Borel–Cantelli Threshold and 2^-k Halting-Probability Digit Deviations — E8 Intelligence Research

FINDING: Borel–Cantelli gives a sharp threshold (summability of probabilities) separating almost-sure finitely-many from infinitely-many occurrences; the 2^-k decay of halting-probability digit deviations sits exactly on the convergent side of this threshold. | MATH: Let (A_k) be events, S = Σ_k P(A_k). BC-1: S < ∞ ⟹ P(A_k i.o.) = 0. BC-2 (independence): S = ∞ ⟹ P(A_k i.o.) = 1. For Ω = halting probability, P(Ω ∈ [0,1]) with binary expansion digits d_k, the event A_k = {first k digits of Ω match a fixed prefix} has P(A_k) = 2^-k, so Σ_k 2^-k = 1 < ∞. Hence almost surely only finitely many prefixes are "hit" — the digit sequence is normal in the BC sense. Moment refinement (arXiv 2204.04369): E[N_n^m] controlled by weighted summability Σ k^{m-1} P(A_k), giving mean-deviation and LIL rates. | CONNECTION: The ratio 2^-k is the binary analogue of the base-60 / dyadic subdivision underlying crystallographic lattice refinement; the geometric series Σ 2^-k = 1 is the unit normalization that m Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com