Certainly! Here's a test case generator for the given problem statement:

```python
import random

def tcgen():
    # Generate N and K within the specified constraints
    N = random.randint(2, 6000)
    K = random.randint(2, N)
    
    vacant_lots = set()
    
    while len(vacant_lots) < N:
        x = random.randint(1, 1000000)
        y = random.randint(1, 1000000)
        
        # Ensure the xi mod 2 = floor(yi / 2) mod 2 condition
        if x % 2 == (y // 2) % 2:
            vacant_lots.add((x, y))
    
    vacant_lots = list(vacant_lots)
    
    # Format the output as required
    result = []
    result.append(f"{N} {K}")
    for lot in vacant_lots:
        result.append(f"{lot[0]} {lot[1]}")
    
    return "\n".join(result)

# Example usage
test_case = tcgen()
print(test_case)
```

This generator ensures that:
1. \(N\) and \(K\) are within the specified constraints.
2. Each coordinate pair \((x_i, y_i)\) satisfies the condition \(x_i \mod 2 = \left\lfloor \frac{y_i}{2} \right\rfloor \mod 2\).
3. No two vacant lots have the same coordinates.

You can use this function to generate random test cases for your problem. Each call to `tcgen()` will produce a new test case formatted as a string, which you can then use to test your solution.