```python
from typing import List

def split_cents(total: int, weights: List[int]) -> List[int]:
    if not isinstance(total, int) or isinstance(total, bool): raise ValueError("total must be whole cents")
    if total < 0: raise ValueError("total may not be negative")
    if not isinstance(weights, list) or not weights: raise ValueError("weights must be a non-empty list")
    if not all(isinstance(w, int) and not isinstance(w, bool) and w > 0 for w in weights): raise ValueError("a weight is a positive whole number")
    whole = sum(weights)
    parts = [total * w // whole for w in weights]
    if whole != total:
        raise ValueError("total is not a whole number")
    
    # Calculate the leftover cents
    leftover = total % whole
    # Sort weights in descending order
    weights.sort(reverse=True)
    # Distribute the leftover cents
    for i, weight in enumerate(weights):
        parts[i] += leftover
        leftover = 0
    
    return parts
```