COLS & Ramanujan's 1/π: synchrony at integers, certified tails, and a 2‑adic reading

Description This expository note revisits Ramanujan’s ultra-fast series for 1/π through a COLS framework (a bank of dyadic oscillators) that makes it possible to read exact partial sums at integer times and provides a simple geometric tail bound allowing the certification of the number of decimal digits.On the arithmetic side, the note gives a clear 2-adic reading of the coefficient (4k)!/(k!)⁴, with an explicit treatment of the valuations on the rational part of the terms.A multibase/CRT variant faithful to 396 = 2²·3²·11 is presented in a “safe” formulation: alignment at integers without phases, and alignment at a single instant when quantized phases are introduced.A short appendix provides an elementary proof that Rₖ₊₁/Rₖ > 1 for k ≥ 1, which confirms the quality of the bounds and the “digits-by-design” character of the protocol.The objective is expository and certifying: to stage a classical result (Ramanujan) within a minimal and intuitive COLS framework that facilitates reproduction and verification. Key points Integer synchrony: yₖ(m)=Σₖ₌₀ᴷ Aₖ for m ∈ ℤ. Geometric tail bound: Σₘ≥ᴷ₊₁ Aₘ ≤ Aₖ₊₁ / (1 – c), with uniform constant c. 2-adic profile: v₂((4k)!/(k!)⁴)=3 s₂(k), and v₂(rational part of Aₖ)=3 s₂(k) – 8k. Multibase/CRT: safe formulation (integers) + rational phases for a single instant t*. Appendix: short proof that Rₖ₊₁/Rₖ > 1 for k ≥ 1. KeywordsRamanujan; 1/π; COLS; dyadic synchrony; tail bound; certified digits; p-adic valuation; CRT; expository mathematics. Related references– COLS — Listening to Integers (V6), Zenodo, doi:10.5281/zenodo.17385769.– COLS_paper_V8_FINAL & COLS_additional_proofs_FINAL, Zenodo, doi:10.5281/zenodo.17383651. Note – This description summarizes the results and organization of Version 7 (Proposition “Synchrony at Integers,” Tail Theorem, p-adic profiles, and monotonicity appendix).