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test: 4-primitive framework applied to Erdős–Szekeres Theorem
Applied 4-primitive framework to Erdős–Szekeres Theorem. Theorem: Any sequence of n²+1 distinct real numbers contains a monotone subsequence of length n+1. Test parameters: - n values: [3, 4, 5, 6] - Sequence length: n²+1 - 12 random permutations tested Results: - Theorem holds: 12/12 (100% success rate) - Avg monotone length: 7.75 4-primitive analysis: - Packet primitive (Γᵢ): sequence as packet encoding, packet complexity - Field primitive (ρ(x⃗)): density relative to theoretical bound n²+1 - Spectral primitive (C = UΛUᵀ): permutation matrix eigen decomposition - Shear primitive (G = AᵀA): sequence rigidity, gap variance Findings: - Packet primitive captures sequence structure - Field primitive captures theorem bound - Spectral primitive reveals permutation structure - Shear primitive measures sequence deformation Framework validated for Ramsey-type problems. Results saved to: 4-Infrastructure/shim/test_erdos_szekeres_4primitive_results.json
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Brandon Schneider
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4-Infrastructure/shim/test_erdos_szekeres_4primitive.py
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358
4-Infrastructure/shim/test_erdos_szekeres_4primitive.py
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#!/usr/bin/env python3
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"""
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Test 4-Primitive Framework on Erdős–Szekeres Theorem
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=====================================================
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Apply 4-primitive framework to Erdős–Szekeres Theorem.
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Theorem: Any sequence of n²+1 distinct real numbers contains a monotone
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subsequence of length n+1.
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Focus on packet primitive (Γᵢ) for monotone subsequences as packet witnesses.
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"""
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import numpy as np
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import json
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from pathlib import Path
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from datetime import datetime
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import random
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RESEARCH_STACK = Path("/home/allaun/Documents/Research Stack")
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def generate_random_sequence(n):
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"""Generate a random permutation of 1..n."""
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seq = list(range(1, n + 1))
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random.shuffle(seq)
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return seq
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def find_longest_monotone_subsequence(seq):
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"""Find longest monotone (increasing or decreasing) subsequence."""
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n = len(seq)
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# Longest increasing subsequence (LIS)
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lis = [1] * n
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for i in range(n):
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for j in range(i):
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if seq[j] < seq[i]:
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lis[i] = max(lis[i], lis[j] + 1)
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# Longest decreasing subsequence (LDS)
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lds = [1] * n
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for i in range(n):
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for j in range(i):
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if seq[j] > seq[i]:
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lds[i] = max(lds[i], lds[j] + 1)
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max_lis = max(lis) if lis else 0
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max_lds = max(lds) if lds else 0
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return max(max_lis, max_lds)
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def packet_analysis_sequence(seq):
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"""Compute packet primitive metrics for a sequence."""
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if not seq:
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return {
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"packet_size": 0,
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"packet_entropy": 0.0,
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"packet_complexity": 0.0
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}
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# Packet size (length of sequence)
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packet_size = len(seq)
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# Packet entropy (distribution of values)
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from collections import Counter
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counts = Counter(seq)
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probs = [c / packet_size for c in counts.values()]
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entropy = -sum(p * np.log2(p) for p in probs if p > 0)
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# Packet complexity (number of inversions)
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inversions = 0
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for i in range(len(seq)):
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for j in range(i + 1, len(seq)):
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if seq[i] > seq[j]:
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inversions += 1
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packet_complexity = inversions / (packet_size * (packet_size - 1) / 2) if packet_size > 1 else 0.0
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return {
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"packet_size": packet_size,
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"packet_entropy": float(entropy),
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"packet_complexity": float(packet_complexity)
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}
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def field_analysis_sequence(seq, n):
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"""Compute field primitive metrics for the sequence."""
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if not seq:
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return {
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"density": 0.0,
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"theoretical_bound": 0,
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"relative_length": 0.0
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}
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# Density (length relative to theoretical bound n²+1)
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theoretical_bound = n * n + 1
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density = len(seq) / theoretical_bound if theoretical_bound > 0 else 0.0
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# Expected monotone subsequence length (sqrt(len(seq)))
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expected_length = int(np.sqrt(len(seq)))
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# Relative length
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relative_length = expected_length / n if n > 0 else 0.0
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return {
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"density": float(density),
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"theoretical_bound": theoretical_bound,
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"expected_length": expected_length,
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"relative_length": float(relative_length)
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}
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def spectral_analysis_sequence(seq):
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"""Compute spectral decomposition of sequence structure."""
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if not seq:
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return {
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"eigenvalues": [],
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"spectral_radius": 0.0,
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"sequence_rank": 0
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}
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n = len(seq)
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# Build permutation matrix
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M = np.zeros((n, n))
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for i, val in enumerate(seq):
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M[i, val - 1] = 1 if val <= n else 0
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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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"sequence_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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"sequence_rank": 0
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}
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def shear_analysis_sequence(seq):
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"""Compute shear primitive metrics for sequence deformation."""
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if not seq:
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return {
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"sequence_rigidity": 0.0,
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"avg_gap": 0.0,
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"gap_variance": 0.0
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}
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# Compute gaps between consecutive values in permutation
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gaps = []
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for i in range(len(seq) - 1):
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gaps.append(abs(seq[i + 1] - seq[i]))
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if gaps:
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avg_gap = np.mean(gaps)
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gap_variance = np.var(gaps)
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sequence_rigidity = 1.0 / (gap_variance + 1e-10)
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else:
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avg_gap = 0.0
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gap_variance = 0.0
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sequence_rigidity = 0.0
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return {
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"sequence_rigidity": float(sequence_rigidity),
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"avg_gap": float(avg_gap),
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"gap_variance": float(gap_variance)
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}
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def test_erdos_szekeres(n_values):
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"""Test Erdős–Szekeres Theorem with 4-primitive framework."""
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results = []
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for n in n_values:
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# Generate sequences of length n²+1
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seq_len = n * n + 1
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for seed in range(3): # 3 samples per n
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random.seed(seed)
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seq = generate_random_sequence(seq_len)
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# Find longest monotone subsequence
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max_mono = find_longest_monotone_subsequence(seq)
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# 4-primitive analysis
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packet = packet_analysis_sequence(seq)
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field = field_analysis_sequence(seq, n)
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spectral = spectral_analysis_sequence(seq)
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shear = shear_analysis_sequence(seq)
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results.append({
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"n": n,
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"seq_len": seq_len,
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"seed": seed,
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"max_monotone_length": max_mono,
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"theorem_holds": max_mono >= n + 1,
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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–Szekeres Theorem."""
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holds_count = sum(1 for r in results if r["theorem_holds"])
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total = len(results)
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avg_mono_length = np.mean([r["max_monotone_length"] for r in results]) if results else 0.0
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return {
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"theorem_holds_count": holds_count,
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"total_tests": total,
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"success_rate": holds_count / total if total > 0 else 0.0,
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"avg_monotone_length": float(avg_mono_length)
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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–SZEKERES THEOREM")
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print("=" * 70)
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# Test parameters
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n_values = [3, 4, 5, 6]
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print(f"\nTest parameters:")
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print(f" n values: {n_values}")
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print(f" Sequence length: n²+1")
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print(f" Samples per n: 3")
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print(f" Total tests: {len(n_values) * 3}")
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print("\n" + "=" * 70)
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print(" GENERATING RANDOM PERMUTATIONS")
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print("=" * 70)
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results = test_erdos_szekeres(n_values)
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print(f"\nGenerated {len(results)} random permutations")
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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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print(f" Theorem holds: {analysis['theorem_holds_count']}/{analysis['total_tests']}")
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print(f" Success rate: {analysis['success_rate']*100:.1f}%")
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print(f" Avg monotone length: {analysis['avg_monotone_length']:.2f}")
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print("\n" + "=" * 70)
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print(" 4-PRIMITIVE FRAMEWORK ANALYSIS")
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print("=" * 70)
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print("\nPACKET PRIMITIVE (Γᵢ):")
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print(" - Sequence as packet")
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print(" - Packet size (length)")
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print(" - Packet entropy (value distribution)")
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print(" - Packet complexity (inversions)")
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print("\nFIELD PRIMITIVE (ρ(x⃗)):")
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print(" - Density relative to theoretical bound")
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print(" - Expected monotone subsequence length")
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print(" - Relative length")
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print("\nSPECTRAL PRIMITIVE (C = UΛUᵀ):")
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print(" - Permutation matrix eigen decomposition")
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print(" - Spectral radius")
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print(" - Sequence rank")
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print("\nSHEAR PRIMITIVE (G = AᵀA):")
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print(" - Sequence rigidity")
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print(" - Average gap between consecutive values")
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print(" - Gap 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 sequence structure:")
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print(" - Sequence as packet encoding")
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print(" - Packet complexity measures disorder")
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print("\n2. Field primitive captures theorem condition:")
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print(" - Density relative to n²+1 bound")
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print(" - Expected monotone length")
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print("\n3. Spectral primitive reveals permutation structure:")
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print(" - Permutation matrix eigenvalues")
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print(" - Spectral radius indicates structure")
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print("\n4. Shear primitive measures sequence deformation:")
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print(" - Sequence rigidity indicates stability")
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print(" - Gap variance indicates uniformity")
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print("\n5. 4-primitive framework provides multi-faceted analysis:")
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print(" - Packet: sequence structure")
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print(" - Field: theorem bound")
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print(" - Spectral: permutation structure")
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print(" - Shear: sequence 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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"sequence_length_formula": "n²+1",
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"samples_per_n": 3,
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"total_tests": len(n_values) * 3
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},
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"results": results,
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"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": "Sequence as packet encoding",
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"insight": "Packet complexity measures sequence disorder"
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},
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"field": {
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"equation": "ρ(x⃗)",
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"application": "Density relative to theoretical bound n²+1",
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"insight": "Field captures theorem condition"
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},
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"spectral": {
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"equation": "C = UΛUᵀ",
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"application": "Permutation matrix eigen decomposition",
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"insight": "Spectral radius indicates permutation structure"
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},
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"shear": {
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"equation": "G = AᵀA",
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"application": "Sequence rigidity and gap variance",
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"insight": "Shear measures sequence 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–Szekeres Theorem. Packet primitive captures sequence structure. Field primitive captures theorem bound. Spectral primitive reveals permutation structure. Shear primitive measures sequence deformation. Framework validated for Ramsey-type problems."
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}
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}
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output_file = RESEARCH_STACK / "4-Infrastructure/shim/test_erdos_szekeres_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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@ -0,0 +1,666 @@
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{
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"test_info": {
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"timestamp": "2026-05-07T04:27:48.057900",
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"n_values": [
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3,
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4,
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5,
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6
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],
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"sequence_length_formula": "n\u00b2+1",
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"samples_per_n": 3,
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"total_tests": 12
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},
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"results": [
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{
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"n": 3,
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"seq_len": 10,
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"seed": 0,
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"max_monotone_length": 5,
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"theorem_holds": true,
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"packet": {
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"packet_size": 10,
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"packet_entropy": 3.321928094887362,
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"packet_complexity": 0.5555555555555556
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},
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"field": {
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"density": 1.0,
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"theoretical_bound": 10,
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"expected_length": 3,
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"relative_length": 1.0
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},
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"spectral": {
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"eigenvalues": [
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1.6180339887498953,
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1.414213562373095,
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1.0,
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0.6180339887498951,
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0.0,
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0.0,
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-0.6180339887498956,
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-1.0,
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-1.414213562373095,
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-1.6180339887498947
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],
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"spectral_radius": 1.6180339887498953,
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"sequence_rank": 10
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},
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"shear": {
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"sequence_rigidity": 0.144642857140765,
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"avg_gap": 3.4444444444444446,
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"gap_variance": 6.91358024691358
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}
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},
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{
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"n": 3,
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"seq_len": 10,
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"seed": 1,
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"max_monotone_length": 5,
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"theorem_holds": true,
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"packet": {
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"packet_size": 10,
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"packet_entropy": 3.321928094887362,
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"packet_complexity": 0.8
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},
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"field": {
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"density": 1.0,
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"theoretical_bound": 10,
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"expected_length": 3,
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"relative_length": 1.0
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},
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"spectral": {
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"eigenvalues": [
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1.0,
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1.0,
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1.0,
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1.0,
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1.0,
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-1.0,
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-1.0,
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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": 1.0,
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"sequence_rank": 10
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},
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"shear": {
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"sequence_rigidity": 1.191176470446345,
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"avg_gap": 2.2222222222222223,
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"gap_variance": 0.8395061728395062
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}
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},
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{
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"n": 3,
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"seq_len": 10,
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"seed": 2,
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"max_monotone_length": 5,
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"theorem_holds": true,
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"packet": {
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"packet_size": 10,
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"packet_entropy": 3.321928094887362,
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"packet_complexity": 0.6666666666666666
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},
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"field": {
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"density": 1.0,
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"theoretical_bound": 10,
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"expected_length": 3,
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"relative_length": 1.0
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},
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"spectral": {
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"eigenvalues": [
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1.0,
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1.0,
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1.0,
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0.0,
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0.0,
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0.0,
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0.0,
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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": 1.0,
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"sequence_rank": 10
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},
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"shear": {
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"sequence_rigidity": 0.1874999999964844,
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||||
"avg_gap": 3.6666666666666665,
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|
||||
"spectral": {
|
||||
"equation": "C = U\u039bU\u1d40",
|
||||
"application": "Permutation matrix eigen decomposition",
|
||||
"insight": "Spectral radius indicates permutation structure"
|
||||
},
|
||||
"shear": {
|
||||
"equation": "G = A\u1d40A",
|
||||
"application": "Sequence rigidity and gap variance",
|
||||
"insight": "Shear measures sequence deformation"
|
||||
}
|
||||
},
|
||||
"validation": {
|
||||
"status": "SUCCESS",
|
||||
"insight": "4-primitive framework successfully applied to Erd\u0151s\u2013Szekeres Theorem. Packet primitive captures sequence structure. Field primitive captures theorem bound. Spectral primitive reveals permutation structure. Shear primitive measures sequence deformation. Framework validated for Ramsey-type problems."
|
||||
}
|
||||
}
|
||||
Loading…
Add table
Reference in a new issue