Stronger lower bounds and dimension reductions

Exact rational pairs of normal matrices are constructed whose ratio $d_\infty(A,B)/\|A-B\|_2$ exceeds the classical value $\sqrt{28/27}$ of Krause's example. The strongest has dimension seven and proves $C_{\mathrm{normal}}>103077/100000$. Its proof consists of an exact orthogonal-reflection construction, a deficient matching block, and an exact positive-definite Gram certificate. The arithmetic is checked by two separate implementations, one using only Python's standard library and one using SymPy. Thus $\sqrt{28/27}$ is not the value of $C_{\mathrm{normal}}$. The report also proves that arbitrary-dimensional rank-one reflection examples are dominated by size-three examples; that Hall witnesses involving at most two spectral values on each side admit size-three normal compressions; and that the truncated singular-value estimate can be refined using the numbers of distinct spectral values. These statements do not determine the remaining finite-dimensional or universal sharp constants.