Sure, let's create a test case generator for the provided problem statement "Equivalent Propositions." The goal is to generate random test cases that adhere to the constraints and input format specified in the problem.

Here's the test case generator:

```python
import random

def generate_test_case():
    # Step 1: Randomly generate N within the allowed range
    N = random.randint(2, 300)
    
    # Step 2: Generate a number of implication pairs, M
    # Ensure M is within the valid range of 1 to N(N-1)
    M = random.randint(1, min(N * (N-1), 5000))
    
    # Step 3: Generate M pairs of implications (a_i, b_i)
    implications = set()
    while len(implications) < M:
        a_i = random.randint(1, N)
        b_i = random.randint(1, N)
        if a_i != b_i:
            implications.add((a_i, b_i))
    
    # Convert set to list for consistent ordering
    implications = list(implications)
    
    # Step 4: Format the output
    output = f"{N} {M}\n"
    for a_i, b_i in implications:
        output += f"{a_i} {b_i}\n"
    
    return output

# Generate a test case and print it
print(generate_test_case())
```

This script generates a random number of propositions `N` and a random number of implication pairs `M` within the specified constraints. Each implication pair is unique and satisfies the conditions \(a_i \neq b_i\) and \(1 \leq a_i, b_i \leq N\).

Here is how it works:
1. **Randomly choose** \(N\) in the range from 2 to 300.
2. **Randomly choose** \(M\) in the range from 1 to \( \min(N(N-1), 5000) \), ensuring it doesn't exceed the maximum possible number of pairs.
3. **Generate unique pairs** \( (a_i, b_i) \), ensuring \( a_i \neq b_i \).
4. **Format the output** to match the required input format for the problem.

This generator ensures that all generated test cases are valid according to the problem constraints.