Sure, I can help you write a test case generator for the given problem statement. The generator will produce random values for the field dimensions, fatigue increments, number of players, and their initial positions.

Here's the test case generator in Python:

```python
import random

def tcgen():
    # Generate random field dimensions H and W
    H = random.randint(1, 500)
    W = random.randint(1, 500)
    
    # Generate random fatigue increments A, B, and C
    A = random.randint(0, 10**9)
    B = random.randint(0, 10**9)
    C = random.randint(0, 10**9)
    
    # Generate random number of players N
    N = random.randint(2, 100000)
    
    # Generate random initial positions for N players
    players = []
    for _ in range(N):
        S_i = random.randint(0, H)
        T_i = random.randint(0, W)
        players.append((S_i, T_i))
    
    # Ensure (S_1, T_1) != (S_N, T_N)
    while players[0] == players[-1]:
        players[-1] = (random.randint(0, H), random.randint(0, W))
    
    return H, W, A, B, C, N, players

# Function to print test case in the required format
def print_test_case(H, W, A, B, C, N, players):
    print(H, W)
    print(A, B, C)
    print(N)
    for player in players:
        print(player[0], player[1])

# Generate a test case
H, W, A, B, C, N, players = tcgen()

# Print the generated test case
print_test_case(H, W, A, B, C, N, players)
```

### Explanation

1. **Field Dimensions (H, W):**
   - Randomly generate the height (H) and width (W) of the field within the constraints \(1 \leq H, W \leq 500\).

2. **Fatigue Increments (A, B, C):**
   - Randomly generate the values for fatigue increments \(A, B, C\) within the constraints \(0 \leq A, B, C \leq 1,000,000,000\).

3. **Number of Players (N):**
   - Randomly generate the number of players \(N\) within the constraints \(2 \leq N \leq 100,000\).

4. **Initial Positions of Players:**
   - For each player, generate a random initial position \((S_i, T_i)\) within the field dimensions \(0 \leq S_i \leq H\) and \(0 \leq T_i \leq W\).
   - Ensure that the initial position of player 1 is not the same as the initial position of player \(N\).

5. **Output:**
   - The function `print_test_case` formats and prints the generated test case data.

This generator ensures that all constraints are met and provides a diverse range of test cases for the problem. You can run the script multiple times to generate different test cases.