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The companion document to the Poster is under preparation. Meanwhile, check out web resources on Bernoulli's triangle, sum of the first k binomial coefficients, Chernoff's bound, tails of a normal distribution, and related topics. Although one can find infinite continued fractions to approximate $s_m(r)$, I have yet to find in the literature anything that implies the Main Result of the poster.

Here is the latest version of the companion document (organizer privilege):