Sum of Two Integers

To add two integers without + or -, use a XOR b for the sum of each bit ignoring carry, and (a AND b) << 1 for the carry bits. Reassign a to the XOR result and b to the carry, repeating until carry is zero. The invariant is that the mathematical total of a + b remains unchanged while carries move left until resolved. In fixed-width two's-complement arithmetic this terminates in at most the word size number of iterations, so it is O(w) time and O(1) space, effectively constant for 32-bit integers. In languages with arbitrary-precision signed integers, negative values can make an unbounded bitwise loop problematic; mask to the required width and convert the final unsigned result back to signed form if the problem specifies 32-bit behavior. Explain this language detail rather than assuming JavaScript or Python bit operations are interchangeable.