Coplanar transit limit ($P_2/P_1<[(1+k_2)/b_1{]}^{3/2}$)

For two exactly coplanar circular orbits around the same star, transit impact parameters scale as semi-major axis, hence as $P^{2/3}$. Any overlap by the outer planet requires $b_2<1+k_2$. In the point-planet limit the period ratio is bounded by $b_1^{-3/2}$. Tangency has zero duration, so this is a geometric supremum; practical detection also requires sufficient signal-to-noise ratio and survey coverage.