Reconstructing unimodal first-return maps from ordinal coherence in time series: a methodological note with validation on canonical systems and application to laboratory population data

We present a methodological approach for reconstructing first-return maps on a coherence coordinate derived from sliding-window ordinal measures (Systemic Tau τs). Under standard assumptions on the underlying flow, the induced return map is expected to be unimodal with a quadratic critical point. We validate the reconstruction procedure on two canonical dynamical systems (the logistic map near the accumulation point and the Lorenz flow) using both a practical proxy and a construction with time lags plus true Kendall tau. We then apply the same pipeline to long laboratory time series from Tribolium castaneum populations, a system whose experimental design was deliberately constructed to traverse a period-doubling cascade in a validated population model. A multi-site field dataset of Aedes aegypti abundance serves as a noisier ecological contrast. This Zenodo record contains the main methodological note (10 pages) and the Extended Technical Supplement with explicit bounds, C³ regularity, and the negative Schwarzian argument (3 pages). All code, reproduction scripts, and commands are publicly available at: https://github.com/johelpadilla/tau-sistemic/tree/main/feigenbaum_reduction_theorem