We give a simplified, self-contained proof of OpenAI's parity lower bound for constant-depth quantum circuits with arbitrary one-qubit gates, unbounded-arity Toffoli gates, and polynomially many ancillas. The proof establishes product-projection localization and uses it to bound the parity Fourier coefficient of the acceptance probability. We remove one circuit layer directly, avoiding intermediate low-count propagation. This yields a leakage recurrence that closes by induction over depth and threshold scales.