Tao's proof via stabilisation of Syracuse first passage
The argument
Earlier results such as Korec's theorem that almost all N descend below N^θ for θ above log 3/log 4 give only local descent, which cannot be iterated directly because the Collatz dynamics do not carry the uniform distribution on one scale to the next. Tao's proof passes to the Syracuse map, builds an approximately invariant family of measures from the Syracuse random variables on Z/3^n Z, and reduces the theorem to a stabilisation property of the associated first-passage distribution, which in turn reduces to high-frequency decay of the characteristic function of a skew random walk on Z/3^n Z, proved through a two-dimensional renewal process escaping a union of well-separated triangles. Because this invariant-measure iteration turns local descent into descent below any function tending to infinity, the theorem follows for almost all N in logarithmic density.
Does it work?
Not yet evaluated.
Premises
- UnassessedNo current assessment. Attention goes where its expected value is highest and someone funds it; nothing has funded an assessment of this claim yet, and anyone can.constitution →For almost all N, Col_min(N) is below N^θ for any fixed θ greater than log3/log4 ≈ 0.7924.
Where it is made
Passages where sources make, criticize or mention this argument. They record where it appears, not how strong it is, and none has precedence over another.
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Earlier ratings
- Sep 16, 2026
The inference goes through: the paper's chain of reductions, from the theorem to stabilisation of the Syracuse first-passage distribution, to fine-scale 3-adic mixing, to Fourier decay of a skew random walk established via a renewal process, was refereed for Forum of Mathematics, Pi and has stood unchallenged since 2019, and later authors have adapted the same route to Collatz-like maps. The argument does not depend on Korec's descent theorem as a premise; that result marks the local-in-time state of the art the proof globalises, and the proof re-derives the descent it uses, so the argument's weight rests entirely on the paper's own analysis.