Digit–Fontaine Generator Hypothesis and the Sakib Index An archimedean perfectoid-inspired complexity invariant from β-digits, with dataset-based illustrations

Perfectoid geometry is built around a Frobenius-like endomorphism and a distinguished “untilt” map whose kernel is principal. On the archimedean side, one can imitate the ring Zp⟨T 1/p∞ ⟩ by allowing real radii r ∈ (0, 1) and studying Z((T ))r together with an evaluation map θr . In this manuscript we propose a digit-normalized candidate for a canonical generator of ker(θr ), built from the greedy β-expansion of 1 with β = 1/r. This leads to a computable complexity invariant, the Sakib Index (SI), defined as an entropy of the first m greedy digits. We give a small suite of data-based plots (using public Gapminder GDP-per-capita values) illustrating how SI behaves under a simple archimedean embedding of real-world scalars into radii.