```python
def base_amount(amount: int, unit: str, defs: dict, base: str) -> int:
    """
    Convert amount of unit into the base unit by unwinding one definition at a time.
    An amount of the base unit is already converted, and otherwise amount of unit
    is amount times factor of the finer unit, converted onward. Reject an amount that
    is not a non-negative whole number, a unit that is neither the base nor defined in defs,
    a factor that is not a positive whole number, and a chain that loops so it can never reach the base.
    """
    if amount < 0 or not isinstance(amount, int):
        raise ValueError("Amount must be a non-negative integer.")
    
    if unit not in defs:
        raise ValueError(f"Unit '{unit}' is not defined in defs.")
    
    if base not in defs:
        raise ValueError(f"Base unit '{base}' is not defined in defs.")
    
    factor, finer = defs[unit]
    if factor <= 0 or not isinstance(factor, int):
        raise ValueError("Factor must be a positive integer.")
    
    current_unit = unit
    while current_unit != base:
        if current_unit not in defs:
            raise ValueError(f"Chain loops and cannot reach the base unit '{base}'.")
        
        factor, finer = defs[current_unit]
        if factor <= 0 or not isinstance(factor, int):
            raise ValueError("Factor must be a positive integer.")
        
        amount *= factor
        current_unit = finer
    
    return amount
```