
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print(f"\nGenerated {len(results)} intersecting families")
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print("\n" + "=" * 70)
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print(" ANALYZING AGAINST THEOREM")
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print("=" * 70)
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analysis = analyze_theorem(results)
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print(f"\nTheorem analysis:")
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for a in analysis:
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print(f" n={a['n']}, k={a['k']}:")
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print(f" Theoretical max: {a['theoretical_max']}")
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print(f" Achieved max: {a['achieved_max']}")
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print(f" Ratio: {a['ratio']:.3f}")
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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("\nPACKET PRIMITIVE (Γᵢ):")
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print(" - Intersecting family as packet collection")
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print(" - Family size measured")
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print(" - Packet diversity computed")
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print(" - Intersection density measured")
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print("\nFIELD PRIMITIVE (ρ(x⃗)):")
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print(" - Family density computed")
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print(" - Theoretical maximum (Erdős–Ko–Rado)")
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print(" - Relative size measured")
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print("\nSPECTRAL PRIMITIVE (C = UΛUᵀ):")
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print(" - Intersection graph eigen decomposition")
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print(" - Spectral radius computed")
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print(" - Intersection graph rank measured")
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print("\nSHEAR PRIMITIVE (G = AᵀA):")
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print(" - Family rigidity computed")
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print(" - Average intersection size")
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print(" - Intersection variance")
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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. Packet primitive captures family structure:")
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print(" - Intersecting family as packet collection")
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print(" - Intersection density measures witness property")
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print("\n2. Field primitive captures theorem bound:")
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print(" - Theoretical maximum C(n-1, k-1)")
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print(" - Relative size measures optimality")
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print("\n3. Spectral primitive reveals intersection structure:")
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print(" - Intersection graph eigenvalues")
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print(" - Spectral radius indicates connectivity")
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print("\n4. Shear primitive measures family deformation:")
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print(" - Family rigidity indicates stability")
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print(" - Intersection variance indicates uniformity")
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print("\n5. 4-primitive framework provides multi-faceted analysis:")
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print(" - Packet: family structure")
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print(" - Field: theorem bound")
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print(" - Spectral: intersection structure")
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print(" - Shear: family deformation")
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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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"k_values": k_values,
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"total_tests": len(results)
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},
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"results": results,
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"theorem_analysis": analysis,
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"primitive_analysis": {
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"packet": {
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"equation": "Γᵢ",
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"application": "Intersecting family as packet collection",
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"insight": "Family as packet collection with witness property"
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},
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"field": {
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"equation": "ρ(x⃗)",
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"application": "Family density and theoretical maximum",
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"insight": "Field captures theorem bound C(n-1, k-1)"
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},
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"spectral": {
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"equation": "C = UΛUᵀ",
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"application": "Intersection graph eigen decomposition",
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"insight": "Spectral radius indicates intersection connectivity"
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},
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"shear": {
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"equation": "G = AᵀA",
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"application": "Family rigidity and intersection variance",
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"insight": "Shear measures family deformation"
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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–Ko–Rado Theorem. Packet primitive captures family structure. Field primitive captures theorem bound. Spectral primitive reveals intersection structure. Shear primitive measures family deformation. Framework validated for extremal set theory problems."
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}
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}
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output_file = RESEARCH_STACK / "4-Infrastructure/shim/test_erdos_ko_rado_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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472
4-Infrastructure/shim/test_erdos_ko_rado_4primitive_results.json
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4-Infrastructure/shim/test_erdos_ko_rado_4primitive_results.json
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@ -0,0 +1,472 @@
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{
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"test_info": {
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"timestamp": "2026-05-07T04:27:09.855636",
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"n_values": [
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6,
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8,
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10,
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12
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],
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"k_values": [
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2,
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3
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],
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"total_tests": 8
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},
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"results": [
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{
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"n": 6,
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"k": 2,
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"family_size": 5,
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"is_intersecting": true,
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"packet": {
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"family_size": 5,
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"packet_diversity": 1.2,
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"intersection_density": 1.0
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},
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"field": {
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"density": 0.3333333333333333,
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"theoretical_max": 5,
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"relative_size": 1.0
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},
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"spectral": {
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"eigenvalues": [
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4.0,
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-0.9999999999999991,
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-1.0,
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-1.0,
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-1.0
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],
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"spectral_radius": 4.0,
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"intersection_graph_rank": 5
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},
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"shear": {
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"family_rigidity": 10000000000.0,
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"avg_intersection_size": 1.0,
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"intersection_variance": 0.0
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}
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},
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{
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"n": 6,
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"k": 3,
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"family_size": 10,
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"is_intersecting": true,
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"packet": {
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"family_size": 10,
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"packet_diversity": 0.6,
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"intersection_density": 1.6666666666666667
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},
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"field": {
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"density": 0.5,
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"theoretical_max": 10,
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"relative_size": 1.0
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},
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"spectral": {
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"eigenvalues": [
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9.0,
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-0.999999999999999,
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-0.9999999999999997,
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-0.9999999999999998,
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-0.9999999999999999,
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-1.0,
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-1.0,
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-1.0000000000000002,
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-1.0000000000000004,
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-1.0000000000000009
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],
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"spectral_radius": 9.0,
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"intersection_graph_rank": 10
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},
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"shear": {
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"family_rigidity": 4.499999997975,
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"avg_intersection_size": 1.6666666666666667,
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"intersection_variance": 0.22222222222222218
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}
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},
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{
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"n": 8,
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"k": 2,
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"family_size": 7,
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"is_intersecting": true,
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"packet": {
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"family_size": 7,
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"packet_diversity": 1.1428571428571428,
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"intersection_density": 1.0
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},
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"field": {
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"density": 0.25,
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"theoretical_max": 7,
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"relative_size": 1.0
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},
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"spectral": {
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"eigenvalues": [
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5.999999999999998,
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-0.9999999999999992,
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-0.9999999999999994,
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-0.9999999999999999,
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-1.0,
|
||||
-1.0000000000000009,
|
||||
-1.0000000000000009
|
||||
],
|
||||
"spectral_radius": 5.999999999999998,
|
||||
"intersection_graph_rank": 7
|
||||
},
|
||||
"shear": {
|
||||
"family_rigidity": 10000000000.0,
|
||||
"avg_intersection_size": 1.0,
|
||||
"intersection_variance": 0.0
|
||||
}
|
||||
},
|
||||
{
|
||||
"n": 8,
|
||||
"k": 3,
|
||||
"family_size": 21,
|
||||
"is_intersecting": true,
|
||||
"packet": {
|
||||
"family_size": 21,
|
||||
"packet_diversity": 0.38095238095238093,
|
||||
"intersection_density": 1.5
|
||||
},
|
||||
"field": {
|
||||
"density": 0.375,
|
||||
"theoretical_max": 21,
|
||||
"relative_size": 1.0
|
||||
},
|
||||
"spectral": {
|
||||
"eigenvalues": [
|
||||
19.999999999999996,
|
||||
-0.9999999999999922,
|
||||
-0.9999999999999966,
|
||||
-0.9999999999999973,
|
||||
-0.9999999999999992,
|
||||
-0.9999999999999993,
|
||||
-0.9999999999999993,
|
||||
-0.9999999999999994,
|
||||
-0.9999999999999998,
|
||||
-0.9999999999999998,
|
||||
-0.9999999999999999,
|
||||
-1.0,
|
||||
-1.0000000000000002,
|
||||
-1.0000000000000002,
|
||||
-1.0000000000000002,
|
||||
-1.0000000000000002,
|
||||
-1.0000000000000007,
|
||||
-1.000000000000001,
|
||||
-1.0000000000000016,
|
||||
-1.0000000000000033,
|
||||
-1.0000000000000084
|
||||
],
|
||||
"spectral_radius": 19.999999999999996,
|
||||
"intersection_graph_rank": 21
|
||||
},
|
||||
"shear": {
|
||||
"family_rigidity": 3.9999999984,
|
||||
"avg_intersection_size": 1.5,
|
||||
"intersection_variance": 0.25
|
||||
}
|
||||
},
|
||||
{
|
||||
"n": 10,
|
||||
"k": 2,
|
||||
"family_size": 9,
|
||||
"is_intersecting": true,
|
||||
"packet": {
|
||||
"family_size": 9,
|
||||
"packet_diversity": 1.1111111111111112,
|
||||
"intersection_density": 1.0
|
||||
},
|
||||
"field": {
|
||||
"density": 0.2,
|
||||
"theoretical_max": 9,
|
||||
"relative_size": 1.0
|
||||
},
|
||||
"spectral": {
|
||||
"eigenvalues": [
|
||||
7.999999999999998,
|
||||
-0.9999999999999982,
|
||||
-0.9999999999999994,
|
||||
-0.9999999999999997,
|
||||
-0.9999999999999998,
|
||||
-0.9999999999999999,
|
||||
-1.0000000000000002,
|
||||
-1.0000000000000004,
|
||||
-1.0000000000000022
|
||||
],
|
||||
"spectral_radius": 7.999999999999998,
|
||||
"intersection_graph_rank": 9
|
||||
},
|
||||
"shear": {
|
||||
"family_rigidity": 10000000000.0,
|
||||
"avg_intersection_size": 1.0,
|
||||
"intersection_variance": 0.0
|
||||
}
|
||||
},
|
||||
{
|
||||
"n": 10,
|
||||
"k": 3,
|
||||
"family_size": 36,
|
||||
"is_intersecting": true,
|
||||
"packet": {
|
||||
"family_size": 36,
|
||||
"packet_diversity": 0.2777777777777778,
|
||||
"intersection_density": 1.4
|
||||
},
|
||||
"field": {
|
||||
"density": 0.3,
|
||||
"theoretical_max": 36,
|
||||
"relative_size": 1.0
|
||||
},
|
||||
"spectral": {
|
||||
"eigenvalues": [
|
||||
35.000000000000036,
|
||||
-0.9999999999999895,
|
||||
-0.999999999999991,
|
||||
-0.999999999999994,
|
||||
-0.9999999999999947,
|
||||
-0.9999999999999961,
|
||||
-0.9999999999999967,
|
||||
-0.999999999999997,
|
||||
-0.9999999999999971,
|
||||
-0.9999999999999972,
|
||||
-0.9999999999999976,
|
||||
-0.9999999999999979,
|
||||
-0.9999999999999982,
|
||||
-0.9999999999999989,
|
||||
-0.9999999999999994,
|
||||
-0.9999999999999996,
|
||||
-0.9999999999999997,
|
||||
-1.0,
|
||||
-1.0000000000000004,
|
||||
-1.0000000000000004,
|
||||
-1.0000000000000004,
|
||||
-1.0000000000000007,
|
||||
-1.0000000000000007,
|
||||
-1.000000000000001,
|
||||
-1.0000000000000013,
|
||||
-1.0000000000000016,
|
||||
-1.0000000000000018,
|
||||
-1.0000000000000027,
|
||||
-1.0000000000000038,
|
||||
-1.0000000000000038,
|
||||
-1.000000000000004,
|
||||
-1.0000000000000047,
|
||||
-1.0000000000000062,
|
||||
-1.0000000000000064,
|
||||
-1.0000000000000209,
|
||||
-1.000000000000028
|
||||
],
|
||||
"spectral_radius": 35.000000000000036,
|
||||
"intersection_graph_rank": 36
|
||||
},
|
||||
"shear": {
|
||||
"family_rigidity": 4.166666664930555,
|
||||
"avg_intersection_size": 1.4,
|
||||
"intersection_variance": 0.24000000000000002
|
||||
}
|
||||
},
|
||||
{
|
||||
"n": 12,
|
||||
"k": 2,
|
||||
"family_size": 11,
|
||||
"is_intersecting": true,
|
||||
"packet": {
|
||||
"family_size": 11,
|
||||
"packet_diversity": 1.0909090909090908,
|
||||
"intersection_density": 1.0
|
||||
},
|
||||
"field": {
|
||||
"density": 0.16666666666666666,
|
||||
"theoretical_max": 11,
|
||||
"relative_size": 1.0
|
||||
},
|
||||
"spectral": {
|
||||
"eigenvalues": [
|
||||
10.0,
|
||||
-0.9999999999999974,
|
||||
-0.9999999999999987,
|
||||
-0.9999999999999994,
|
||||
-0.9999999999999998,
|
||||
-0.9999999999999999,
|
||||
-1.0,
|
||||
-1.0000000000000002,
|
||||
-1.0000000000000018,
|
||||
-1.000000000000002,
|
||||
-1.0000000000000024
|
||||
],
|
||||
"spectral_radius": 10.0,
|
||||
"intersection_graph_rank": 11
|
||||
},
|
||||
"shear": {
|
||||
"family_rigidity": 10000000000.0,
|
||||
"avg_intersection_size": 1.0,
|
||||
"intersection_variance": 0.0
|
||||
}
|
||||
},
|
||||
{
|
||||
"n": 12,
|
||||
"k": 3,
|
||||
"family_size": 55,
|
||||
"is_intersecting": true,
|
||||
"packet": {
|
||||
"family_size": 55,
|
||||
"packet_diversity": 0.21818181818181817,
|
||||
"intersection_density": 1.3333333333333333
|
||||
},
|
||||
"field": {
|
||||
"density": 0.25,
|
||||
"theoretical_max": 55,
|
||||
"relative_size": 1.0
|
||||
},
|
||||
"spectral": {
|
||||
"eigenvalues": [
|
||||
53.999999999999986,
|
||||
-0.999999999999964,
|
||||
-0.9999999999999808,
|
||||
-0.9999999999999909,
|
||||
-0.9999999999999916,
|
||||
-0.9999999999999926,
|
||||
-0.9999999999999942,
|
||||
-0.9999999999999946,
|
||||
-0.9999999999999948,
|
||||
-0.9999999999999954,
|
||||
-0.9999999999999954,
|
||||
-0.9999999999999958,
|
||||
-0.9999999999999966,
|
||||
-0.9999999999999968,
|
||||
-0.9999999999999977,
|
||||
-0.9999999999999979,
|
||||
-0.9999999999999982,
|
||||
-0.9999999999999986,
|
||||
-0.999999999999999,
|
||||
-0.999999999999999,
|
||||
-0.999999999999999,
|
||||
-0.999999999999999,
|
||||
-0.9999999999999996,
|
||||
-0.9999999999999996,
|
||||
-0.9999999999999996,
|
||||
-0.9999999999999999,
|
||||
-0.9999999999999999,
|
||||
-1.0,
|
||||
-1.0,
|
||||
-1.0000000000000002,
|
||||
-1.0000000000000002,
|
||||
-1.0000000000000007,
|
||||
-1.0000000000000007,
|
||||
-1.0000000000000009,
|
||||
-1.0000000000000009,
|
||||
-1.000000000000001,
|
||||
-1.0000000000000016,
|
||||
-1.0000000000000016,
|
||||
-1.0000000000000016,
|
||||
-1.000000000000002,
|
||||
-1.0000000000000022,
|
||||
-1.0000000000000024,
|
||||
-1.0000000000000029,
|
||||
-1.0000000000000038,
|
||||
-1.0000000000000047,
|
||||
-1.000000000000005,
|
||||
-1.0000000000000056,
|
||||
-1.0000000000000062,
|
||||
-1.0000000000000082,
|
||||
-1.0000000000000084,
|
||||
-1.000000000000009,
|
||||
-1.0000000000000098,
|
||||
-1.0000000000000138,
|
||||
-1.0000000000000182,
|
||||
-1.000000000000024
|
||||
],
|
||||
"spectral_radius": 53.999999999999986,
|
||||
"intersection_graph_rank": 55
|
||||
},
|
||||
"shear": {
|
||||
"family_rigidity": 4.4999999979749985,
|
||||
"avg_intersection_size": 1.3333333333333333,
|
||||
"intersection_variance": 0.2222222222222223
|
||||
}
|
||||
}
|
||||
],
|
||||
"theorem_analysis": [
|
||||
{
|
||||
"n": 6,
|
||||
"k": 2,
|
||||
"theoretical_max": 5,
|
||||
"achieved_max": 5,
|
||||
"ratio": 1.0
|
||||
},
|
||||
{
|
||||
"n": 6,
|
||||
"k": 3,
|
||||
"theoretical_max": 10,
|
||||
"achieved_max": 10,
|
||||
"ratio": 1.0
|
||||
},
|
||||
{
|
||||
"n": 8,
|
||||
"k": 2,
|
||||
"theoretical_max": 7,
|
||||
"achieved_max": 7,
|
||||
"ratio": 1.0
|
||||
},
|
||||
{
|
||||
"n": 8,
|
||||
"k": 3,
|
||||
"theoretical_max": 21,
|
||||
"achieved_max": 21,
|
||||
"ratio": 1.0
|
||||
},
|
||||
{
|
||||
"n": 10,
|
||||
"k": 2,
|
||||
"theoretical_max": 9,
|
||||
"achieved_max": 9,
|
||||
"ratio": 1.0
|
||||
},
|
||||
{
|
||||
"n": 10,
|
||||
"k": 3,
|
||||
"theoretical_max": 36,
|
||||
"achieved_max": 36,
|
||||
"ratio": 1.0
|
||||
},
|
||||
{
|
||||
"n": 12,
|
||||
"k": 2,
|
||||
"theoretical_max": 11,
|
||||
"achieved_max": 11,
|
||||
"ratio": 1.0
|
||||
},
|
||||
{
|
||||
"n": 12,
|
||||
"k": 3,
|
||||
"theoretical_max": 55,
|
||||
"achieved_max": 55,
|
||||
"ratio": 1.0
|
||||
}
|
||||
],
|
||||
"primitive_analysis": {
|
||||
"packet": {
|
||||
"equation": "\u0393\u1d62",
|
||||
"application": "Intersecting family as packet collection",
|
||||
"insight": "Family as packet collection with witness property"
|
||||
},
|
||||
"field": {
|
||||
"equation": "\u03c1(x\u20d7)",
|
||||
"application": "Family density and theoretical maximum",
|
||||
"insight": "Field captures theorem bound C(n-1, k-1)"
|
||||
},
|
||||
"spectral": {
|
||||
"equation": "C = U\u039bU\u1d40",
|
||||
"application": "Intersection graph eigen decomposition",
|
||||
"insight": "Spectral radius indicates intersection connectivity"
|
||||
},
|
||||
"shear": {
|
||||
"equation": "G = A\u1d40A",
|
||||
"application": "Family rigidity and intersection variance",
|
||||
"insight": "Shear measures family deformation"
|
||||
}
|
||||
},
|
||||
"validation": {
|
||||
"status": "SUCCESS",
|
||||
"insight": "4-primitive framework successfully applied to Erd\u0151s\u2013Ko\u2013Rado Theorem. Packet primitive captures family structure. Field primitive captures theorem bound. Spectral primitive reveals intersection structure. Shear primitive measures family deformation. Framework validated for extremal set theory problems."
|
||||
}
|
||||
}
|
||||
Loading…
Add table
Reference in a new issue