Maximum Sum Circular Subarray

Split into two cases

A best circular subarray either does not wrap, which is ordinary Kadane's maximum, or it wraps across the ends. In the wrapped case, the elements excluded from the circle form one contiguous minimum-sum subarray, so its sum is total - minSubarray. Run Kadane twice in one scan: once for maximum and once for minimum, while accumulating total. Return max(maxSubarray, total - minSubarray) only when the second candidate is nonempty.

Important guard

If every value is negative, the minimum subarray is the entire array and total - minSubarray becomes zero, incorrectly selecting an empty subarray. In that situation return the ordinary maximum, the least-negative element. Explain that this guard follows the nonempty requirement. Do not duplicate the array and run a length-limited window unless constraints specifically call for it. Time is O(n) and auxiliary space is O(1).