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results: Erdős–Mollin–Walsh investigation with DAG + FAMM complete
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
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Brandon Schneider
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4-Infrastructure/shim/investigate_erdos_mollin_walsh_refined.py
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4-Infrastructure/shim/investigate_erdos_mollin_walsh_refined.py
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#!/usr/bin/env python3
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"""
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Refined Investigation of Erdős–Mollin–Walsh Conjecture with DAG + FAMM
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======================================================================
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Investigate Erdős–Mollin–Walsh Conjecture with DAG + FAMM components.
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Conjecture: There are no consecutive triples of powerful numbers.
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Previous test found consecutive triples (conjecture holds: False).
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This investigation uses DAG + FAMM for powerful number sequence analysis.
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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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RESEARCH_STACK = Path("/home/allaun/Documents/Research Stack")
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def is_powerful(n):
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"""Check if n is a powerful number (all prime factors have exponent >= 2)."""
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if n < 1:
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return False
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for p in range(2, int(np.sqrt(n)) + 1):
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if n % p == 0:
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count = 0
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while n % p == 0:
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n //= p
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count += 1
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if count == 1:
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return False
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return n == 1 or n > 1
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def generate_powerful_numbers(max_n):
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"""Generate all powerful numbers up to max_n."""
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powerful = []
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for n in range(1, max_n + 1):
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if is_powerful(n):
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powerful.append(n)
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return powerful
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def find_consecutive_triples(powerful_numbers):
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"""Find consecutive triples of powerful numbers."""
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triples = []
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for i in range(len(powerful_numbers) - 2):
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if powerful_numbers[i + 1] == powerful_numbers[i] + 1 and powerful_numbers[i + 2] == powerful_numbers[i] + 2:
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triples.append((powerful_numbers[i], powerful_numbers[i + 1], powerful_numbers[i + 2]))
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return triples
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def dag_powerful_sequence(powerful_numbers):
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"""Build DAG structure for powerful number sequence."""
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n = len(powerful_numbers)
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# Assign layers based on prime factor complexity
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layers = []
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for num in powerful_numbers:
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# Layer based on number of distinct prime factors
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temp = num
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distinct_primes = 0
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for p in range(2, int(np.sqrt(temp)) + 1):
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if temp % p == 0:
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distinct_primes += 1
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while temp % p == 0:
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temp //= p
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if temp > 1:
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distinct_primes += 1
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layers.append(distinct_primes % 4)
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# Build DAG with edges based on divisibility
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A = np.zeros((n, n))
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for i in range(n):
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for j in range(i + 1, n):
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# Edge if j is a multiple of i (divisibility relation)
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if powerful_numbers[j] % powerful_numbers[i] == 0:
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if layers[j] >= layers[i]: # Forward in complexity
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A[i, j] = 1
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return A, layers
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def famm_powerful_sequence(A, layers, delay_steps=3):
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"""Apply FAMM delay lines for powerful number temporal sequencing."""
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n = A.shape[0]
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# Create delay line matrices for each delay step
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delay_matrices = []
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for delay in range(1, delay_steps + 1):
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# Delay matrix: information flows with delay
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D = 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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if A[i, j] == 1:
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# Check if j is exactly 'delay' complexity layers ahead of i
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if layers[j] - layers[i] == delay:
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D[i, j] = 1
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delay_matrices.append(D)
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# Engram consolidation: weighted sum of delay lines
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engram_matrix = np.zeros((n, n))
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for i, D in enumerate(delay_matrices):
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weight = 1.0 / (i + 1) # Decreasing weight for longer delays
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engram_matrix += weight * D
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return {
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"delay_matrices": [D.tolist() for D in delay_matrices],
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"engram_matrix": engram_matrix.tolist(),
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"delay_steps": delay_steps
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}
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def field_analysis_powerful(powerful_numbers, max_n):
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"""Compute field primitive metrics for powerful numbers."""
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if not powerful_numbers:
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return {
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"density": 0.0,
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"asymptotic_density": 0.0,
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"gap_distribution": []
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}
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density = len(powerful_numbers) / max_n
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asymptotic_density = density
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gaps = [powerful_numbers[i + 1] - powerful_numbers[i] for i in range(len(powerful_numbers) - 1)]
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return {
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"density": float(density),
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"asymptotic_density": float(asymptotic_density),
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"avg_gap": float(np.mean(gaps)) if gaps else 0.0,
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"max_gap": float(np.max(gaps)) if gaps else 0.0,
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"gap_distribution": gaps[:10]
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}
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def spectral_analysis_powerful(A):
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"""Compute spectral decomposition of powerful number DAG."""
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eigenvalues, _ = np.linalg.eigh(A)
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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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"structure_rank": int(np.linalg.matrix_rank(A))
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}
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def shear_analysis_powerful(powerful_numbers):
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"""Compute shear primitive metrics for powerful number deformation."""
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if not powerful_numbers:
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return {
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"powerful_rigidity": 0.0,
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"gap_variance": 0.0,
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"clustering_score": 0.0
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}
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gaps = [powerful_numbers[i + 1] - powerful_numbers[i] for i in range(len(powerful_numbers) - 1)]
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gap_variance = np.var(gaps)
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powerful_rigidity = 1.0 / (gap_variance + 1e-10)
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small_gaps = sum(1 for g in gaps if g <= 2)
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clustering_score = small_gaps / len(gaps) if gaps else 0.0
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return {
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"powerful_rigidity": float(powerful_rigidity),
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"gap_variance": float(gap_variance),
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"clustering_score": float(clustering_score)
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}
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def packet_analysis_powerful(powerful_numbers, triples):
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"""Compute packet primitive metrics for powerful number encoding."""
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if not powerful_numbers:
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return {
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"packet_size": 0,
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"triple_count": 0,
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"encoding_efficiency": 0.0
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}
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packet_size = len(powerful_numbers)
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triple_count = len(triples)
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max_n = powerful_numbers[-1] if powerful_numbers else 1
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encoding_efficiency = packet_size / max_n if max_n > 0 else 0.0
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return {
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"packet_size": packet_size,
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"triple_count": triple_count,
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"encoding_efficiency": float(encoding_efficiency)
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}
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def dag_analysis_powerful(A, layers):
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"""Compute DAG-specific metrics for powerful numbers."""
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n = A.shape[0]
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topological_depth = len(set(layers))
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has_directed_cycle = False
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for i in range(n):
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for j in range(n):
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if A[i, j] == 1 and A[j, i] == 1:
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has_directed_cycle = True
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break
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if has_directed_cycle:
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break
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cross_layer_edges = 0
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total_edges = 0
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for i in range(n):
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for j in range(n):
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if A[i, j] == 1:
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total_edges += 1
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if layers[i] != layers[j]:
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cross_layer_edges += 1
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temporal_density = cross_layer_edges / total_edges if total_edges > 0 else 0.0
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return {
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"topological_depth": topological_depth,
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"has_directed_cycle": has_directed_cycle,
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"temporal_density": float(temporal_density),
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"is_acyclic": not has_directed_cycle
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}
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def famm_analysis_powerful(famm_result):
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"""Compute FAMM-specific metrics for powerful numbers."""
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delay_matrices = famm_result["delay_matrices"]
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engram_matrix = np.array(famm_result["engram_matrix"])
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engram_strength = np.sum(engram_matrix)
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delay_diversity = sum(1 for D in delay_matrices if np.sum(D) > 0)
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temporal_integration = np.trace(engram_matrix) / engram_strength if engram_strength > 0 else 0.0
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return {
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"engram_strength": float(engram_strength),
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"delay_diversity": delay_diversity,
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"temporal_integration": float(temporal_integration),
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"delay_steps": famm_result["delay_steps"]
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}
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def investigate_erdos_mollin_walsh_refined(max_n_values):
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"""Investigate Erdős–Mollin–Walsh Conjecture with DAG + FAMM."""
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results = []
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for max_n in max_n_values:
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# Generate powerful numbers
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powerful_numbers = generate_powerful_numbers(max_n)
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# Find consecutive triples
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triples = find_consecutive_triples(powerful_numbers)
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# Build DAG structure
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A, layers = dag_powerful_sequence(powerful_numbers)
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# Apply FAMM delay lines
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famm_result = famm_powerful_sequence(A, layers, delay_steps=3)
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# 4-primitive analysis
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field = field_analysis_powerful(powerful_numbers, max_n)
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spectral = spectral_analysis_powerful(A)
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shear = shear_analysis_powerful(powerful_numbers)
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packet = packet_analysis_powerful(powerful_numbers, triples)
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# DAG + FAMM analysis
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dag = dag_analysis_powerful(A, layers)
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famm = famm_analysis_powerful(famm_result)
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results.append({
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"max_n": max_n,
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"num_powerful": len(powerful_numbers),
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"consecutive_triples": triples,
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"triple_count": len(triples),
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"conjecture_holds": len(triples) == 0,
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"field": field,
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"spectral": spectral,
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"shear": shear,
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"packet": packet,
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"dag": dag,
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"famm": famm
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})
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return results
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def analyze_investigation(results):
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"""Analyze investigation results with DAG + FAMM."""
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total = len(results)
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holds_count = sum(1 for r in results if r["conjecture_holds"])
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# DAG metrics
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avg_acyclic = np.mean([1 if r["dag"]["is_acyclic"] else 0 for r in results]) if results else 0.0
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avg_temporal_density = np.mean([r["dag"]["temporal_density"] for r in results]) if results else 0.0
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# FAMM metrics
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avg_engram_strength = np.mean([r["famm"]["engram_strength"] for r in results]) if results else 0.0
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avg_delay_diversity = np.mean([r["famm"]["delay_diversity"] for r in results]) if results else 0.0
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return {
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"total_tests": total,
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"conjecture_holds_count": holds_count,
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"conjecture_holds": holds_count == total if total > 0 else True,
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"dag_metrics": {
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"avg_acyclic_rate": float(avg_acyclic),
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"avg_temporal_density": float(avg_temporal_density)
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},
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"famm_metrics": {
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"avg_engram_strength": float(avg_engram_strength),
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"avg_delay_diversity": float(avg_delay_diversity)
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},
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"note": "Refined investigation using DAG structure + FAMM delay lines for powerful number sequence analysis"
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}
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def main():
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print("=" * 70)
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print(" REFINED INVESTIGATION OF ERDŐS–MOLLIN–WALSH WITH DAG + FAMM")
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print("=" * 70)
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# Test parameters
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max_n_values = [100, 1000, 10000]
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print(f"\nTest parameters:")
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print(f" max_n values: {max_n_values}")
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print(f" Total tests: {len(max_n_values)}")
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print(f" Graph construction: DAG (divisibility-based)")
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print(f" Temporal sequencing: FAMM delay lines")
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print(f" Layer assignment: Prime factor complexity")
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print("\n" + "=" * 70)
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print(" GENERATING POWERFUL NUMBER DAG + FAMM")
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print("=" * 70)
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results = investigate_erdos_mollin_walsh_refined(max_n_values)
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print(f"\nGenerated {len(results)} powerful number DAG + FAMM analyses")
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print("\n" + "=" * 70)
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print(" ANALYZING INVESTIGATION RESULTS")
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print("=" * 70)
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analysis = analyze_investigation(results)
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print(f"\nInvestigation analysis:")
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print(f" Total tests: {analysis['total_tests']}")
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print(f" Conjecture holds: {analysis['conjecture_holds_count']}/{analysis['total_tests']}")
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print(f" Conjecture holds: {analysis['conjecture_holds']}")
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print(f"\n DAG metrics:")
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print(f" Avg acyclic rate: {analysis['dag_metrics']['avg_acyclic_rate']:.2%}")
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print(f" Avg temporal density: {analysis['dag_metrics']['avg_temporal_density']:.2%}")
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print(f"\n FAMM metrics:")
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print(f" Avg engram strength: {analysis['famm_metrics']['avg_engram_strength']:.4f}")
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print(f" Avg delay diversity: {analysis['famm_metrics']['avg_delay_diversity']:.2f}")
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print(f" Note: {analysis['note']}")
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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. DAG structure based on divisibility:")
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print(" - Layers encode prime factor complexity")
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print(" - Edges represent divisibility relations")
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print(" - Temporal density measures cross-complexity connectivity")
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print("\n2. FAMM delay lines capture temporal dynamics:")
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print(" - Delay matrices capture complexity-level flow")
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print(" - Engram consolidation integrates weighted delays")
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print(" - Temporal integration measures cross-complexity coherence")
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print("\n3. 4-primitive framework provides structural insight:")
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print(" - Field: density and gap distribution")
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print(" - Spectral: divisibility structure")
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print(" - Shear: gap variance and clustering")
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print(" - Packet: triple encoding")
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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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"max_n_values": max_n_values,
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"total_tests": len(max_n_values),
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"graph_construction": "DAG (divisibility-based)",
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"temporal_sequencing": "FAMM delay lines",
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"layer_assignment": "Prime factor complexity"
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},
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"results": results,
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"investigation_analysis": analysis,
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"primitive_insights": {
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"field": "Density and gap distribution",
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"spectral": "Divisibility structure",
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"shear": "Gap variance and clustering",
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"packet": "Triple encoding"
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},
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"dag_insights": {
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"divisibility_structure": "Edges represent divisibility relations",
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"complexity_layers": "Prime factor complexity encoding",
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"temporal_density": "Cross-complexity connectivity"
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},
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"famm_insights": {
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"complexity_flow": "Delay matrices capture complexity-level flow",
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"engram_consolidation": "Weighted delay integration",
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"temporal_integration": "Cross-complexity coherence"
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},
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"validation": {
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"status": "INVESTIGATION_COMPLETE",
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"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."
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}
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}
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output_file = RESEARCH_STACK / "4-Infrastructure/shim/investigate_erdos_mollin_walsh_refined_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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