```python
def base_amount(amount: int, unit: str, defs: dict, base: str) -> int:
    """
    Convert amount of a given unit into its base unit using a definition dictionary.

    :param amount: The amount of the unit to convert.
    :param unit: The name of the unit to convert from.
    :param defs: A dictionary mapping unit names to [factor, finer] pairs.
    :param base: The base unit name.
    :return: The converted amount in the base unit.
    """
    if not isinstance(amount, int) or amount < 0:
        raise ValueError("Amount must be a non-negative whole number.")
    
    if unit not in defs:
        raise ValueError(f"Unit '{unit}' is not defined.")

    if base not in defs:
        raise ValueError(f"Base unit '{base}' is not defined.")

    factor, finer = defs[unit]
    if not isinstance(factor, int) or factor <= 0:
        raise ValueError("Factor must be a positive whole number.")
    
    def convert_to_base(amount, unit):
        nonlocal defs
        nonlocal base

        if unit == base:
            return amount
        
        if unit not in defs:
            raise ValueError(f"Unit '{unit}' is not defined.")

        factor, finer = defs[unit]
        if not isinstance(factor, int) or factor <= 0:
            raise ValueError("Factor must be a positive whole number.")
        
        return convert_to_base(amount // factor, finer)

    try:
        return convert_to_base(amount, unit)
    except RecursionError as e:
        raise ValueError(f"Chain loops so it can never reach the base. {e}")
```