A fourteen-term identity and a sharp orthogonal obstruction bound

After a common unitary change of basis, write the normal pair as $D_a$ and $UD_bU^*$ with $U$ unitary. A direct fourteen-term identity is established for every complex unitary of order four and arbitrary complex spectra. It writes $\det(D_a+UD_bU^*)$ as a real combination of the products $\prod_i(a_i+b_{\sigma(i)})$ with weights independent of the spectra. On each of 48 explicitly defined obstructed regions, it has exactly one negative coefficient and attains the least possible negative mass among all spectrum-independent real representations. No two of these regions can occur simultaneously. The identity also yields a sufficient condition for common convex weights, and hence for the determinantal inclusion for all spectra. This includes an operator-norm ball of radius $1/100$ around a specified Hadamard matrix; block sums extend these classes to every dimension. A separate analytic maximization gives the sharp bound on the 48 obstructions over all real orthogonal matrices of order four, extending an earlier bound for a quaternion family of rotations.