Analytical Proofs for Jensen–Shannon Divergence Maximization in the AWGN Channel

This work provides a fully analytic and self-contained proof that, under individual power constraints, the output Jensen–Shannon divergence of the scalar AWGN channel is uniquely maximized (up to permutation) by antipodal deterministic inputs. The proof avoids heuristic reductions and relies on convexity, concavity, and moment bounds with explicit equality conditions. In addition, the exact high-SNR asymptotic of the normalized JSD is derived with a closed-form constant. No numerical arguments are used.