The Fourier coefficients of the Syracuse random variable mod 3^n at frequencies not divisible by 3 decay faster than any power of n.
Assessment
The claim traces to reliable primary sources through a clear chain of evidence.
This is Proposition 1.17 of Terence Tao's paper "Almost all orbits of the Collatz map attain almost bounded values" (arXiv 2019, Forum of Mathematics, Pi, 2022). The Syracuse random variable mod 3^n is the law, on the residues modulo 3^n, of the offset produced by n steps of the Syracuse map when the 2-adic valuations along the way are independent geometric variables of mean 2; the proposition says that its Fourier coefficient at any frequency not divisible by 3 is at most C_A n^{-A} for every A, with C_A independent of n and of the frequency. The restriction on the frequency is essential, since the variable never takes values divisible by 3 and its coefficients at multiples of 3^{n-1} do not decay.
The estimate is a theorem with an accepted proof. Tao calls it the most difficult step of his argument and proves it in Section 7 of the paper by pairing adjacent valuations so that, after conditioning, the characteristic function becomes an average of products of cosines along a two-dimensional renewal process, and then showing that the process meets enough points where the cosine is small. The paper was refereed, the method has been reused and generalised by other authors (Gonçalves, Greenfeld and Madrid extended the renewal-process argument to Collatz-like maps), and by Remark 1.18 the proposition is equivalent to the fine-scale mixing of the Syracuse random variables, from which the main theorem follows. In 2026 two Lean 4 formalizations of the entire paper were announced, both following this proof; whether they are complete and axiom-clean has not yet been independently confirmed, so they corroborate the proof without yet raising it to the machine-checked grade. A numerical slip that a reader found in the paper in July 2026 lay in the derivation of fine-scale mixing from this estimate, not in the proof of the estimate itself, and was corrected in the revised text.
Tao notes that the true decay is probably exponential in n, as an entropy heuristic predicts; the proposition claims only the superpolynomial rate needed for the application.
Full reasoning: the evidence and decisions behind this verdict
Statement. The paper (arxiv.org/pdf/1909.03562, v7 of 16 July 2026) states Proposition 1.17 as: for n ≥ 1 and ξ in Z/3^nZ not divisible by 3, E exp(-2πi ξ Syrac(Z/3^nZ)/3^n) ≪_A n^{-A} for any fixed A > 0, with the implied constant uniform in n and ξ. Tao's blog announcement (terrytao.wordpress.com/2019/09/10/almost-all-collatz-orbits-attain-almost-bounded-values/) restates it as display (4). Both recorded instances affirm the claim and are one voice (same author); no source denies it or poses it as open.
The proof, read in full in this pass. Section 7 reduces the proposition to Proposition 7.1 (the same bound for the sum 2^{-a_1} + 3·2^{-a_[1,2]} + ... with iid Geom(2) valuations, order reversed). Pairing adjacent valuations into b_j = a_{2j-1} + a_{2j} (iid Pascal variables) and conditioning on the b_j makes the remaining variables independent, giving |S_χ(n)| ≤ E ∏_j |f(3^{2j-2} 2^{-b_[1,j]}, b_j)| (display 7.5). Lemma 7.2 computes that when b_j = 3 the factor equals cos(π θ(j, l)), hence is at most exp(-ε^3) at "white" points where |θ(j,l)| > ε. This reduces the task to Proposition 7.3: the renewal process of points (j, b_[1,j]) with b_j = 3, whose holding time Hold has mean (4, 16) and exponential tail (Lemma 7.6), visits white points often enough that E exp(-ε^3 #white) ≪_A n^{-A}. Lemma 7.4 proves by elementary number theory (the relations θ(j+1,l) = 9θ(j,l) and θ(j,l-1) = 2θ(j,l) mod 1, and the fact that ξ not divisible by 3 forces black points into the strip j ≤ n/2 - (1/10) log(1/ε)) that the black set is a disjoint union of triangles separated by at least (1/10) log(1/ε). The remainder is a monotonicity induction (Proposition 7.8) on the maximal weighted quantity Q_m, split into three cases: a white starting point (immediate gain), a start near the top of a triangle (Lemma 7.7 puts the exit point within O(1) of the triangle with probability ≫ 1, hence white), and a start deep inside a large triangle, where Lemma 7.9 converts many triangle encounters into many white points and Lemma 7.10 shows large triangles are rarely met shortly after a long crossing, so that R = ⌊A^2/ε^4⌋ triangles and hence ≫ A^2/ε^3 white points are found with probability at least 1 - 10^{-A-2}. The steps checked here are internally consistent; the argument is long but each case was followed.
Standing. Refereed (Forum of Mathematics, Pi 10 (2022) e12; received 2019, revised April 2022). Reused: Gonçalves, Greenfeld and Madrid (arXiv:2111.06170) adapt the renewal-process Fourier decay argument to generalised Collatz maps, which is independent exposition of the method. Remark 1.18 gives the converse inequality |E e^{-2πiξ Syrac/3^n}| ≤ Osc_{n-1,n}, so the proposition is equivalent to fine-scale mixing (Proposition 1.14), which the graph holds as verified.
Scrutiny in 2026. A comment of 5 July 2026 on the blog reported that arXiv revisions v4 and v5 had tightened a buffer from (C_A)^3 log n to (C_A)^2 log n without enough margin; Tao replied on 6 July that replacing 1/2 by 0.99 from display (6.7) onward restores the numerical bounds. Display (6.7) sits in Section 6, the derivation of Proposition 1.14 from Proposition 1.17 (the window on a_[1,k+1] and the injectivity argument of Corollary 6.3), so the slip did not touch the proof of this claim; v7 read here carries 0.99 in (6.7), (6.8) and Corollary 6.3. That a constant survived refereeing lowers verdict confidence slightly, from what would otherwise be near-certain, to 0.9.
Formal evidence. Two Lean 4 formalizations of the whole paper were announced in July 2026: gotrevor/tao-collatz (github.com/gotrevor/tao-collatz), an AI-authored repository whose README reports both headline theorems sorry-free with axioms exactly propext, Classical.choice and Quot.sound, and whose blueprint (gotrevor.github.io/tao-collatz/blueprint/) has a chapter "The crux: character decay via renewal versus triangles" ending in "Assembly: Prop 1.17"; and a formalization of roughly 124,000 lines at proofatlas.ai reported by Lech Mazur, who told the blog thread on 5 September 2026 that later work "combines the fine-scale mixing theorem (Proposition 1.14) and its Fourier-renewal machinery". Tao remarked on 23 August 2026 that "My original argument now has some Lean formalizations". Neither has been checked by this graph against a published formal statement of this claim, and neither has had wide independent audit, so they enter as supporting evidence whose own standing is unassessed; the verdict of verified rests on the accepted refereed proof, not on them.
What would change the verdict: a demonstrated gap in Case 3 of Proposition 7.8 (Lemmas 7.9 and 7.10) that resists repair, or a failure of the announced formalizations at the Section 7 nodes. A counterexample is implausible: Tao reports numerics consistent with mixing up to moderate n, and the entropy heuristic (Remark 1.15) predicts exponential rather than merely superpolynomial decay. Credence 0.98.
Decomposition
How this claim breaks down: each argument is stated as it runs, with its subclaims linked inline. ↗︎ opens a subclaim; the map shows how they fit together.
Pairing adjacent geometric valuations and conditioning on the pair sums makes the summands of the Syracuse random variable independent, so its characteristic function becomes an average, over a two-dimensional renewal process, of products of factors of modulus at most one, each at most exp(-ε^3) at a "white" lattice point where the relevant phase is not near an integer. Because the "black" points form well-separated triangles and the renewal process, by a monotone induction with the large triangles handled by a separation estimate, meets enough white points, the average is O_A(n^{-A}) uniformly in n and in the frequency; the announced machine-checked formalization of the whole paper follows this same route and corroborates that the inference goes through.
The inference goes through: the conditional factorisation, the cosine bound at white points, the triangle structure of the black set, and the three-case monotonicity induction were each followed in the published text and fit together without gaps, and the argument has been refereed and adapted by other authors to generalised Collatz maps. It rests on no named external result beyond standard local limit and Chernoff bounds for random walks, so it does not stand or fall with any other claim in the graph. The announced machine-checked formalization of the paper would, if confirmed, raise this from an accepted to a kernel-checked proof; its current unassessed standing neither adds to nor subtracts from the argument's validity.
Provenance
Where this claim has been said, linked to its canonical form.
\mathbb{E} e^{-2\pi i \xi \mathbf{Syrac}({\bf Z}/3^n{\bf Z}) / 3^n} \ll_A n^{-A}
non-trivial decay of the characteristic function for ξ not divisible by 3
The source's own evidence bears what it asserts. The blog post is the author's summary of his own paper and states the decay bound with a sketch of its proof; the proof itself lives in the paper. The stored text of the post drops its typeset formulas, so the recorded display cannot be matched against the copy word for word; it was checked against Proposition 1.17 of the paper, which it restates faithfully. The comment thread, in July and September 2026, also records two announced Lean formalizations of the paper and the author's remark that his original argument now has Lean formalizations. The quoted passage was not found in the stored copy of this source.
Proposition 1.17 (Decay of characteristic function). Let n ≥ 1, and let ξ ∈ Z/3nZ be not divisible by 3. Then Ee−2πiξSyrac(Z/3nZ)/3n ≪A n−A (1.28) for any fixed A > 0.
The paper states the decay estimate as Proposition 1.17, stresses that the implied constant is uniform in n and ξ, calls it "the most difficult step in the argument", proves it in Section 7 via a two-dimensional renewal process and the triangle structure of the black set, and notes in Remark 1.18 that it is equivalent to the fine-scale mixing Proposition 1.14.
The source's own evidence bears what it asserts. This is the primary source and contains the complete proof. The introduction, Section 6 and the whole of Section 7 (the reduction to Proposition 7.1, the white-point bound of Lemma 7.2, the triangle structure of Lemma 7.4, the holding-time renewal process of Lemmas 7.6 and 7.7, and the three-case proof of the monotonicity Proposition 7.8) were read in this pass. The version read is the seventh arXiv revision of July 2026; the numerical slip a reader reported in July 2026 concerned Section 6, the derivation of fine-scale mixing from this estimate, not the proof of this estimate itself. Worth reading closely: Section 7 is the entire proof and the paper's hardest part; a line-by-line check of Case 3 of Proposition 7.8 (Lemmas 7.9 and 7.10) is where any remaining gap would be.
How these sources relate
- https://terrytao.wordpress.com/2019/09/10/almost-all-collatz-orbits-attain-almost-bounded-values/ draws its statement from https://arxiv.org/pdf/1909.03562, faithfully. The post announces the paper and restates its Proposition 1.17 with the same hypothesis (frequency not divisible by 3) and the same conclusion (decay faster than any power of n); the proof is deferred to the paper. Same author, so this is one voice, not independent corroboration.
- https://terrytao.wordpress.com/2019/09/10/almost-all-collatz-orbits-attain-almost-bounded-values/ and https://arxiv.org/pdf/1909.03562 share an author. Both are by Terence Tao: the arXiv paper carries his name and UCLA address, and the blog post is on his own blog "What's new", bylined "by Terence Tao", announcing "my paper".
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Created by extractor · Sep 13, 2026. Every judgment on this page is accompanied by a reasoning trace.