Contact order governs the onset of entanglement cascades

Entangling rates describe a direct entangling channel, but give no information when that channel is forbidden. For finite-dimensional analytic pure-state dynamics, we show that the onset of genuine $(n+1)$-partite entanglement is determined by the contact order $m_{\star }$ between the physical trajectory and the $S\mathrm{\mid }E$ product manifold. It is the first Taylor order that cannot be reproduced by any product curve, or equivalently the first nonvanishing order of the Fubini–Study distance from the product manifold. We derive a time-ordered recursion that removes curvature-induced kinematic terms and computes $m_{\star }$ from the Taylor coefficients of $H(t)$. This provides a geometric description of an entanglement cascade, in which new subsystems can join multipartite entanglement only after one or more interaction steps. A symmetry-protected three-qubit model realizes $m_{\star }=2$, and a three-mode bosonic example shows that genuine tripartite entanglement can arise even when the two newly formed reduced pairs remain separable.