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This approach leverages operator overloading (or language-specific methods) to define how instances of ArrayWrapper behave when used with the '+' operator and the String conversion function. We'll override/add appropriate methods to customize these operations:
ArrayWrapper instances.Time Complexity: O(n + m), where n and m are the lengths of the two arrays in the instances being added. 
Space Complexity: O(1), as we do not use any additional space.
#include <vector>
#include <numeric>
#include <sstream>
class ArrayWrapper {
    std::vector<int> nums;
public:
    ArrayWrapper(std::vector<int> nums) : nums(nums) {}
    int operator+(const ArrayWrapper& other) {
        return std::accumulate(this->nums.begin(), this->nums.end(), 0) +
               std::accumulate(other.nums.begin(), other.nums.end(), 0);
    }
    friend std::ostream& operator<<(std::ostream& os, const ArrayWrapper& aw) {
        os << "[";
        for(size_t i = 0; i < aw.nums.size(); ++i) {
            os << aw.nums[i];
            if (i != aw.nums.size() - 1) os << ",";
        }
        os << "]";
        return os;
    }
};
// Usage Example
int main() {
    ArrayWrapper obj1({1, 2});
    ArrayWrapper obj2({3, 4});
    std::cout << (obj1 + obj2) << std::endl;  // Output: 10
    std::cout << obj1 << std::endl;            // Output: [1,2]
    return 0;
}In C++, we overloaded the '+' operator to handle the sum of elements from two ArrayWrapper instances, and provided an operator<< overload to handle converting the instance to a string for output.
This approach focuses on languages that support the implementation of interfaces to achieve the desired functionality. We utilize relevant interfaces or methods required to override the behavior of arithmetic operations and object-to-string conversion.
Time Complexity: O(n + m), for summing elements of both arrays. 
Space Complexity: O(1), no additional space required.
1class ArrayWrapper {
2    private int[] nums;
3
4    public ArrayWrapper(int[] nums) {
5        this.nums = nums;
6    }
7
8    public int add(ArrayWrapper other) {
9        return Arrays.stream(this.nums).sum() + Arrays.stream(other.nums).sum();
10    }
11
12    public String toString() {
13        return Arrays.toString(nums);
14    }
15}
16
17// Usage Example
18ArrayWrapper obj1 = new ArrayWrapper(new int[]{1,2});
19ArrayWrapper obj2 = new ArrayWrapper(new int[]{3,4});
20System.out.println(obj1.add(obj2));  // Output: 10
21System.out.println(obj1.toString()); // Output: [1, 2]In Java, implementing an interface is less visible here but we use standard methods. Despite not directly implementing an interface, the method pattern acts as such.