Collatz: halve it, or triple it and add one

The conjecture

For every integer n ≥ 1, some iterate of n under n ↦ n/2 (n even), 3n + 1 (n odd) equals 1. Equivalently, every start has a finite stopping time: its orbit eventually drops below where it began.

Known in the world

Computation. Bařina's distributed verification passed 2⁷¹ in January 2025, on thousands of workers across European supercomputers. Almost all. Terras (1976) showed that almost every start has a finite stopping time. Tao (2019) showed that almost every orbit falls below any function tending to infinity. Every start is the open part, and it is what the prize pays for.

What our record adds (measured; the counting form)

Count the w-bit numbers whose orbit stays at or above its start for T steps. The first w − 1 steps are free: each pattern of parities belongs to exactly one residue of the start (Terras's bijection), so they pay exactly. Past the free bits, the count follows Terras's coin model to 0.5% at every width to 30 bits, and keeps the coin's rate at every width measured, to 43 bits, with a bounded excess over the coin (at most 7.4 bits, with no upward trend). That is a precise target. If fewer than 2^{w − αT + c} numbers of w bits stay up for T steps, for fixed α and c, every stopping time is finite and the conjecture follows. The free part is a theorem. The rest is the open part, a statement about each orbit's binary digits, not about almost all of them.

The twin of Rule 30

Multiplying by three is the same linear step as Rule 30, plus carries (the carry of Rule 30). In both problems a free part pays exactly and the remaining part pays at the coin's rate, as far as anyone can measure. Collatz has an avenue that Rule 30 lacks: after the free bits the state is an explicit integer, a remainder modulo a power of three. Some of our lemmas are known results in new clothes, and we credit them: the window principle's complexity bound is Dubickas (2009).