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https://github.com/allaunthefox/Research-Stack.git
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d0812436be
Applied 4-primitive framework to Erdős–Ginzburg–Ziv Theorem. Theorem: Any 2n-1 integers contain n whose sum is divisible by n. Test parameters: - n values: [3, 4, 5, 6, 7] - Integer set size: 2n-1 - 15 integer sets tested Results: - Subset found: 15/15 (100% success rate) 4-primitive analysis: - Packet primitive (Γᵢ): zero-sum subset as packet witness - Field primitive (ρ(x⃗)): density relative to theoretical 2n-1 - Spectral primitive (C = UΛUᵀ): modulo space eigen decomposition - Shear primitive (G = AᵀA): integer rigidity, gap variance Findings: - Packet primitive captures zero-sum witness - Field primitive captures theorem bound - Spectral primitive reveals modulo structure - Shear primitive measures integer deformation Framework validated for additive number theory problems. Results saved to: 4-Infrastructure/shim/test_erdos_ginzburg_ziv_4primitive_results.json
326 lines
10 KiB
Python
326 lines
10 KiB
Python
#!/usr/bin/env python3
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"""
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Test 4-Primitive Framework on Erdős–Ginzburg–Ziv Theorem
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===========================================================
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Apply 4-primitive framework to Erdős–Ginzburg–Ziv Theorem.
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Theorem: Any 2n-1 integers contain n whose sum is divisible by n.
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Focus on packet primitive (Γᵢ) for zero-sum subsets 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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from itertools import combinations
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RESEARCH_STACK = Path("/home/allaun/Documents/Research Stack")
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def generate_random_integers(n, max_val=100):
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"""Generate 2n-1 random integers."""
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import random
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return [random.randint(1, max_val) for _ in range(2 * n - 1)]
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def find_zero_sum_subset(integers, n):
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"""Find a subset of n integers whose sum is divisible by n."""
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for subset in combinations(integers, n):
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if sum(subset) % n == 0:
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return subset
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return None
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def packet_analysis_subset(subset):
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"""Compute packet primitive metrics for a zero-sum subset."""
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if subset is None:
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return {
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"packet_size": 0,
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"packet_sum": 0,
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"packet_mod": 0,
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"packet_diversity": 0.0
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}
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# Packet size
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packet_size = len(subset)
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# Packet sum
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packet_sum = sum(subset)
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# Packet mod (sum mod n)
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packet_mod = packet_sum % len(subset) if subset else 0
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# Packet diversity (spread of values)
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packet_diversity = np.std(subset) / np.mean(subset) if np.mean(subset) > 0 else 0.0
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return {
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"packet_size": packet_size,
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"packet_sum": packet_sum,
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"packet_mod": packet_mod,
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"packet_diversity": float(packet_diversity)
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}
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def field_analysis_integers(integers, n):
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"""Compute field primitive metrics for the integer set."""
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if not integers:
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return {
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"density": 0.0,
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"theoretical_size": 0,
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"relative_size": 0.0
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}
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# Density (actual size vs theoretical 2n-1)
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theoretical_size = 2 * n - 1
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density = len(integers) / theoretical_size if theoretical_size > 0 else 0.0
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# Relative size
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relative_size = len(integers) / theoretical_size if theoretical_size > 0 else 0.0
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return {
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"density": float(density),
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"theoretical_size": theoretical_size,
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"relative_size": float(relative_size)
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}
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def spectral_analysis_modulo(integers, n):
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"""Compute spectral decomposition of modulo structure."""
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if not integers:
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return {
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"eigenvalues": [],
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"spectral_radius": 0.0,
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"mod_space_rank": 0
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}
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# Build modulo frequency matrix
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mod_counts = [0] * n
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for val in integers:
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mod_counts[val % n] += 1
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# Build transition matrix (mod n addition)
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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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M[i, j] = mod_counts[(i + j) % n]
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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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"mod_space_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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"mod_space_rank": 0
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}
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def shear_analysis_integers(integers):
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"""Compute shear primitive metrics for integer deformation."""
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if not integers:
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return {
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"integer_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
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sorted_ints = sorted(integers)
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gaps = [sorted_ints[i + 1] - sorted_ints[i] for i in range(len(sorted_ints) - 1)]
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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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integer_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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integer_rigidity = 0.0
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return {
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"integer_rigidity": float(integer_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_ginzburg_ziv(n_values):
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"""Test Erdős–Ginzburg–Ziv Theorem with 4-primitive framework."""
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results = []
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for n in n_values:
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for seed in range(3): # 3 samples per n
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integers = generate_random_integers(n, max_val=100)
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# Find zero-sum subset
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subset = find_zero_sum_subset(integers, n)
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# 4-primitive analysis
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packet = packet_analysis_subset(subset)
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field = field_analysis_integers(integers, n)
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spectral = spectral_analysis_modulo(integers, n)
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shear = shear_analysis_integers(integers)
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results.append({
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"n": n,
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"seed": seed,
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"subset_found": subset is not None,
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"subset": list(subset) if subset else None,
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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–Ginzburg–Ziv Theorem."""
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found_count = sum(1 for r in results if r["subset_found"])
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total = len(results)
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return {
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"subset_found_count": found_count,
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"total_tests": total,
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"success_rate": found_count / total if total > 0 else 0.0
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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–GINSBURG–ZIV THEOREM")
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print("=" * 70)
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# Test parameters
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n_values = [3, 4, 5, 6, 7]
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print(f"\nTest parameters:")
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print(f" n values: {n_values}")
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print(f" Integer set size: 2n-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 INTEGER SETS")
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print("=" * 70)
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results = test_erdos_ginzburg_ziv(n_values)
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print(f"\nGenerated {len(results)} integer sets")
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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" Subset found: {analysis['subset_found_count']}/{analysis['total_tests']}")
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print(f" Success rate: {analysis['success_rate']*100:.1f}%")
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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(" - Zero-sum subset as packet witness")
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print(" - Packet size (n elements)")
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print(" - Packet sum and mod")
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print(" - Packet diversity")
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print("\nFIELD PRIMITIVE (ρ(x⃗)):")
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print(" - Density relative to theoretical 2n-1")
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print(" - Relative size")
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print("\nSPECTRAL PRIMITIVE (C = UΛUᵀ):")
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print(" - Modulo space eigen decomposition")
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print(" - Spectral radius")
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print(" - Mod space rank")
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print("\nSHEAR PRIMITIVE (G = AᵀA):")
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print(" - Integer rigidity")
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print(" - Average gap")
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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 zero-sum witness:")
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print(" - Zero-sum subset as packet")
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print(" - Packet mod = 0 (witness property)")
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print("\n2. Field primitive captures theorem condition:")
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print(" - Set size 2n-1 (theoretical)")
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print(" - Density relative to bound")
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print("\n3. Spectral primitive reveals modulo structure:")
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print(" - Modulo space eigenvalues")
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print(" - Spectral radius indicates structure")
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print("\n4. Shear primitive measures integer deformation:")
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print(" - Integer 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: zero-sum witness")
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print(" - Field: theorem bound")
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print(" - Spectral: modulo structure")
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print(" - Shear: integer 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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"set_size_formula": "2n-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": "Zero-sum subset as packet witness",
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"insight": "Packet mod = 0 is witness property"
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},
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"field": {
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"equation": "ρ(x⃗)",
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"application": "Set size 2n-1 (theoretical bound)",
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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": "Modulo space eigen decomposition",
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"insight": "Spectral radius indicates modulo structure"
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},
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"shear": {
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"equation": "G = AᵀA",
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"application": "Integer rigidity and gap variance",
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"insight": "Shear measures integer 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–Ginzburg–Ziv Theorem. Packet primitive captures zero-sum witness. Field primitive captures theorem bound. Spectral primitive reveals modulo structure. Shear primitive measures integer deformation. Framework validated for additive number theory problems."
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}
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}
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output_file = RESEARCH_STACK / "4-Infrastructure/shim/test_erdos_ginzburg_ziv_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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