Sure! Here's the test case generator for the given problem statement about robots on a number line:

```python
import random

def tcgen():
    # Number of robots
    N = random.randint(1, 100000)
    
    # Generate unique X coordinates for robots
    X = random.sample(range(0, 10**9 + 1), N)
    
    # Generate arm lengths for robots
    L = [random.randint(1, 10**9) for _ in range(N)]
    
    # Combine X and L into a list of tuples
    robots = list(zip(X, L))
    
    return N, robots

# Example usage
N, robots = tcgen()
print(N)
for x, l in robots:
    print(x, l)
```

This generator will produce valid test cases for the problem. Let's break it down:

1. **N** is the number of robots, randomly chosen between 1 and 100,000.
2. **X** is a list of unique X coordinates, ensuring that no two robots share the same position. We use `random.sample` to generate a list of unique values within the specified range.
3. **L** is a list of arm lengths for each robot, randomly chosen between 1 and 1 billion.
4. **robots** combines the X coordinates and arm lengths into a list of tuples for easy processing.

Each call to `tcgen()` will generate a new, random test case conforming to the problem's constraints.