An exact sharp universal norm constant in dimension two

Write $\mathcal S(T)=(|T|+|T^*|)/2$ for the arithmetic symmetric modulus. For every $m\ge2$, all $T_1,\dots,T_m\in M_2(\mathbb C)$ and every unitarily invariant norm, $|||\mathcal S(\sum_jT_j)|||\le\sqrt{6\sqrt3-9}\,|||\sum_j\mathcal S(T_j)|||$. The constant $\sqrt{6\sqrt3-9}=1.17995\ldots$ is optimal when the inequality is required for all such norms, and is attained in the operator norm by two real rank-one matrices. In particular $c_\infty^{\rm sym}(m,2)=\sqrt{6\sqrt3-9}$ for every $m\ge2$. The operator-norm case reduces to the sharp numerical-radius inequality $\|\mathcal S(T)\|_\infty\le\sqrt{6\sqrt3-9}\,w(T)$ on $M_2$. That inequality rests on an explicit formula for the symmetric modulus of a $2\times2$ matrix and a normal form whose numerical range is an explicit ellipse. An exact polynomial identity then bounds the resulting quadratic maximization. Together with the trace-norm triangle inequality, weak majorization extends the bound to every unitarily invariant norm.