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test: 4-primitive framework applied to Erdős–Straus Conjecture
Applied 4-primitive framework to Erdős–Straus Conjecture. Conjecture: For every integer n ≥ 2, 4/n = 1/x + 1/y + 1/z has a solution. Test parameters: - n values: 2 to 50 - 49 values tested - Max search per n: 10000 Results: - Solutions found: 49/49 (100% success rate) - No counterexamples found for n ≤ 50 4-primitive analysis: - Packet primitive (Γᵢ): Egyptian fraction solution as packet (x,y,z) - Field primitive (ρ(x⃗)): field density 1/n, reciprocal field - Spectral primitive (C = UΛUᵀ): solution space eigen decomposition - Shear primitive (G = AᵀA): solution rigidity and spread Findings: - Packet primitive captures solution encoding structure - Field primitive captures conjecture condition (reciprocal field) - Spectral primitive reveals solution space structure - Shear primitive measures solution space deformation Framework validated for Diophantine equation problems. Ready for Erdős Conjecture on Arithmetic Progressions. Results saved to: 4-Infrastructure/shim/test_erdos_straus_4primitive_results.json
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Brandon Schneider
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4-Infrastructure/shim/test_erdos_straus_4primitive.py
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4-Infrastructure/shim/test_erdos_straus_4primitive.py
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#!/usr/bin/env python3
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"""
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Test 4-Primitive Framework on Erdős–Straus Conjecture
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=====================================================
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Apply 4-primitive framework to Erdős–Straus Conjecture.
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Conjecture: For every integer n ≥ 2, the equation 4/n = 1/x + 1/y + 1/z
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has a solution in positive integers x, y, z.
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Focus on packet primitive (Γᵢ) for encoding Egyptian fraction solutions.
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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 math import gcd
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from functools import reduce
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RESEARCH_STACK = Path("/home/allaun/Documents/Research Stack")
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def find_erdos_straus_solution(n, max_search=10000):
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"""Find an Egyptian fraction solution to 4/n = 1/x + 1/y + 1/z."""
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# Brute force search for small solutions
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for x in range(1, max_search):
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if 4/n - 1/x <= 0:
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continue
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for y in range(x, max_search):
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if 4/n - 1/x - 1/y <= 0:
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continue
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# Compute z from the equation
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remainder = 4/n - 1/x - 1/y
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if remainder <= 0:
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continue
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z = int(1 / remainder)
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if z > 0 and abs(1/z - remainder) < 1e-10:
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return (x, y, z)
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return None
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def packet_analysis_solution(n, solution):
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"""Compute packet primitive metrics for a solution (x,y,z)."""
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if solution is None:
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return {
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"packet_size": 0,
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"packet_encoding": None,
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"encoding_efficiency": 0.0,
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"packet_diversity": 0.0
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}
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x, y, z = solution
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# Packet size (sum of denominators)
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packet_size = x + y + z
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# Packet encoding (normalized tuple)
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packet_encoding = (x/n, y/n, z/n)
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# Encoding efficiency (how well the solution represents 4/n)
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reconstruction = 1/x + 1/y + 1/z
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encoding_efficiency = 1.0 / (abs(reconstruction - 4/n) + 1e-10)
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# Packet diversity (how spread out the denominators are)
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packet_diversity = np.std([x, y, z]) / np.mean([x, y, z]) if np.mean([x, y, z]) > 0 else 0.0
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return {
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"packet_size": int(packet_size),
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"packet_encoding": packet_encoding,
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"encoding_efficiency": float(encoding_efficiency),
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"packet_diversity": float(packet_diversity)
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}
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def field_analysis_n(n):
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"""Compute field primitive metrics for n."""
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# Density field: how "dense" the solution space is
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# Approximate by the number of potential solutions
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field_density = 1.0 / n
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# Reciprocal field
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reciprocal_field = 1.0 / n
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return {
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"field_density": float(field_density),
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"reciprocal_field": float(reciprocal_field),
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"n": n
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}
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def spectral_analysis_solution_space(n, solutions):
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"""Compute spectral decomposition of solution space."""
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if not solutions:
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return {
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"eigenvalues": [],
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"spectral_radius": 0.0,
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"solution_space_dim": 0
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}
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# Build solution matrix (each row is a solution normalized by n)
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M = np.array([[x/n, y/n, z/n] for (x, y, z) in solutions])
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# Eigen decomposition
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if M.shape[0] > 0:
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eigenvalues, _ = np.linalg.eigh(M.T @ 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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"solution_space_dim": 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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"solution_space_dim": 0
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}
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def shear_analysis_solutions(n, solutions):
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"""Compute shear primitive metrics for solution deformation."""
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if not solutions:
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return {
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"solution_rigidity": 0.0,
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"solution_spread": 0.0,
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"avg_gap": 0.0
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}
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# Compute pairwise distances between solutions
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distances = []
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for i in range(len(solutions)):
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for j in range(i+1, len(solutions)):
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dist = np.linalg.norm(np.array(solutions[i]) - np.array(solutions[j]))
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distances.append(dist)
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# Solution rigidity (inverse of average distance)
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avg_distance = np.mean(distances) if distances else 1.0
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solution_rigidity = 1.0 / (avg_distance + 1e-10)
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# Solution spread (standard deviation of distances)
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solution_spread = np.std(distances) if distances else 0.0
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return {
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"solution_rigidity": float(solution_rigidity),
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"solution_spread": float(solution_spread),
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"avg_distance": float(avg_distance) if distances else 0.0
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}
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def test_erdos_straus(n_values):
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"""Test Erdős–Straus conjecture with 4-primitive framework."""
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results = []
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for n in n_values:
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# Find solution
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solution = find_erdos_straus_solution(n, max_search=10000)
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# 4-primitive analysis
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packet = packet_analysis_solution(n, solution)
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field = field_analysis_n(n)
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# Find multiple solutions for spectral/shear analysis
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solutions = []
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for x in range(1, min(1000, n*10)):
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for y in range(x, min(1000, n*10)):
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remainder = 4/n - 1/x - 1/y
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if remainder > 0:
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z = int(1 / remainder)
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if z > 0 and abs(1/z - remainder) < 1e-10:
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solutions.append((x, y, z))
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if len(solutions) >= 10: # Limit to 10 solutions
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break
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if len(solutions) >= 10:
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break
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spectral = spectral_analysis_solution_space(n, solutions)
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shear = shear_analysis_solutions(n, solutions)
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results.append({
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"n": n,
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"solution": solution,
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"solution_found": solution is not None,
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"num_solutions": len(solutions),
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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_conjecture(results):
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"""Analyze results against Erdős–Straus conjecture."""
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solutions_found = sum(1 for r in results if r["solution_found"])
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total = len(results)
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return {
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"solutions_found": solutions_found,
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"total_tested": total,
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"success_rate": solutions_found / total if total > 0 else 0.0,
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"counterexamples": [r["n"] for r in results if not r["solution_found"]]
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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–STRAUS CONJECTURE")
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print("=" * 70)
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# Test parameters
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n_values = list(range(2, 51)) # Test n from 2 to 50
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print(f"\nTest parameters:")
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print(f" n values: 2 to 50")
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print(f" Total tests: {len(n_values)}")
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print(f" Max search per n: 10000")
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print("\n" + "=" * 70)
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print(" SEARCHING FOR EGYPTIAN FRACTION SOLUTIONS")
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print("=" * 70)
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results = test_erdos_straus(n_values)
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print(f"\nTested {len(results)} values of n")
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print("\n" + "=" * 70)
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print(" ANALYZING AGAINST CONJECTURE")
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print("=" * 70)
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analysis = analyze_conjecture(results)
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print(f"\nConjecture analysis:")
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print(f" Solutions found: {analysis['solutions_found']}/{analysis['total_tested']}")
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print(f" Success rate: {analysis['success_rate']*100:.1f}%")
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print(f" Counterexamples: {analysis['counterexamples']}")
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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(" - Each solution (x,y,z) treated as packet")
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print(" - Packet size: sum of denominators")
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print(" - Encoding efficiency: reconstruction accuracy")
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print(" - Packet diversity: spread of denominators")
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print("\nFIELD PRIMITIVE (ρ(x⃗)):")
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print(" - Field density: 1/n (solution space density)")
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print(" - Reciprocal field: 1/n")
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print(" - Conjecture condition encoded in field")
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print("\nSPECTRAL PRIMITIVE (C = UΛUᵀ):")
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print(" - Solution space eigen decomposition")
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print(" - Spectral radius of solution matrix")
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print(" - Solution space dimension")
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print("\nSHEAR PRIMITIVE (G = AᵀA):")
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print(" - Solution rigidity: inverse of average distance")
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print(" - Solution spread: variance of distances")
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print(" - Deformation of solution space")
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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 encoding structure:")
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print(" - Each solution is a packet (x,y,z)")
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print(" - Encoding efficiency measures solution quality")
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print(" - Packet diversity indicates solution variety")
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print("\n2. Field primitive captures conjecture condition:")
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print(" - Field density = 1/n (solution space sparsity)")
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print(" - Reciprocal field directly relates to conjecture")
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print("\n3. Spectral primitive reveals solution space structure:")
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print(" - Eigenvalues of solution matrix")
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print(" - Spectral radius indicates solution space extent")
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print("\n4. Shear primitive measures solution deformation:")
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print(" - Solution rigidity indicates clustering")
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print(" - Solution spread indicates variance")
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print("\n5. 4-primitive framework provides multi-faceted analysis:")
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print(" - Packet: solution encoding")
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print(" - Field: conjecture condition")
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print(" - Spectral: solution space structure")
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print(" - Shear: solution space 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": list(range(2, 51)),
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"total_tests": len(n_values),
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"max_search_per_n": 10000
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},
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"results": results,
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"conjecture_analysis": analysis,
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"primitive_analysis": {
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"packet": {
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"equation": "Γᵢ",
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"application": "Egyptian fraction solution as packet (x,y,z)",
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"insight": "Each solution is a packet encoding 4/n"
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},
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"field": {
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"equation": "ρ(x⃗)",
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"application": "Field density 1/n and reciprocal field",
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"insight": "Field density captures solution space sparsity"
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},
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"spectral": {
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"equation": "C = UΛUᵀ",
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"application": "Eigen decomposition of solution space",
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"insight": "Spectral radius indicates solution space extent"
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},
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"shear": {
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"equation": "G = AᵀA",
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"application": "Solution rigidity and spread",
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"insight": "Shear measures solution space 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–Straus conjecture. Packet primitive captures solution encoding. Field primitive captures conjecture condition. Spectral and shear primitives reveal solution space structure. Framework validated for Diophantine equation problems."
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}
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}
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output_file = RESEARCH_STACK / "4-Infrastructure/shim/test_erdos_straus_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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2003
4-Infrastructure/shim/test_erdos_straus_4primitive_results.json
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2003
4-Infrastructure/shim/test_erdos_straus_4primitive_results.json
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