Certainly! To create a test case generator for the given problem statement "Infallibly Crack Perplexing Cryptarithm", we need to generate valid encrypted arithmetic equations based on the specified rules. Here is how we can approach it:

1. **Generate a valid equation based on the provided grammar:**
   - Equations must have one equal sign `=`.
   - Each side of the equation must be a valid expression involving binary digits, `+`, `-`, `*`, and parentheses `()`.
   - Use context-free grammar to ensure the expressions are valid.

2. **Randomly replace some characters with letters:**
   - The letters used can be any letter from the Roman alphabet, either lowercase or uppercase.
   - Ensure the replacements are consistent, i.e., the same character is always replaced by the same letter.
   - Ensure that different characters are replaced by different letters.

3. **Make sure the test cases include edge cases:**
   - Equations with minimal and maximal numbers of characters.
   - Different combinations of operators and parentheses.
   - Ensure some cases where no valid equation can be decrypted.

Here's the Python code for generating test cases:

```python
import random
import string

def generate_expression():
    """Generate a valid binary arithmetic expression based on the given grammar."""
    def generate_N():
        length = random.randint(1, 10)  # Generate a binary number of random length.
        return ''.join(random.choice('01') for _ in range(length)).lstrip('0') or '0'
    
    def generate_F():
        choice = random.choice(['N', '-F', '(E)'])
        if choice == 'N':
            return generate_N()
        elif choice == '-F':
            return '-' + generate_F()
        elif choice == '(E)':
            return '(' + generate_E() + ')'
    
    def generate_T():
        result = generate_F()
        while random.choice([True, False]):
            result += '*' + generate_F()
        return result
    
    def generate_E():
        result = generate_T()
        while random.choice([True, False]):
            result += random.choice(['+', '-']) + generate_T()
        return result
    
    return generate_E()

def generate_equation():
    """Generate a full equation Q ::= E=E"""
    left_expr = generate_expression()
    right_expr = generate_expression()
    return left_expr + '=' + right_expr

def encrypt_equation(equation):
    """Encrypt the equation by replacing some characters with letters."""
    chars = list(set(equation))  # Unique characters in the equation.
    letters = list(string.ascii_letters)  # Possible letters for replacement.
    replacements = {}
    
    for char in chars:
        if random.choice([True, False]):
            letter = random.choice(letters)
            replacements[char] = letter
            letters.remove(letter)
    
    encrypted_equation = ''.join(replacements.get(char, char) for char in equation)
    return encrypted_equation

def generate_test_case():
    equation = generate_equation()
    encrypted_equation = encrypt_equation(equation)
    return encrypted_equation

# Example usage:
for _ in range(10):  # Generate 10 test cases
    print(generate_test_case())
```

### Explanation
1. **generate_expression**:
   - This generates a valid arithmetic expression recursively based on the given grammar.
   - It handles `N`, `F`, `T`, and `E` productions as described.

2. **generate_equation**:
   - Combines two expressions with an equal sign `=` to form a valid equation.

3. **encrypt_equation**:
   - Randomly replaces some characters in the equation with letters while ensuring consistency.

4. **generate_test_case**:
   - Combines the above functions to produce a fully encrypted test case.

This code will produce valid and varied test cases for the given problem statement.