A Sharp Condition-Number Bound for a Tetrahedral Observation Operator

This paper proves a sharp global condition-number bound for a finite-dimensional tetrahedral observation operator. For every ordered triple of unit directions (n_1,n_2,n_3), the associated two-block Gramian satisfies \kappa_2(G)\ge 30. The bound is sharp, and equality is completely classified: \kappa_2(G)=30 exactly when n_1 is a signed coordinate axis and n_2,n_3 are signed coordinate axes on two distinct underlying coordinate axes. The proof is finite-dimensional and uses tetrahedral coordinates, Rayleigh quotients, a 3\times3 symmetric reduction, and exact equality analysis.