```ts
/**
 * Cent-exact money helpers for order totals.
 */

export function splitEvenly(totalCents: number, ways: number): number[] {
  if (!Number.isInteger(totalCents) || totalCents < 0) {
    throw new Error("totalCents must be a non-negative integer");
  }
  if (!Number.isInteger(ways) || ways <= 0) {
    throw new Error("ways must be a positive integer");
  }

  const base = Math.floor(totalCents / ways);
  const remainder = totalCents % ways;
  const parts: number[] = new Array(ways).fill(base);
  for (let i = 0; i < remainder; i++) {
    parts[i]++;
  }
  return parts;
}

export function applyBps(cents: number, bps: number): number {
  if (!Number.isInteger(cents) || cents < 0) {
    throw new Error("cents must be a non-negative integer");
  }
  if (!Number.isInteger(bps) || bps < 0) {
    throw new Error("bps must be a non-negative integer");
  }
  return Math.floor((cents * bps + 5000) / 10000);
}

export function sumParts(parts: number[]): number {
  let total = 0;
  for (const part of parts) {
    if (!Number.isInteger(part)) {
      throw new Error("parts must be integers");
    }
    total += part;
  }
  return total;
}
```

Key fixes in `splitEvenly`:
- **Validation added**: rejects non-integer or negative `totalCents`, and non-positive `ways`.
- **No per-part rounding**: uses `Math.floor` for the base share and adds 1 to the first `remainder` parts, so the sum is exactly `totalCents`.
- **Non-increasing order**: parts are initialized equal and only the first `remainder` are incremented, so the array is naturally non-increasing.
- **No mutation of input**: uses a new `Array(ways).fill(base)` instead of mutating a pre-allocated array.