```python
"""Cent-exact money helpers for order totals."""


def split_evenly(total_cents: int, ways: int) -> list:
    """Split a total into `ways` parts that sum to exactly `total_cents`,
    are in non-increasing order, and differ by at most one cent.
    """
    if not isinstance(total_cents, int) or total_cents < 0:
        raise ValueError("total_cents must be a non-negative integer")
    if not isinstance(ways, int) or ways <= 0:
        raise ValueError("ways must be a positive integer")

    base = total_cents // ways
    remainder = total_cents % ways
    parts = [base + 1] * remainder + [base] * (ways - remainder)
    return parts


def apply_bps(cents: int, bps: int) -> int:
    """Convert cents to basis points, rounding up at 0.5%."""
    if isinstance(cents, bool) or not isinstance(cents, int) or cents < 0:
        raise ValueError("cents must be a non-negative integer")
    if isinstance(bps, bool) or not isinstance(bps, int) or bps < 0:
        raise ValueError("bps must be a non-negative integer")
    return (cents * bps + 5000) // 10000


def sum_parts(parts: list) -> int:
    """Sum a list of integer parts, validating each element."""
    total = 0
    for part in parts:
        if isinstance(part, bool) or not isinstance(part, int):
            raise ValueError("parts must be integers")
        total += part
    return total
```

Key fixes to `split_evenly`:
- Replaced `round(total_cents / ways)` (which rounds each share independently and can cause the parts to not sum to the total) with integer division `total_cents // ways` plus a remainder distribution.
- The `remainder` extra cents are distributed one each to the *earliest* (largest) parts, ensuring non-increasing order and exact summation.
- Added validation for `total_cents` (non-negative integer) and `ways` (positive integer), raising `ValueError` with a short message for invalid inputs.
- `apply_bps` and `sum_parts` are unchanged.