Every number checked reaches 1. No one has proved they all do.

Nothing but the rule. Each number is placed one step from the number it goes to next, turned a little one way if it is even and the other way if it is odd.

Because every path that reaches a number continues exactly as that number does, paths merge and share their line from there to 1. The trunk is the handful of numbers almost every path passes through on the way down (16, 8, 4, 2, 1); the branches are where they part. The turn angles and the orientation are choices of the drawing; which numbers lie on which branch is not.

Every number below 2⁶⁸, about 2.95 × 10²⁰, has been checked by computer and reaches 1 (D. Barina, The Journal of Supercomputing, 2021).

In 2019 Terence Tao proved that almost all starting numbers, in the sense of logarithmic density, eventually fall below any function that grows to infinity, however slowly: almost all paths reach almost bounded values. That is the strongest result so far, and it is not the conjecture.

John Conway showed in 1972 that a natural generalisation of the problem is undecidable: no algorithm can settle every instance of it. That says nothing about whether this instance is decidable.

Anything about the numbers it does not reach. A counterexample would be a number whose path climbs forever or enters a loop other than 4 → 2 → 1; no such number is known, and none below 2⁶⁸ exists. A check, however large, is not a proof.

That distinction is why this is here: it is the line Ostrium's own records keep between what has been measured and what is claimed.