Sum Two Numbers Without + or sum

Reply data-processing code may need a low-level integer operation that does not rely on Python's arithmetic addition. Implement a function that adds two signed integers without using the + operator, sum(), .sum(), operator.add, or equivalent arithmetic helpers.

Formal Specification

Implement add_integers(a, b):

• Input: Two Python integers a and b in the signed 32-bit range.
• Output: Their mathematical sum as a Python integer.
• Use bitwise operations such as XOR, AND, left shift, and masking to simulate binary addition.
• The implementation must correctly handle positive values, negative values, zero, and sums outside the signed 32-bit range.
• Do not convert the integers to strings or lists of bits.

Python integers use unlimited precision and conceptually extend negative values with infinite leading ones. Explain how your implementation uses a fixed-width two's-complement representation to ensure termination and then converts the result back to a Python integer.