pCHA: p-adic Categorical Reconstruction and Forecasting of Chaos Under Extreme Data Shortage

pCHA (p-adic Categorical Hyperchaos Approximation) is a framework for long-horizon forecasting of chaotic trajectories under extreme data scarcity. It requires a minimum of 15 observed anchor states for a uniquely determined reconstruction; below this threshold, a canonical envelope of admissible reconstructions is returned. No knowledge of the governing equations is assumed, and no noise filtering or denoising is applied in the preprocessing pipeline. The framework operates entirely in a p-adic ultrametric state space Q_p³, where the strong triangle inequality d(x,z) ≤ max(d(x,y), d(y,z)) replaces the additive structure of the Archimedean continuum. Local reconstructions on p-adic balls form a categorical layer: objects are local states, morphisms are transitions between them, and the global trajectory is assembled from compatible morphisms through piecewise-adelic reconstruction. Standard Archimedean data-driven approaches - neural operators, recurrent networks, sparse regression, Koopman methods - do not satisfy the strict PASS criterion under this protocol. Results on the 117-system dysts benchmark: PASS rates 99.15% / 99.15% / 95.73% at ε ∈ {0.1, 0.01, 0.001} evaluated on the 95th-percentile error, and 99.15% / 98.29% / 79.49% evaluated on the maximum error. On HyperRossler (4D hyperchaos), pCHA is the only method that satisfies PASS at all three thresholds. On seven real-world datasets (laser, space weather, ECG, financial equities), pCHA retains PASS on all sources at n_train = 15. On experimental TMDB fusion-plasma data, pCHA satisfies PASS on 88 of 92 discharges, whereas EDMD, NVAR, and SINDy produce no successful discharges. In the long-horizon regime, the maximum error remains below 2.14 × 10⁻⁴ over 9.9 × 10⁶ consecutive steps. Documented boundaries include discrete maps (ChaosNetBench-CML), scalar delay-differential systems (MGAB), strongly non-stationary processes (Streamflow), and high-dimensional systems without an adequate 3D representation (KF256). Two boundary cases are reported separately: ScrollDelay (discontinuous DDE, dysts) as a structural out-of-scope case, and MSFT at ε = 0.01σ as a failure at the stricter real-world threshold. Reproducibility is provided through a public blind-test protocol. The mathematical specification and evaluation protocol are disclosed; the operational implementation is withheld. Contents:- main.pdf - compiled article (42 pages)- main.tex - LaTeX source- tables/ - LaTeX table files (T1–T9, Appendix A/B)- figures/ - PDF figures (fig1–fig7)GitHub: https://github.com/vallais-dev/AET-Systems