```ts
export function applyCharges(caps: Record<string, [number, number]>, charges: [string, string, number][]): { left: Record<string, number>; refused: [string, string][] } {
  const result = { left: { ...caps }, refused: [] };

  for (const [id, bucket, amount] of charges) {
    if (!Number.isInteger(amount) || amount <= 0) {
      throw new Error("Invalid charge amount");
    }

    const cap = caps[bucket];
    if (!cap) {
      result.refused.push([id, "unknown"]);
      continue;
    }

    const [allowance, singleCap] = cap;
    if (amount > singleCap) {
      result.refused.push([id, "single"]);
      continue;
    }

    if (result.left[bucket] < amount) {
      result.refused.push([id, "cap"]);
      continue;
    }

    result.left[bucket] -= amount;
  }

  return result;
}
```