This section from Archimedes Palimpsest has a mixture of ancient geometry and Byzantine prayers. The missing page still has traces of geometric diagrams based on Greek mathematician Archimedes of Syracuse's work. Breakthroughs, discoveries, and DIY tips sent six days a week. The Archimedes Palimpsest is a Byzantine prayerbook written in 1229, but the artifact holds more than what immediately meets the eye. The original writing on its pages was erased and replaced--making it a palimpsest--a common practice during the medieval period for expensive writing materials made from animal-skin like parchment.

This paper introduces a distillation framework for an ensemble of entropy-optimal Sparse Probabilistic Approximation (eSPA) models, trained exclusively on satellite-era observational and reanalysis data to predict ENSO phase up to 24 months in advance. While eSPA ensembles yield state-of-the-art forecast skill, they are harder to interpret than individual eSPA models. We show how to compress the ensemble into a compact set of "distilled" models by aggregating the structure of only those ensemble members that make correct predictions. This process yields a single, diagnostically tractable model for each forecast lead time that preserves forecast performance while also enabling diagnostics that are impractical to implement on the full ensemble. An analysis of the regime persistence of the distilled model "superclusters", as well as cross-lead clustering consistency, shows that the discretised system accurately captures the spatiotemporal dynamics of ENSO. By considering the effective dimension of the feature importance vectors, the complexity of the input space required for correct ENSO phase prediction is shown to peak when forecasts must cross the boreal spring predictability barrier. Spatial importance maps derived from the feature importance vectors are introduced to identify where predictive information resides in each field and are shown to include known physical precursors at certain lead times. Case studies of key events are also presented, showing how fields reconstructed from distilled model centroids trace the evolution from extratropical and inter-basin precursors to the mature ENSO state. Overall, the distillation framework enables a rigorous investigation of long-range ENSO predictability that complements real-time data-driven operational forecasts.

The comparing methods in Table 1 of the manuscript consist of two parts.

We present the Manuscripts of Handwritten Arabic (Muharaf) dataset, which is a machine learning dataset consisting of more than 1,600 historic handwritten page images transcribed by experts in archival Arabic.