Certainly! Let's create a test case generator for the given problem statement in Python.

The problem involves generating an \(N \times N\) matrix \(A\) where \(A[i][j]\) represents the shortest distances between cities. The constraints and conditions given need to be respected.

Here's a test case generator for this problem:

```python
import random

def generate_matrix(N):
    A = [[0] * N for _ in range(N)]
    
    # Ensure the diagonal is zero
    for i in range(N):
        A[i][i] = 0
    
    # Populate the upper triangle and mirror it to the lower triangle
    for i in range(N):
        for j in range(i + 1, N):
            if random.choice([True, False]):
                dist = random.randint(1, 10**9)
                A[i][j] = dist
                A[j][i] = dist
            else:
                # For ensuring some -1 cases, let's create some inconsistent paths
                dist = random.randint(1, 10**9)
                A[i][j] = dist
                A[j][i] = dist + random.randint(1, 100)  # Inconsistent distance

    return A

def tcgen():
    N = random.randint(1, 300)
    A = generate_matrix(N)
    
    # Convert matrix to the input format
    inputs = [str(N)]
    for row in A:
        inputs.append(" ".join(map(str, row)))
    
    return "\n".join(inputs)

# Example usage:
if __name__ == "__main__":
    print(tcgen())
```

### Explanation:
1. **Generating N**: Generate a random integer \(N\) between 1 and 300.
2. **Generating Matrix**:
   - Create an \(N \times N\) matrix initialized with zeros.
   - Ensure the diagonal elements are zero since \(A[i][i] = 0\).
   - Populate the upper triangle of the matrix with random distances.
   - Mirror the upper triangle to the lower triangle to maintain symmetry.
   - Introduce some inconsistencies deliberately to ensure some test cases will fail to meet the shortest path conditions.
3. **Formatting the Output**: Convert the matrix into the required input format.

This generator will provide various test cases, some of which will have valid road networks, while others will not, helping to thoroughly test the solution.

Feel free to modify the generator as needed or add more constraints to better fit specific testing scenarios.