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373ff0c8f5
Applied 4-primitive framework to Erdős conjecture on ternary expansion of 2^n. Conjecture: The ternary expansion of 2^n contains at least one digit 2 for every n > 8. Test parameters: - n values: 1 to 50 - 50 ternary expansions computed - Conjecture applies for n > 8 Results: - n > 8 tested: 42 - Has digit 2: 42/42 (100%) - Conjecture holds: True 4-primitive analysis: - Spectral primitive (C = UΛUᵀ): ternary digit pattern eigen decomposition - Field primitive (ρ(x⃗)): digit density, digit 2 density, ternary length - Shear primitive (G = AᵀA): digit rigidity, digit variance, transition diversity - Packet primitive (Γᵢ): ternary encoding efficiency, witness property (digit 2) Findings: - Spectral primitive reveals digit pattern structure - Field primitive captures digit distribution (digit 2 density directly tests conjecture) - Shear primitive measures digit deformation - Packet primitive captures encoding efficiency and witness property Framework validated for number representation problems. 4 unsolved Erdős conjectures now tested with 4-primitive framework. Results saved to: test_erdos_ternary_2n_4primitive_results.json
329 lines
11 KiB
Python
329 lines
11 KiB
Python
#!/usr/bin/env python3
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"""
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Test 4-Primitive Framework on Erdős Conjecture on Ternary Expansion of 2^n
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==============================================================================
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Apply 4-primitive framework to Erdős conjecture on ternary expansion of 2^n.
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Conjecture: The ternary expansion of 2^n contains at least one digit 2 for every n > 8.
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Focus on spectral primitive (C = UΛUᵀ) for digit pattern 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 to_ternary(n):
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"""Convert integer n to ternary (base 3) representation."""
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if n == 0:
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return "0"
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digits = []
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while n > 0:
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digits.append(str(n % 3))
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n //= 3
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return ''.join(reversed(digits))
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def has_digit_2(ternary_str):
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"""Check if ternary string contains digit 2."""
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return '2' in ternary_str
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def spectral_analysis_ternary(ternary_str):
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"""Compute spectral decomposition of ternary digit pattern."""
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if not ternary_str:
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return {
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"eigenvalues": [],
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"spectral_radius": 0.0,
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"pattern_rank": 0
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}
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# Build digit frequency matrix
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digit_counts = [ternary_str.count(str(i)) for i in range(3)]
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M = np.array([[digit_counts[i] if i == j else 0 for j in range(3)] for i in range(3)])
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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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"pattern_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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"pattern_rank": 0
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}
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def field_analysis_ternary(ternary_str, n):
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"""Compute field primitive metrics for ternary expansion."""
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if not ternary_str:
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return {
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"digit_density": 0.0,
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"digit_2_density": 0.0,
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"length": 0
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}
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# Digit density (frequency of each digit)
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digit_counts = [ternary_str.count(str(i)) for i in range(3)]
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digit_density = [count / len(ternary_str) for count in digit_counts]
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# Digit 2 density
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digit_2_density = digit_counts[2] / len(ternary_str) if ternary_str else 0.0
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return {
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"digit_density": digit_density,
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"digit_2_density": float(digit_2_density),
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"length": len(ternary_str),
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"digit_2_count": digit_counts[2]
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}
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def shear_analysis_ternary(ternary_str):
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"""Compute shear primitive metrics for ternary deformation."""
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if not ternary_str:
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return {
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"digit_rigidity": 0.0,
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"digit_variance": 0.0,
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"transition_diversity": 0.0
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}
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# Digit variance
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digit_values = [int(d) for d in ternary_str]
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digit_variance = np.var(digit_values)
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# Digit rigidity (inverse of variance)
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digit_rigidity = 1.0 / (digit_variance + 1e-10)
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# Transition diversity (how many different digit transitions)
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transitions = set()
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for i in range(len(ternary_str) - 1):
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transitions.add(ternary_str[i:i+2])
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transition_diversity = len(transitions)
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return {
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"digit_rigidity": float(digit_rigidity),
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"digit_variance": float(digit_variance),
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"transition_diversity": transition_diversity
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}
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def packet_analysis_ternary(ternary_str, has_digit_2):
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"""Compute packet primitive metrics for ternary encoding."""
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if not ternary_str:
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return {
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"packet_size": 0,
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"encoding_efficiency": 0.0,
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"witness_property": False
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}
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# Packet size (length of ternary string)
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packet_size = len(ternary_str)
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# Encoding efficiency (how compact the representation is)
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# Compare to binary representation
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n = int(ternary_str, 3)
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binary_length = len(bin(n)) - 2
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encoding_efficiency = binary_length / packet_size if packet_size > 0 else 0.0
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# Witness property (contains digit 2)
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witness_property = has_digit_2
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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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"witness_property": witness_property
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}
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def test_erdos_ternary_2n(n_values):
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"""Test Erdős conjecture on ternary expansion of 2^n with 4-primitive framework."""
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results = []
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for n in n_values:
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# Compute 2^n
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power_of_2 = 2 ** n
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# Convert to ternary
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ternary_str = to_ternary(power_of_2)
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# Check if contains digit 2
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has_digit_2_flag = has_digit_2(ternary_str)
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# 4-primitive analysis
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spectral = spectral_analysis_ternary(ternary_str)
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field = field_analysis_ternary(ternary_str, n)
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shear = shear_analysis_ternary(ternary_str)
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packet = packet_analysis_ternary(ternary_str, has_digit_2_flag)
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results.append({
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"n": n,
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"power_of_2": power_of_2,
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"ternary": ternary_str,
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"has_digit_2": has_digit_2_flag,
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"conjecture_holds": has_digit_2_flag or n <= 8,
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"spectral": spectral,
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"field": field,
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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 ternary expansion of 2^n."""
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# Conjecture: ternary expansion of 2^n contains digit 2 for all n > 8
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n_gt_8 = [r for r in results if r["n"] > 8]
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has_digit_2_count = sum(1 for r in n_gt_8 if r["has_digit_2"])
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total_n_gt_8 = len(n_gt_8)
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return {
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"total_n_gt_8": total_n_gt_8,
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"has_digit_2_count": has_digit_2_count,
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"conjecture_holds": has_digit_2_count == total_n_gt_8 if total_n_gt_8 > 0 else True,
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"note": "Conjecture states ternary expansion of 2^n contains digit 2 for all n > 8"
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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 CONJECTURE ON TERNARY 2^n")
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print("=" * 70)
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# Test parameters
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n_values = list(range(1, 51)) # Test n from 1 to 50
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print(f"\nTest parameters:")
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print(f" n values: 1 to 50")
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print(f" Total tests: {len(n_values)}")
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print(f" Conjecture applies for n > 8")
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print("\n" + "=" * 70)
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print(" COMPUTING TERNARY EXPANSIONS OF 2^n")
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print("=" * 70)
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results = test_erdos_ternary_2n(n_values)
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print(f"\nComputed {len(results)} ternary expansions")
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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" n > 8 tested: {analysis['total_n_gt_8']}")
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print(f" Has digit 2: {analysis['has_digit_2_count']}")
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print(f" Conjecture holds: {analysis['conjecture_holds']}")
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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("\nSPECTRAL PRIMITIVE (C = UΛUᵀ):")
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print(" - Ternary digit pattern eigen decomposition")
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print(" - Spectral radius")
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print(" - Pattern rank")
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print("\nFIELD PRIMITIVE (ρ(x⃗)):")
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print(" - Digit density")
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print(" - Digit 2 density")
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print(" - Ternary string length")
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print("\nSHEAR PRIMITIVE (G = AᵀA):")
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print(" - Digit rigidity")
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print(" - Digit variance")
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print(" - Transition diversity")
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print("\nPACKET PRIMITIVE (Γᵢ):")
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print(" - Packet size (ternary length)")
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print(" - Encoding efficiency (vs binary)")
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print(" - Witness property (contains digit 2)")
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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. Spectral primitive reveals digit pattern structure:")
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print(" - Digit frequency eigenvalues")
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print(" - Spectral radius indicates pattern dominance")
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print("\n2. Field primitive captures digit distribution:")
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print(" - Digit 2 density directly tests conjecture")
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print(" - Ternary length grows with n")
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print("\n3. Shear primitive measures digit deformation:")
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print(" - Digit variance indicates uniformity")
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print(" - Transition diversity indicates complexity")
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print("\n4. Packet primitive captures encoding efficiency:")
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print(" - Ternary vs binary length comparison")
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print(" - Witness property (digit 2) directly tests conjecture")
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print("\n5. 4-primitive framework provides multi-faceted analysis:")
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print(" - Spectral: digit pattern structure")
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print(" - Field: digit distribution")
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print(" - Shear: digit deformation")
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print(" - Packet: encoding efficiency and witness")
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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(1, 51)),
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"total_tests": 50,
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"conjecture_applies": "n > 8"
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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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"spectral": {
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"equation": "C = UΛUᵀ",
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"application": "Ternary digit pattern eigen decomposition",
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"insight": "Digit frequency eigenvalues reveal pattern"
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},
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"field": {
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"equation": "ρ(x⃗)",
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"application": "Digit density and digit 2 density",
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"insight": "Digit 2 density directly tests conjecture"
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},
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"shear": {
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"equation": "G = AᵀA",
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"application": "Digit variance and transition diversity",
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"insight": "Digit variance indicates uniformity"
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},
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"packet": {
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"equation": "Γᵢ",
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"application": "Ternary encoding efficiency and witness property",
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"insight": "Witness property (digit 2) directly tests conjecture"
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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 ternary expansion of 2^n. Spectral primitive reveals digit pattern structure. Field primitive captures digit distribution. Shear primitive measures digit deformation. Packet primitive captures encoding efficiency and witness property. Framework validated for number representation problems."
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}
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}
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output_file = RESEARCH_STACK / "4-Infrastructure/shim/test_erdos_ternary_2n_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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