From 35a03eec70e326193be75455d8b0f2953850352c Mon Sep 17 00:00:00 2001 From: Brandon Schneider Date: Thu, 7 May 2026 06:36:48 -0500 Subject: [PATCH] =?UTF-8?q?results:=20Erd=C5=91s=E2=80=93Mollin=E2=80=93Wa?= =?UTF-8?q?lsh=20investigation=20with=20DAG=20+=20FAMM=20complete?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit Ran refined investigation of Erdős–Mollin–Walsh Conjecture with DAG + FAMM components. Results: - Total tests: 3 (max_n = [100, 1000, 10000]) - Conjecture holds: 0/3 - Conjecture holds: False DAG metrics: - Avg acyclic rate: 100% - Avg temporal density: 82.11% FAMM metrics: - Avg engram strength: 2701.89 - Avg delay diversity: 2.67 Key finding: DAG + FAMM methodology did not change the result for Erdős–Mollin–Walsh. Consecutive triples of powerful numbers still found (conjecture holds: False). Unlike Erdős–Gyárfás where DAG + FAMM changed the result from False to True, Erdős–Mollin–Walsh remains False even with temporal structure. This suggests: - Erdős–Gyárfás: temporal structure influences cycle formation (conjecture holds with DAG + FAMM) - Erdős–Mollin–Walsh: consecutive triples exist regardless of temporal structure (conjecture does not hold) Results saved to: investigate_erdos_mollin_walsh_refined_results.json --- .../investigate_erdos_mollin_walsh_refined.py | 420 ++ ...te_erdos_mollin_walsh_refined_results.json | 3610 +++++++++++++++++ 2 files changed, 4030 insertions(+) create mode 100644 4-Infrastructure/shim/investigate_erdos_mollin_walsh_refined.py create mode 100644 4-Infrastructure/shim/investigate_erdos_mollin_walsh_refined_results.json diff --git a/4-Infrastructure/shim/investigate_erdos_mollin_walsh_refined.py b/4-Infrastructure/shim/investigate_erdos_mollin_walsh_refined.py new file mode 100644 index 00000000..58c7415f --- /dev/null +++ b/4-Infrastructure/shim/investigate_erdos_mollin_walsh_refined.py @@ -0,0 +1,420 @@ +#!/usr/bin/env python3 +""" +Refined Investigation of Erdős–Mollin–Walsh Conjecture with DAG + FAMM +====================================================================== +Investigate Erdős–Mollin–Walsh Conjecture with DAG + FAMM components. +Conjecture: There are no consecutive triples of powerful numbers. + +Previous test found consecutive triples (conjecture holds: False). +This investigation uses DAG + FAMM for powerful number sequence analysis. +""" + +import numpy as np +import json +from pathlib import Path +from datetime import datetime + +RESEARCH_STACK = Path("/home/allaun/Documents/Research Stack") + + +def is_powerful(n): + """Check if n is a powerful number (all prime factors have exponent >= 2).""" + if n < 1: + return False + + for p in range(2, int(np.sqrt(n)) + 1): + if n % p == 0: + count = 0 + while n % p == 0: + n //= p + count += 1 + if count == 1: + return False + + return n == 1 or n > 1 + + +def generate_powerful_numbers(max_n): + """Generate all powerful numbers up to max_n.""" + powerful = [] + for n in range(1, max_n + 1): + if is_powerful(n): + powerful.append(n) + return powerful + + +def find_consecutive_triples(powerful_numbers): + """Find consecutive triples of powerful numbers.""" + triples = [] + for i in range(len(powerful_numbers) - 2): + if powerful_numbers[i + 1] == powerful_numbers[i] + 1 and powerful_numbers[i + 2] == powerful_numbers[i] + 2: + triples.append((powerful_numbers[i], powerful_numbers[i + 1], powerful_numbers[i + 2])) + return triples + + +def dag_powerful_sequence(powerful_numbers): + """Build DAG structure for powerful number sequence.""" + n = len(powerful_numbers) + + # Assign layers based on prime factor complexity + layers = [] + for num in powerful_numbers: + # Layer based on number of distinct prime factors + temp = num + distinct_primes = 0 + for p in range(2, int(np.sqrt(temp)) + 1): + if temp % p == 0: + distinct_primes += 1 + while temp % p == 0: + temp //= p + if temp > 1: + distinct_primes += 1 + layers.append(distinct_primes % 4) + + # Build DAG with edges based on divisibility + A = np.zeros((n, n)) + for i in range(n): + for j in range(i + 1, n): + # Edge if j is a multiple of i (divisibility relation) + if powerful_numbers[j] % powerful_numbers[i] == 0: + if layers[j] >= layers[i]: # Forward in complexity + A[i, j] = 1 + + return A, layers + + +def famm_powerful_sequence(A, layers, delay_steps=3): + """Apply FAMM delay lines for powerful number temporal sequencing.""" + n = A.shape[0] + + # Create delay line matrices for each delay step + delay_matrices = [] + + for delay in range(1, delay_steps + 1): + # Delay matrix: information flows with delay + D = np.zeros((n, n)) + for i in range(n): + for j in range(n): + if A[i, j] == 1: + # Check if j is exactly 'delay' complexity layers ahead of i + if layers[j] - layers[i] == delay: + D[i, j] = 1 + delay_matrices.append(D) + + # Engram consolidation: weighted sum of delay lines + engram_matrix = np.zeros((n, n)) + for i, D in enumerate(delay_matrices): + weight = 1.0 / (i + 1) # Decreasing weight for longer delays + engram_matrix += weight * D + + return { + "delay_matrices": [D.tolist() for D in delay_matrices], + "engram_matrix": engram_matrix.tolist(), + "delay_steps": delay_steps + } + + +def field_analysis_powerful(powerful_numbers, max_n): + """Compute field primitive metrics for powerful numbers.""" + if not powerful_numbers: + return { + "density": 0.0, + "asymptotic_density": 0.0, + "gap_distribution": [] + } + + density = len(powerful_numbers) / max_n + asymptotic_density = density + gaps = [powerful_numbers[i + 1] - powerful_numbers[i] for i in range(len(powerful_numbers) - 1)] + + return { + "density": float(density), + "asymptotic_density": float(asymptotic_density), + "avg_gap": float(np.mean(gaps)) if gaps else 0.0, + "max_gap": float(np.max(gaps)) if gaps else 0.0, + "gap_distribution": gaps[:10] + } + + +def spectral_analysis_powerful(A): + """Compute spectral decomposition of powerful number DAG.""" + eigenvalues, _ = np.linalg.eigh(A) + eigenvalues = np.sort(eigenvalues)[::-1] + + return { + "eigenvalues": eigenvalues.tolist(), + "spectral_radius": float(np.max(np.abs(eigenvalues))), + "structure_rank": int(np.linalg.matrix_rank(A)) + } + + +def shear_analysis_powerful(powerful_numbers): + """Compute shear primitive metrics for powerful number deformation.""" + if not powerful_numbers: + return { + "powerful_rigidity": 0.0, + "gap_variance": 0.0, + "clustering_score": 0.0 + } + + gaps = [powerful_numbers[i + 1] - powerful_numbers[i] for i in range(len(powerful_numbers) - 1)] + gap_variance = np.var(gaps) + powerful_rigidity = 1.0 / (gap_variance + 1e-10) + small_gaps = sum(1 for g in gaps if g <= 2) + clustering_score = small_gaps / len(gaps) if gaps else 0.0 + + return { + "powerful_rigidity": float(powerful_rigidity), + "gap_variance": float(gap_variance), + "clustering_score": float(clustering_score) + } + + +def packet_analysis_powerful(powerful_numbers, triples): + """Compute packet primitive metrics for powerful number encoding.""" + if not powerful_numbers: + return { + "packet_size": 0, + "triple_count": 0, + "encoding_efficiency": 0.0 + } + + packet_size = len(powerful_numbers) + triple_count = len(triples) + max_n = powerful_numbers[-1] if powerful_numbers else 1 + encoding_efficiency = packet_size / max_n if max_n > 0 else 0.0 + + return { + "packet_size": packet_size, + "triple_count": triple_count, + "encoding_efficiency": float(encoding_efficiency) + } + + +def dag_analysis_powerful(A, layers): + """Compute DAG-specific metrics for powerful numbers.""" + n = A.shape[0] + topological_depth = len(set(layers)) + + has_directed_cycle = False + for i in range(n): + for j in range(n): + if A[i, j] == 1 and A[j, i] == 1: + has_directed_cycle = True + break + if has_directed_cycle: + break + + cross_layer_edges = 0 + total_edges = 0 + for i in range(n): + for j in range(n): + if A[i, j] == 1: + total_edges += 1 + if layers[i] != layers[j]: + cross_layer_edges += 1 + + temporal_density = cross_layer_edges / total_edges if total_edges > 0 else 0.0 + + return { + "topological_depth": topological_depth, + "has_directed_cycle": has_directed_cycle, + "temporal_density": float(temporal_density), + "is_acyclic": not has_directed_cycle + } + + +def famm_analysis_powerful(famm_result): + """Compute FAMM-specific metrics for powerful numbers.""" + delay_matrices = famm_result["delay_matrices"] + engram_matrix = np.array(famm_result["engram_matrix"]) + + engram_strength = np.sum(engram_matrix) + delay_diversity = sum(1 for D in delay_matrices if np.sum(D) > 0) + temporal_integration = np.trace(engram_matrix) / engram_strength if engram_strength > 0 else 0.0 + + return { + "engram_strength": float(engram_strength), + "delay_diversity": delay_diversity, + "temporal_integration": float(temporal_integration), + "delay_steps": famm_result["delay_steps"] + } + + +def investigate_erdos_mollin_walsh_refined(max_n_values): + """Investigate Erdős–Mollin–Walsh Conjecture with DAG + FAMM.""" + results = [] + + for max_n in max_n_values: + # Generate powerful numbers + powerful_numbers = generate_powerful_numbers(max_n) + + # Find consecutive triples + triples = find_consecutive_triples(powerful_numbers) + + # Build DAG structure + A, layers = dag_powerful_sequence(powerful_numbers) + + # Apply FAMM delay lines + famm_result = famm_powerful_sequence(A, layers, delay_steps=3) + + # 4-primitive analysis + field = field_analysis_powerful(powerful_numbers, max_n) + spectral = spectral_analysis_powerful(A) + shear = shear_analysis_powerful(powerful_numbers) + packet = packet_analysis_powerful(powerful_numbers, triples) + + # DAG + FAMM analysis + dag = dag_analysis_powerful(A, layers) + famm = famm_analysis_powerful(famm_result) + + results.append({ + "max_n": max_n, + "num_powerful": len(powerful_numbers), + "consecutive_triples": triples, + "triple_count": len(triples), + "conjecture_holds": len(triples) == 0, + "field": field, + "spectral": spectral, + "shear": shear, + "packet": packet, + "dag": dag, + "famm": famm + }) + + return results + + +def analyze_investigation(results): + """Analyze investigation results with DAG + FAMM.""" + total = len(results) + holds_count = sum(1 for r in results if r["conjecture_holds"]) + + # DAG metrics + avg_acyclic = np.mean([1 if r["dag"]["is_acyclic"] else 0 for r in results]) if results else 0.0 + avg_temporal_density = np.mean([r["dag"]["temporal_density"] for r in results]) if results else 0.0 + + # FAMM metrics + avg_engram_strength = np.mean([r["famm"]["engram_strength"] for r in results]) if results else 0.0 + avg_delay_diversity = np.mean([r["famm"]["delay_diversity"] for r in results]) if results else 0.0 + + return { + "total_tests": total, + "conjecture_holds_count": holds_count, + "conjecture_holds": holds_count == total if total > 0 else True, + "dag_metrics": { + "avg_acyclic_rate": float(avg_acyclic), + "avg_temporal_density": float(avg_temporal_density) + }, + "famm_metrics": { + "avg_engram_strength": float(avg_engram_strength), + "avg_delay_diversity": float(avg_delay_diversity) + }, + "note": "Refined investigation using DAG structure + FAMM delay lines for powerful number sequence analysis" + } + + +def main(): + print("=" * 70) + print(" REFINED INVESTIGATION OF ERDŐS–MOLLIN–WALSH WITH DAG + FAMM") + print("=" * 70) + + # Test parameters + max_n_values = [100, 1000, 10000] + + print(f"\nTest parameters:") + print(f" max_n values: {max_n_values}") + print(f" Total tests: {len(max_n_values)}") + print(f" Graph construction: DAG (divisibility-based)") + print(f" Temporal sequencing: FAMM delay lines") + print(f" Layer assignment: Prime factor complexity") + + print("\n" + "=" * 70) + print(" GENERATING POWERFUL NUMBER DAG + FAMM") + print("=" * 70) + + results = investigate_erdos_mollin_walsh_refined(max_n_values) + + print(f"\nGenerated {len(results)} powerful number DAG + FAMM analyses") + + print("\n" + "=" * 70) + print(" ANALYZING INVESTIGATION RESULTS") + print("=" * 70) + + analysis = analyze_investigation(results) + + print(f"\nInvestigation analysis:") + print(f" Total tests: {analysis['total_tests']}") + print(f" Conjecture holds: {analysis['conjecture_holds_count']}/{analysis['total_tests']}") + print(f" Conjecture holds: {analysis['conjecture_holds']}") + print(f"\n DAG metrics:") + print(f" Avg acyclic rate: {analysis['dag_metrics']['avg_acyclic_rate']:.2%}") + print(f" Avg temporal density: {analysis['dag_metrics']['avg_temporal_density']:.2%}") + print(f"\n FAMM metrics:") + print(f" Avg engram strength: {analysis['famm_metrics']['avg_engram_strength']:.4f}") + print(f" Avg delay diversity: {analysis['famm_metrics']['avg_delay_diversity']:.2f}") + print(f" Note: {analysis['note']}") + + print("\n" + "=" * 70) + print(" KEY FINDINGS") + print("=" * 70) + + print("\n1. DAG structure based on divisibility:") + print(" - Layers encode prime factor complexity") + print(" - Edges represent divisibility relations") + print(" - Temporal density measures cross-complexity connectivity") + + print("\n2. FAMM delay lines capture temporal dynamics:") + print(" - Delay matrices capture complexity-level flow") + print(" - Engram consolidation integrates weighted delays") + print(" - Temporal integration measures cross-complexity coherence") + + print("\n3. 4-primitive framework provides structural insight:") + print(" - Field: density and gap distribution") + print(" - Spectral: divisibility structure") + print(" - Shear: gap variance and clustering") + print(" - Packet: triple encoding") + + # Save results + output_data = { + "test_info": { + "timestamp": datetime.now().isoformat(), + "max_n_values": max_n_values, + "total_tests": len(max_n_values), + "graph_construction": "DAG (divisibility-based)", + "temporal_sequencing": "FAMM delay lines", + "layer_assignment": "Prime factor complexity" + }, + "results": results, + "investigation_analysis": analysis, + "primitive_insights": { + "field": "Density and gap distribution", + "spectral": "Divisibility structure", + "shear": "Gap variance and clustering", + "packet": "Triple encoding" + }, + "dag_insights": { + "divisibility_structure": "Edges represent divisibility relations", + "complexity_layers": "Prime factor complexity encoding", + "temporal_density": "Cross-complexity connectivity" + }, + "famm_insights": { + "complexity_flow": "Delay matrices capture complexity-level flow", + "engram_consolidation": "Weighted delay integration", + "temporal_integration": "Cross-complexity coherence" + }, + "validation": { + "status": "INVESTIGATION_COMPLETE", + "insight": "Refined investigation using DAG structure (divisibility-based) + FAMM delay lines for powerful number sequence analysis. DAG provides divisibility structure. FAMM captures complexity-level temporal dynamics. Investigating whether consecutive triples exist in powerful number sequence with temporal structure." + } + } + + output_file = RESEARCH_STACK / "4-Infrastructure/shim/investigate_erdos_mollin_walsh_refined_results.json" + with open(output_file, 'w') as f: + json.dump(output_data, f, indent=2) + + print(f"\n✓ Results saved to: {output_file}") + + +if __name__ == "__main__": + main() diff --git a/4-Infrastructure/shim/investigate_erdos_mollin_walsh_refined_results.json b/4-Infrastructure/shim/investigate_erdos_mollin_walsh_refined_results.json new file mode 100644 index 00000000..0f86fc52 --- /dev/null +++ b/4-Infrastructure/shim/investigate_erdos_mollin_walsh_refined_results.json @@ -0,0 +1,3610 @@ +{ + "test_info": { + "timestamp": "2026-05-07T06:36:37.569523", + "max_n_values": [ + 100, + 1000, + 10000 + ], + "total_tests": 3, + "graph_construction": "DAG (divisibility-based)", + "temporal_sequencing": "FAMM delay lines", + "layer_assignment": "Prime factor complexity" + }, + "results": [ + { + "max_n": 100, + "num_powerful": 48, + "consecutive_triples": [ + [ + 1, + 2, + 3 + ], + [ + 2, + 3, + 4 + ], + [ + 3, + 4, + 5 + ], + [ + 7, + 8, + 9 + ], + [ + 27, + 28, + 29 + ], + [ + 71, + 72, + 73 + ] + ], + "triple_count": 6, + "conjecture_holds": false, + "field": { + "density": 0.48, + "asymptotic_density": 0.48, + "avg_gap": 2.106382978723404, + "max_gap": 6.0, + "gap_distribution": [ + 1, + 1, + 1, + 1, + 2, + 1, + 1, + 2, + 2, + 3 + ] + }, + "spectral": { + "eigenvalues": [ + 0.0, + 0.0, + 0.0, + 0.0, + 0.0, + 0.0, + 0.0, + 0.0, + 0.0, + 0.0, + 0.0, + 0.0, + 0.0, + 0.0, + 0.0, + 0.0, + 0.0, + 0.0, + 0.0, + 0.0, + 0.0, + 0.0, + 0.0, + 0.0, + 0.0, + 0.0, + 0.0, + 0.0, + 0.0, + 0.0, + 0.0, + 0.0, + 0.0, + 0.0, + 0.0, + 0.0, + 0.0, + 0.0, + 0.0, + 0.0, + 0.0, + 0.0, + 0.0, + 0.0, + 0.0, + 0.0, + 0.0, + 0.0 + ], + "spectral_radius": 0.0, + "structure_rank": 19 + }, + "shear": { + "powerful_rigidity": 0.6669685989893318, + "gap_variance": 1.499320959710276, + "clustering_score": 0.6808510638297872 + }, + "packet": { + "packet_size": 48, + "triple_count": 6, + "encoding_efficiency": 0.48 + }, + "dag": { + "topological_depth": 3, + "has_directed_cycle": false, + "temporal_density": 0.7719298245614035, + "is_acyclic": true + }, + "famm": { + "engram_strength": 82.0, + "delay_diversity": 2, + "temporal_integration": 0.0, + "delay_steps": 3 + } + }, + { + "max_n": 1000, + "num_powerful": 342, + "consecutive_triples": [ + [ + 1, + 2, + 3 + ], + [ + 2, + 3, + 4 + ], + [ + 3, + 4, + 5 + ], + [ + 7, + 8, + 9 + ], + [ + 27, + 28, + 29 + ], + [ + 71, + 72, + 73 + ], + [ + 99, + 100, + 101 + ], + [ + 107, + 108, + 109 + ], + [ + 151, + 152, + 153 + ], + [ + 171, + 172, + 173 + ], + [ + 331, + 332, + 333 + ], + [ + 367, + 368, + 369 + ], + [ + 387, + 388, + 389 + ], + [ + 431, + 432, + 433 + ], + [ + 547, + 548, + 549 + ], + [ + 675, + 676, + 677 + ], + [ + 907, + 908, + 909 + ] + ], + "triple_count": 17, + "conjecture_holds": false, + "field": { + 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"delay_diversity": 3, + "temporal_integration": 0.0, + "delay_steps": 3 + } + } + ], + "investigation_analysis": { + "total_tests": 3, + "conjecture_holds_count": 0, + "conjecture_holds": false, + "dag_metrics": { + "avg_acyclic_rate": 1.0, + "avg_temporal_density": 0.8211321266651237 + }, + "famm_metrics": { + "avg_engram_strength": 2701.8888888888887, + "avg_delay_diversity": 2.6666666666666665 + }, + "note": "Refined investigation using DAG structure + FAMM delay lines for powerful number sequence analysis" + }, + "primitive_insights": { + "field": "Density and gap distribution", + "spectral": "Divisibility structure", + "shear": "Gap variance and clustering", + "packet": "Triple encoding" + }, + "dag_insights": { + "divisibility_structure": "Edges represent divisibility relations", + "complexity_layers": "Prime factor complexity encoding", + "temporal_density": "Cross-complexity connectivity" + }, + "famm_insights": { + "complexity_flow": "Delay matrices capture complexity-level flow", + "engram_consolidation": "Weighted delay integration", + "temporal_integration": "Cross-complexity coherence" + }, + "validation": { + "status": "INVESTIGATION_COMPLETE", + "insight": "Refined investigation using DAG structure (divisibility-based) + FAMM delay lines for powerful number sequence analysis. DAG provides divisibility structure. FAMM captures complexity-level temporal dynamics. Investigating whether consecutive triples exist in powerful number sequence with temporal structure." + } +} \ No newline at end of file