Lie-derivative organizers of ordinal-pattern regimes in chaotic systems

Classical ordinal-pattern methods quantify complexity from a scalar time series without requiring a model, but they are usually reported as global statistics and say little about where along a chaotic trajectory ordinal regimes change. For a smooth flow x˙=F(x) observed through a scalar ϕ(x), we define the first two material derivatives ϕ˙=LFϕ and ϕ¨=LF2ϕ and the associated organizer sets M1(ϕ)={x:ϕ˙=0} and M0(ϕ)={x:ϕ¨=0}. An exact integral criterion shows that if LFϕ keeps a strict sign on an ordinal window, then the corresponding ordinal pattern is monotone; thus, M1 marks where that sufficient sign-persistence mechanism can fail, while M1∩M0 identifies tangencies of M1 at its regular points. A small-delay expansion further shows that ordinal ranks in delay embeddings are controlled at leading orders by (ϕ˙,ϕ¨). We quantify organizer effects with an ordinal sensitivity index (OSI), defined as the Jensen-Shannon divergence between ordinal-pattern distributions in near and far sets. For Lorenz-63 with ϕ = x, we derive a closed-form expression for M0(x). Across Lorenz-63 and the Aizawa system, OSI remains large over broad embedding and delay ranges; near-M1 conditioning increases switching and transition entropy in 98.8%-100% of tested configurations, whereas M0 produces strong but delay-dependent separation and often increased monotone-pattern prevalence. As a secondary delay-space refinement, a boundary-clearance score improves switching prediction beyond |ϕ˙| in all tested configurations.