A Shift-Operator Proof of Strong Convergence for Boţ–Nguyen Acceleration

Bauschke and Gao conjectured that the Boţ–Nguyen accelerated iteration associated with any linearnonexpansive operator on a real Hilbert space converges strongly for every parameter α > 2. Their proofat α = 4 proceeded through a Lorentz condition for universal weighted-mean coefficients. We prove thefull conjecture by applying the already known fixed-point residual estimate to one operator: the unilateralright shift on ℓ2. The shift places the universal coefficients in distinct orthogonal coordinates, and a singleCauchy–Schwarz inequality gives the stronger quantitative bound TV (cn) = o(n−1/2). We state a reusableshift criterion and provide reproducible 80-digit Python computations checking the coefficient recurrenceand its residual identity. These finite computations supplement rather than replace the proof.