### Clone Repository and Install Dependencies

Source: https://github.com/bbarclay/collatzconjecture/blob/main/docs/README.md

Clone the repository to your local machine and install the necessary Python dependencies using pip.

```bash
git clone https://github.com/yourusername/collatzcrypto.git
cd collatzcrypto
```

```bash
pip install -r requirements.txt
```

--------------------------------

### Collatz Sequence Entropy Analysis Setup

Source: https://github.com/bbarclay/collatzconjecture/blob/main/collatz_residue_analysis.ipynb

Imports necessary libraries for numerical operations, plotting, and precise decimal calculations. Sets up the plotting style and color palette.

```python
import numpy as np
import matplotlib.pyplot as plt
from collections import defaultdict
import seaborn as sns
from math import log2
from decimal import Decimal, getcontext

getcontext().prec = 50
plt.style.use('seaborn')
sns.set_palette('husl')
```

--------------------------------

### Install Project Dependencies

Source: https://github.com/bbarclay/collatzconjecture/blob/main/README.md

Install the necessary Python dependencies for the project using pip.

```bash
pip install -r requirements.txt
```

--------------------------------

### Clone Collatz Conjecture Repository

Source: https://github.com/bbarclay/collatzconjecture/blob/main/README.md

Clone the repository to get started with the Collatz Conjecture project.

```bash
git clone https://github.com/bbarclay/collatzconjecture.git
```

--------------------------------

### Plot Collatz Entropy Analysis Results

Source: https://github.com/bbarclay/collatzconjecture/blob/main/collatz_residue_analysis.ipynb

Visualizes the entropy changes and total entropy over steps for a Collatz sequence starting from 7. Highlights the net loss of information per step.

```python
# Let's analyze a simple number: 7
steps, changes, totals = analyze_step_entropy(7)

# Plot the results
fig, (ax1, ax2) = plt.subplots(2, 1, figsize=(12, 8))

# Plot entropy changes
ax1.bar(range(len(changes)), changes, alpha=0.6)
ax1.axhline(y=0, color='r', linestyle='--', alpha=0.3)
ax1.set_title('Entropy Change at Each Step')
ax1.set_ylabel('Bits of Information')
ax1.text(0.02, 0.95, 'Gain: log₂(3) ≈ 1.58 bits
Loss: τ bits', 
         transform=ax1.transAxes, bbox=dict(facecolor='white', alpha=0.8))

# Plot total entropy
ax2.plot(totals, '-o', alpha=0.6)
ax2.set_title('Total Entropy Over Time')
ax2.set_ylabel('Total Bits')
ax2.text(0.02, 0.95, 'Even temporary gains
can\'t prevent overall loss', 
         transform=ax2.transAxes, bbox=dict(facecolor='white', alpha=0.8))

plt.tight_layout()
plt.show()
```

--------------------------------

### Contributing to the Project

Source: https://github.com/bbarclay/collatzconjecture/blob/main/docs/README.md

Follow these steps to contribute to the project: fork the repository, create a new branch, commit your changes, and open a Pull Request.

```bash
git checkout -b feature/amazing-feature
```

```bash
git commit -m 'Add amazing feature'
```

```bash
git push origin feature/amazing-feature
```

--------------------------------

### Run Test Suite

Source: https://github.com/bbarclay/collatzconjecture/blob/main/docs/README.md

Execute the project's test suite to ensure all components are functioning correctly.

```bash
python run_tests.py
```

--------------------------------

### Generate Verification Report

Source: https://github.com/bbarclay/collatzconjecture/blob/main/docs/README.md

Generate a detailed report from the verification results by providing the path to the results file.

```bash
python generate_report.py <results_file>
```

--------------------------------

### Extract Code Snippets from LaTeX

Source: https://github.com/bbarclay/collatzconjecture/blob/main/docs/VERIFICATION_PLAN.md

Use this bash command to extract code listings from LaTeX files. It searches for \begin{lstlisting} and captures the following 20 lines, saving them to a text file for review.

```bash
for file in sections/*.tex; do
  grep -A 20 "\\begin{lstlisting}" "$file" > "code_review/${file%.tex}_code.txt"
done
```

--------------------------------

### Visualize Residue Class Mixing in Collatz Sequences

Source: https://github.com/bbarclay/collatzconjecture/blob/main/collatz_residue_analysis.ipynb

Visualizes the transition probabilities between residue classes modulo a given number for the Collatz step (3n+1). Useful for demonstrating perfect mixing. Requires numpy and seaborn.

```python
def show_mixing(modulus=8, samples=1000):
    """Show how numbers mix across residue classes"""
    transitions = np.zeros((modulus, modulus))
    row_counts = np.zeros(modulus)
    
    for n in range(1, samples, 2):
        start = n % modulus
        next_n = (3 * n + 1) % modulus
        transitions[start][next_n] += 1
        row_counts[start] += 1
    
    # Safe normalization avoiding division by zero
    normalized = np.zeros_like(transitions)
    for i in range(modulus):
        if row_counts[i] > 0:
            normalized[i] = transitions[i] / row_counts[i]
    
    plt.figure(figsize=(10, 8))
    sns.heatmap(normalized, annot=True, fmt='.2f', cmap='viridis')
    plt.title(f'Transition Probabilities (mod {modulus})')
    plt.xlabel('Next Residue')
    plt.ylabel('Current Residue')
    plt.text(1.2, 0.5, 'Perfect mixing means:\nNo residue class can\nform a closed loop', 
             transform=plt.gca().transAxes, bbox=dict(facecolor='white', alpha=0.8))
    plt.show()

show_mixing()
```

--------------------------------

### Run Collatz Conjecture Verification

Source: https://github.com/bbarclay/collatzconjecture/blob/main/docs/README.md

Execute the main verification script to run all verification components, generate results, save logs, and produce a summary report.

```bash
python verify_proof.py
```

--------------------------------

### Analyze Gaps Between Powers of 2 and 3

Source: https://github.com/bbarclay/collatzconjecture/blob/main/collatz_residue_analysis.ipynb

Visualizes the gaps between powers of 2 and 3 and plots them against Baker's bounds. Requires matplotlib and numpy. The function returns the top 10 closest pairs.

```python
from math import log2
import matplotlib.pyplot as plt
import numpy as np

def analyze_power_gaps(max_power=20):
    """Analyze gaps between powers of 2 and 3"""
    powers_2 = [2**n for n in range(max_power)]
    powers_3 = [3**n for n in range(max_power)]
    
    gaps = []
    pairs = []
    
    # Find closest pairs of powers
    for p2 in powers_2:
        for p3 in powers_3:
            gap = abs(p2 - p3)
            gaps.append((gap, p2, p3))
    
    # Sort by gap size
    gaps.sort()
    top_10 = gaps[:10]
    
    # Plot the results
    fig, (ax1, ax2) = plt.subplots(2, 1, figsize=(12, 10))
    
    # Plot 1: Powers on log scale
    ax1.scatter([log2(x[1]) for x in top_10], [log2(x[2]) for x in top_10], 
               alpha=0.6, s=100, label='Closest Pairs')
    ax1.plot([0, max_power], [0, max_power * log2(3)], 'r--', alpha=0.3, 
            label='y = x * log₂(3)')
    ax1.set_title('Closest Power Pairs (Log Scale)')
    ax1.set_xlabel('Power of 2 (log₂)')
    ax1.set_ylabel('Power of 3 (log₂)')
    ax1.legend()
    ax1.grid(True, alpha=0.3)
    
    # Plot 2: Relative gaps
    relative_gaps = [gap[0]/max(gap[1], gap[2]) for gap in top_10]
    ax2.bar(range(len(relative_gaps)), relative_gaps, alpha=0.6)
    ax2.set_title('Relative Gap Sizes for Closest Pairs')
    ax2.set_ylabel('Relative Gap (Difference / Max Value)')
    ax2.set_xlabel('Pair Index')
    
    # Add Baker's bound line
    C = min(relative_gaps) * 1.1  # Use observed minimum as reference
    kappa = 2.5  # Example value
    x = np.arange(1, len(relative_gaps) + 1)
    bound = C / (x**kappa)
    ax2.plot(range(len(relative_gaps)), bound, 'r--', 
            label=f'Baker\'s Bound (C/x^κ)', alpha=0.7)
    ax2.legend()
    
    plt.tight_layout()
    plt.show()
    
    # Print the closest pairs
    print('\nClosest power pairs (2^a ≈ 3^b):')
    for gap, p2, p3 in top_10:
        a = round(log2(p2))
        b = round(log2(p3) / log2(3))
        print(f'2^{a} = {p2:,} ≈ 3^{b} = {p3:,} (gap: {gap:,})')
    
    return top_10

# Analyze gaps between powers
closest_pairs = analyze_power_gaps(30)
```

--------------------------------

### Generate Collatz Visualizations

Source: https://github.com/bbarclay/collatzconjecture/blob/main/README.md

Run Python scripts to generate various visualizations for the Collatz Conjecture project, including core visualizations, measure theory, information theory, and cover art.

```python
python py_visuals/collatz_core_viz.py
```

```python
python py_visuals/measure_theory_viz.py
```

```python
python py_visuals/information_theory_viz.py
```

```python
python py_visuals/cover_art.py
```

--------------------------------

### Analyze Collatz Step Entropy Changes

Source: https://github.com/bbarclay/collatzconjecture/blob/main/collatz_residue_analysis.ipynb

Calculates the entropy change at each step of the Collatz sequence for a given number. Handles both even (division by 2) and odd (3n+1) steps, tracking bit changes.

```python
def analyze_step_entropy(n, max_steps=50):
    """Analyze entropy changes step by step"""
    steps = []
    entropy_changes = []
    total_entropy = [log2(n)]
    current = n
    
    for _ in range(max_steps):
        if current == 1:
            break
            
        steps.append(current)
        if current % 2 == 0:
            # Even step: always loses exactly 1 bit
            next_num = current // 2
            entropy_changes.append(-1)
        else:
            # Odd step: gains log2(3) bits, then loses τ bits
            next_num = current * 3 + 1
            tau = 0
            while next_num % 2 == 0:
                tau += 1
                next_num //= 2
            entropy_changes.append(log2(3) - tau)
        
        current = next_num
        total_entropy.append(log2(current))
    
    return steps, entropy_changes, total_entropy
```

--------------------------------

### Compare Collatz Trajectories With and Without +1

Source: https://github.com/bbarclay/collatzconjecture/blob/main/collatz_residue_analysis.ipynb

Compares the behavior of Collatz sequences with and without the '+1' term. Useful for visualizing the impact of the '+1' on trajectory mixing. Requires matplotlib and numpy.

```python
def compare_with_without_one(start_n=15, steps=10):
    """Compare trajectories with and without the +1 term"""
    with_one = [start_n]
    without_one = [start_n]
    
    n1, n2 = start_n, start_n
    for _ in range(steps):
        # With +1
        if n1 % 2 == 0:
            n1 //= 2
        else:
            n1 = 3 * n1 + 1
        with_one.append(n1)
        
        # Without +1
        if n2 % 2 == 0:
            n2 //= 2
        else:
            n2 = 3 * n2
        without_one.append(n2)
    
    plt.figure(figsize=(12, 6))
    plt.plot(with_one, '-o', label='3n+1', alpha=0.7)
    plt.plot(without_one, '-o', label='3n', alpha=0.7)
    plt.yscale('log')
    plt.title('The Critical Role of +1')
    plt.ylabel('Value (log scale)')
    plt.xlabel('Step')
    plt.legend()
    plt.grid(True, alpha=0.3)
    plt.text(0.02, 0.95, 'Without +1:\nStays in multiplicative group\n\nWith +1:\nForces mixing across all numbers', 
             transform=plt.gca().transAxes, bbox=dict(facecolor='white', alpha=0.8))
    plt.show()

compare_with_without_one()
```

--------------------------------

### Analyze Collatz Trajectory Power Balance

Source: https://github.com/bbarclay/collatzconjecture/blob/main/collatz_residue_analysis.ipynb

Analyzes how a number's Collatz trajectory balances divisions by 2 and multiplications by 3. It plots the trajectory and the cumulative counts of these operations. Requires matplotlib.

```python
import matplotlib.pyplot as plt

def analyze_power_balance(start_n, steps=50):
    """Analyze how a number's trajectory balances powers of 2 and 3"""
    trajectory = [start_n]
    power2_counts = [0]  # Cumulative divisions by 2
    power3_counts = [0]  # Cumulative multiplications by 3
    
    n = start_n
    p2 = 0  # Cumulative power of 2
    p3 = 0  # Cumulative power of 3
    
    for _ in range(steps):
        if n == 1:
            break
            
        if n % 2 == 0:
            n //= 2
            p2 += 1
        else:
            n = 3 * n + 1
            p3 += 1
            # Count trailing zeros
            while n % 2 == 0:
                n //= 2
                p2 += 1
        
        trajectory.append(n)
        power2_counts.append(p2)
        power3_counts.append(p3)
    
    # Plot the results
    fig, (ax1, ax2) = plt.subplots(2, 1, figsize=(12, 8))
    
    # Plot trajectory
    ax1.plot(trajectory, '-o', alpha=0.6)
    ax1.set_yscale('log')
    ax1.set_title(f'Trajectory for n = {start_n}')
    ax1.set_ylabel('Value')
    
    # Plot power balance
    ax2.plot(power2_counts, label='Divisions by 2', alpha=0.6)
    ax2.plot(power3_counts, label='Multiplications by 3', alpha=0.6)
    ax2.set_title('Cumulative Power Balance')
    ax2.set_ylabel('Count')
    ax2.legend()
    
    plt.tight_layout()
    plt.show()
    
    # Print summary
    print(f'\nFinal balance for {start_n}:')
    print(f'Total divisions by 2: {power2_counts[-1]}')
    print(f'Total multiplications by 3: {power3_counts[-1]}')
    print(f'Ratio (should never perfectly balance): {power2_counts[-1]/power3_counts[-1]:.3f}')

# Analyze a few different starting numbers
for n in [27, 31, 41]:
    analyze_power_balance(n)
```

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