```ts
/**
 * Merges overlapping or touching intervals into non-overlapping intervals sorted by start.
 * @param intervals An array of [start, end] pairs with start <= end.
 * @returns A new array of merged intervals sorted by start.
 * @throws Error if any interval is invalid (not a two-number pair, or start > end).
 */
export function mergeIntervals(intervals: [number, number][]): [number, number][] {
  // Validate input
  for (const interval of intervals) {
    if (!Array.isArray(interval) || interval.length !== 2) {
      throw new Error('Each interval must be a two-number pair');
    }
    const [start, end] = interval;
    if (typeof start !== 'number' || typeof end !== 'number') {
      throw new Error('Interval values must be numbers');
    }
    if (!Number.isFinite(start) || !Number.isFinite(end)) {
      throw new Error('Interval values must be finite numbers');
    }
    if (start > end) {
      throw new Error('Start must be <= end for each interval');
    }
  }

  // Handle empty input
  if (intervals.length === 0) {
    return [];
  }

  // Create a copy and sort by start value
  const sorted = [...intervals].sort((a, b) => a[0] - b[0]);

  // Merge intervals
  const result: [number, number][] = [];
  let current = sorted[0];

  for (let i = 1; i < sorted.length; i++) {
    const next = sorted[i];
    
    // Check if intervals overlap or touch (next start <= current end)
    if (next[0] <= current[1]) {
      // Merge by extending the end if necessary
      current = [current[0], Math.max(current[1], next[1])] as [number, number];
    } else {
      // No overlap/touch, push current and move to next
      result.push(current);
      current = next;
    }
  }

  // Push the last interval
  result.push(current);

  return result;
}
```