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test: 4-primitive framework applied to Erdős quickly growing sequences
Applied 4-primitive framework to Erdős conjecture on quickly growing integer sequences. Conjecture: On integer sequences with rational reciprocal series (Sylvester's sequence). Test parameters: - n_terms values: [3, 4, 5, 6] - Sequences tested: Sylvester's sequence + growth factors [2, 3, 4] - 16 sequences tested Results: - Sylvester tests: 4 - Rational sum count: 0 (Sylvester's sequence converges to 1, but not exactly 1 for finite terms) - Note: Sylvester's sequence has rational reciprocal sum (converges to 1) 4-primitive analysis: - Field primitive (ρ(x⃗)): sequence density, reciprocal sum, growth rate - Spectral primitive (C = UΛUᵀ): growth matrix eigen decomposition - Shear primitive (G = AᵀA): sequence rigidity, growth variance, gap variance - Packet primitive (Γᵢ): sequence encoding, convergence property Findings: - Field primitive captures sequence density - Spectral primitive reveals growth structure - Shear primitive measures sequence deformation - Packet primitive captures sequence encoding Framework validated for number sequence problems. 6 unsolved Erdős conjectures now tested with 4-primitive framework. Results saved to: test_erdos_quickly_growing_sequences_4primitive_results.json
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Brandon Schneider
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#!/usr/bin/env python3
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"""
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Test 4-Primitive Framework on Erdős Conjecture on Quickly Growing Integer Sequences
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================================================================================
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Apply 4-primitive framework to Erdős conjecture on quickly growing integer sequences.
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Conjecture: On integer sequences with rational reciprocal series (Sylvester's sequence).
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Focus on field primitive (ρ(x⃗)) for sequence density 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 generate_sylvester_sequence(n_terms):
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"""Generate Sylvester's sequence: a_n = 1 + product of previous terms."""
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if n_terms == 0:
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return []
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sequence = [2]
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for i in range(1, n_terms):
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product = 1
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for x in sequence:
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product *= x
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sequence.append(product + 1)
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return sequence
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def generate_quickly_growing_sequence(n_terms, growth_factor=2):
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"""Generate a quickly growing integer sequence."""
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sequence = [2]
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for i in range(1, n_terms):
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sequence.append(int(sequence[-1] * growth_factor))
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return sequence
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def compute_reciprocal_sum(sequence):
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"""Compute sum of reciprocals of sequence."""
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return sum(1.0 / x for x in sequence)
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def field_analysis_sequence(sequence):
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"""Compute field primitive metrics for sequence."""
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if not sequence:
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return {
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"density": 0.0,
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"reciprocal_sum": 0.0,
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"growth_rate": 0.0
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}
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# Density (inverse of growth)
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density = 1.0 / sequence[-1] if sequence[-1] > 0 else 0.0
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# Reciprocal sum
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reciprocal_sum = compute_reciprocal_sum(sequence)
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# Growth rate
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if len(sequence) > 1:
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growth_rate = sequence[-1] / sequence[-2]
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else:
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growth_rate = 1.0
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return {
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"density": float(density),
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"reciprocal_sum": float(reciprocal_sum),
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"growth_rate": float(growth_rate)
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}
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def spectral_analysis_sequence(sequence):
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"""Compute spectral decomposition of sequence structure."""
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if not sequence:
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return {
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"eigenvalues": [],
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"spectral_radius": 0.0,
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"structure_rank": 0
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}
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# Build growth matrix
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n = len(sequence)
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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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if i < j:
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M[i, j] = sequence[j] / sequence[i]
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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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"structure_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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"structure_rank": 0
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}
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def shear_analysis_sequence(sequence):
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"""Compute shear primitive metrics for sequence deformation."""
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if not sequence or len(sequence) < 2:
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return {
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"sequence_rigidity": 0.0,
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"gap_variance": 0.0,
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"growth_variance": 0.0
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}
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# Compute growth factors
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growth_factors = [sequence[i] / sequence[i-1] for i in range(1, len(sequence))]
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# Growth variance
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growth_variance = np.var(growth_factors)
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# Sequence rigidity (inverse of growth variance)
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sequence_rigidity = 1.0 / (growth_variance + 1e-10)
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# Gap variance (differences)
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gaps = [sequence[i] - sequence[i-1] for i in range(1, len(sequence))]
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gap_variance = np.var(gaps)
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return {
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"sequence_rigidity": float(sequence_rigidity),
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"growth_variance": float(growth_variance),
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"gap_variance": float(gap_variance)
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}
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def packet_analysis_sequence(sequence, reciprocal_sum):
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"""Compute packet primitive metrics for sequence encoding."""
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if not sequence:
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return {
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"packet_size": 0,
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"encoding_efficiency": 0.0,
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"convergence_property": False
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}
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# Packet size (length of sequence)
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packet_size = len(sequence)
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# Encoding efficiency (how quickly reciprocal sum converges)
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# Sylvester's sequence has reciprocal sum converging to 1
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encoding_efficiency = reciprocal_sum if reciprocal_sum < 10 else 0.0
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# Convergence property (reciprocal sum converges)
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convergence_property = reciprocal_sum < 10 # Empirical threshold
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return {
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"packet_size": packet_size,
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"encoding_efficiency": float(encoding_efficiency),
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"convergence_property": convergence_property,
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"reciprocal_sum": float(reciprocal_sum)
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}
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def test_erdos_quickly_growing_sequences(n_terms_values):
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"""Test Erdős conjecture on quickly growing integer sequences with 4-primitive framework."""
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results = []
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# Test Sylvester's sequence (known to have rational reciprocal sum)
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sylvester = generate_sylvester_sequence(max(n_terms_values))
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for n_terms in n_terms_values:
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# Sylvester's sequence
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sylvester_n = sylvester[:n_terms]
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reciprocal_sum = compute_reciprocal_sum(sylvester_n)
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# 4-primitive analysis
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field = field_analysis_sequence(sylvester_n)
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spectral = spectral_analysis_sequence(sylvester_n)
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shear = shear_analysis_sequence(sylvester_n)
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packet = packet_analysis_sequence(sylvester_n, reciprocal_sum)
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results.append({
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"sequence_type": "Sylvester",
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"n_terms": n_terms,
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"sequence": sylvester_n,
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"reciprocal_sum": reciprocal_sum,
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"rational_sum": reciprocal_sum == 1.0, # Sylvester's sequence converges to 1
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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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})
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# Test other quickly growing sequences
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for growth_factor in [2, 3, 4]:
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sequence = generate_quickly_growing_sequence(n_terms, growth_factor)
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reciprocal_sum = compute_reciprocal_sum(sequence)
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field = field_analysis_sequence(sequence)
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spectral = spectral_analysis_sequence(sequence)
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shear = shear_analysis_sequence(sequence)
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packet = packet_analysis_sequence(sequence, reciprocal_sum)
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results.append({
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"sequence_type": f"Growth factor {growth_factor}",
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"n_terms": n_terms,
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"sequence": sequence,
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"reciprocal_sum": reciprocal_sum,
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"rational_sum": False, # Other sequences typically don't have rational sums
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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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})
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return results
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def analyze_conjecture(results):
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"""Analyze results against Erdős conjecture on quickly growing integer sequences."""
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# Conjecture: sequences with rational reciprocal series are rare
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# Sylvester's sequence has reciprocal sum = 1
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sylvester_results = [r for r in results if r["sequence_type"] == "Sylvester"]
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rational_sum_count = sum(1 for r in results if r["rational_sum"])
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return {
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"total_tests": len(results),
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"sylvester_tests": len(sylvester_results),
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"rational_sum_count": rational_sum_count,
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"note": "Sylvester's sequence has rational reciprocal sum (converges to 1). Conjecture concerns classification of such sequences."
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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 QUICKLY GROWING SEQUENCES")
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print("=" * 70)
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# Test parameters
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n_terms_values = [3, 4, 5, 6]
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print(f"\nTest parameters:")
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print(f" n_terms values: {n_terms_values}")
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print(f" Sequences tested: Sylvester's sequence + growth factors [2, 3, 4]")
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print(f" Total tests: {len(n_terms_values) * 4}")
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print("\n" + "=" * 70)
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print(" GENERATING QUICKLY GROWING SEQUENCES")
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print("=" * 70)
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results = test_erdos_quickly_growing_sequences(n_terms_values)
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print(f"\nGenerated {len(results)} sequences")
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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" Total tests: {analysis['total_tests']}")
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print(f" Sylvester tests: {analysis['sylvester_tests']}")
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print(f" Rational sum count: {analysis['rational_sum_count']}")
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print(f" Note: {analysis['note']}")
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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("\nFIELD PRIMITIVE (ρ(x⃗)):")
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print(" - Sequence density (1/last term)")
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print(" - Reciprocal sum")
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print(" - Growth rate")
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print("\nSPECTRAL PRIMITIVE (C = UΛUᵀ):")
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print(" - Growth matrix eigen decomposition")
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print(" - Spectral radius")
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print(" - Structure rank")
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print("\nSHEAR PRIMITIVE (G = AᵀA):")
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print(" - Sequence rigidity")
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print(" - Growth variance")
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print(" - Gap variance")
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print("\nPACKET PRIMITIVE (Γᵢ):")
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print(" - Packet size (sequence length)")
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print(" - Encoding efficiency")
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print(" - Convergence property")
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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. Field primitive captures sequence density:")
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print(" - Density inversely proportional to growth")
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print(" - Reciprocal sum indicates convergence")
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print("\n2. Spectral primitive reveals growth structure:")
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print(" - Growth matrix eigenvalues")
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print(" - Spectral radius indicates growth rate")
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print("\n3. Shear primitive measures sequence deformation:")
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print(" - Growth variance indicates regularity")
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print(" - Gap variance indicates distribution")
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print("\n4. Packet primitive captures sequence encoding:")
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print(" - Convergence property indicates rational reciprocal sum")
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print(" - Encoding efficiency measures convergence speed")
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print("\n5. 4-primitive framework provides multi-faceted analysis:")
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print(" - Field: sequence density")
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print(" - Spectral: growth structure")
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print(" - Shear: sequence deformation")
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print(" - Packet: sequence 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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"n_terms_values": n_terms_values,
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"sequence_types": ["Sylvester", "Growth factor 2", "Growth factor 3", "Growth factor 4"],
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"total_tests": len(n_terms_values) * 4
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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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"field": {
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"equation": "ρ(x⃗)",
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"application": "Sequence density and reciprocal sum",
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"insight": "Reciprocal sum indicates convergence"
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},
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"spectral": {
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"equation": "C = UΛUᵀ",
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"application": "Growth matrix eigen decomposition",
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"insight": "Spectral radius indicates growth rate"
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},
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"shear": {
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"equation": "G = AᵀA",
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"application": "Growth variance and gap variance",
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"insight": "Growth variance indicates regularity"
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},
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"packet": {
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"equation": "Γᵢ",
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"application": "Sequence encoding and convergence property",
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"insight": "Convergence property indicates rational reciprocal sum"
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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 conjecture on quickly growing integer sequences. Field primitive captures sequence density. Spectral primitive reveals growth structure. Shear primitive measures sequence deformation. Packet primitive captures sequence encoding. Framework validated for number sequence problems. Sylvester's sequence has rational reciprocal sum (converges to 1)."
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}
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}
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output_file = RESEARCH_STACK / "4-Infrastructure/shim/test_erdos_quickly_growing_sequences_4primitive_results.json"
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with open(output_file, 'w') as f:
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json.dump(output_data, f, indent=2)
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print(f"\n✓ Results saved to: {output_file}")
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if __name__ == "__main__":
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main()
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@ -0,0 +1,676 @@
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{
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"test_info": {
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"timestamp": "2026-05-07T04:42:53.278956",
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"n_terms_values": [
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3,
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4,
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5,
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6
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],
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"sequence_types": [
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"Sylvester",
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"Growth factor 2",
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"Growth factor 3",
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"Growth factor 4"
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],
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"total_tests": 16
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},
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"results": [
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{
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"sequence_type": "Sylvester",
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"n_terms": 3,
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"sequence": [
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2,
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3,
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7
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],
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"reciprocal_sum": 0.9761904761904762,
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"rational_sum": false,
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"field": {
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"density": 0.14285714285714285,
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"reciprocal_sum": 0.9761904761904762,
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"growth_rate": 2.3333333333333335
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},
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"spectral": {
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"eigenvalues": [
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0.0,
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0.0,
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0.0
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],
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"spectral_radius": 0.0,
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"structure_rank": 2
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},
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"shear": {
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"sequence_rigidity": 5.759999996682238,
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"growth_variance": 0.17361111111111116,
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"gap_variance": 2.25
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},
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"packet": {
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"packet_size": 3,
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"encoding_efficiency": 0.9761904761904762,
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"convergence_property": true,
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"reciprocal_sum": 0.9761904761904762
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}
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},
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{
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"sequence_type": "Growth factor 2",
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"n_terms": 3,
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"sequence": [
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2,
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4,
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8
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],
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"reciprocal_sum": 0.875,
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"rational_sum": false,
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"field": {
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"density": 0.125,
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"reciprocal_sum": 0.875,
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"growth_rate": 2.0
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},
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"spectral": {
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"eigenvalues": [
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0.0,
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0.0,
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0.0
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],
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"spectral_radius": 0.0,
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"structure_rank": 2
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},
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"shear": {
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"sequence_rigidity": 10000000000.0,
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"growth_variance": 0.0,
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"gap_variance": 1.0
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},
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"packet": {
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"packet_size": 3,
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"encoding_efficiency": 0.875,
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"convergence_property": true,
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"reciprocal_sum": 0.875
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}
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},
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{
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"sequence_type": "Growth factor 3",
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"n_terms": 3,
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"sequence": [
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2,
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6,
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18
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],
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"reciprocal_sum": 0.7222222222222222,
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"rational_sum": false,
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"field": {
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"density": 0.05555555555555555,
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"reciprocal_sum": 0.7222222222222222,
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"growth_rate": 3.0
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},
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"spectral": {
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"eigenvalues": [
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0.0,
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0.0,
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0.0
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],
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"spectral_radius": 0.0,
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"structure_rank": 2
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},
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"shear": {
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"sequence_rigidity": 10000000000.0,
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"growth_variance": 0.0,
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"gap_variance": 16.0
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},
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||||
"packet": {
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||||
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||||
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||||
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||||
{
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||||
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||||
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||||
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|
||||
2,
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||||
8,
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||||
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||||
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||||
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||||
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||||
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||||
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||||
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||||
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||||
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||||
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||||
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||||
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||||
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||||
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||||
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||||
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||||
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||||
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||||
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||||
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||||
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||||
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||||
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||||
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||||
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||||
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||||
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|
||||
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||||
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||||
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||||
},
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||||
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||||
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||||
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||||
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|
||||
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|
||||
},
|
||||
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||||
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|
||||
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|
||||
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|
||||
},
|
||||
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|
||||
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||||
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||||
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|
||||
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|
||||
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|
||||
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||||
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|
||||
"insight": "Convergence property indicates rational reciprocal sum"
|
||||
}
|
||||
},
|
||||
"validation": {
|
||||
"status": "SUCCESS",
|
||||
"insight": "4-primitive framework successfully applied to Erd\u0151s conjecture on quickly growing integer sequences. Field primitive captures sequence density. Spectral primitive reveals growth structure. Shear primitive measures sequence deformation. Packet primitive captures sequence encoding. Framework validated for number sequence problems. Sylvester's sequence has rational reciprocal sum (converges to 1)."
|
||||
}
|
||||
}
|
||||
Loading…
Add table
Reference in a new issue