The Syracuse random variables mod 3^n are equidistributed at fine 3-adic scales, with total variation error decaying faster than any power of the scale exponent m.
Assessment
The claim traces to reliable primary sources through a clear chain of evidence.
The Syracuse random variable on the integers mod 3^n is the distribution of the 3-adic offset produced by n steps of the Syracuse map applied to a "typical" odd integer, built from n independent geometric variables of mean 2. It is far from uniform at coarse scales (it never takes a value divisible by 3), but Terence Tao proved in his 2019 paper on Collatz orbits, refereed and published in Forum of Mathematics, Pi in 2022, that it is nearly uniform at fine scales: conditioning on the residue mod 3^m, the distribution on the finer cosets differs from uniform by a total-variation error smaller than any fixed power of m, uniformly in n. This estimate, Proposition 1.14 of the paper, is the key ingredient in constructing the approximately invariant family of measures that lets local descent results be iterated, and hence in proving that almost all Collatz orbits attain almost bounded values.
The proof runs through an equivalent statement, that the characteristic function of the Syracuse random variable decays superpolynomially at every frequency not divisible by 3. Given that decay, the mixing estimate follows from Plancherel's theorem together with an elementary injectivity property of the Syracuse offset map; the decay bound itself, the hardest step of the paper, is proved by analysing how a two-dimensional renewal process interacts with a union of well-separated "black" triangles. The result has stood since 2019 and has been reused and generalised by other authors. In July 2026 a reader reported that a constant in the Section 6 derivation had been sharpened too far in later arXiv revisions; the author supplied a one-line correction, which the current version incorporates. In the same month two independent projects announced complete Lean 4 formalizations of the whole paper, following its architecture and reporting a clean axiom footprint; whether those formalizations are complete and faithful has not yet been widely audited, so they corroborate rather than carry the verdict. Heuristically the true rate is expected to be exponential in m rather than merely superpolynomial, but that stronger bound is not claimed and remains open.
Full reasoning: the evidence and decisions behind this verdict
Identity of the claim. The blog display (2) bounds the sum over Y in Z/3^nZ of |P(Syrac(Z/3^nZ)=Y) − 3^{m−n} P(Syrac(Z/3^mZ) = Y mod 3^m)| by O_A(m^{-A}) for every A > 0 and 1 ≤ m ≤ n. In the paper (arxiv.org/pdf/1909.03562, v7 of 16 July 2026) the same statement is Proposition 1.14, written as an oscillation Osc_{m,n} at 3-adic scale 3^{-m}; the two forms coincide because Syrac(Z/3^nZ) mod 3^m has the law of Syrac(Z/3^mZ) (identity (1.23)). Both recorded instances, the paper and Tao's announcement, affirm it; no source denies it.
The proof. Section 6 of the paper derives Proposition 1.14 from Proposition 1.17, the superpolynomial decay of the characteristic function at frequencies not divisible by 3. The derivation, read in full in this pass, restricts to an event on which the partial valuation sums stay within a window of width about (C_A)^2 log n of n log 3/log 2 (displays (6.6) to (6.8)), splits the offset as a sum of a term depending on the first k+1 valuations and an independent rescaled Syracuse variable of order 3^{n−k−1}, applies Plancherel so that the fine-scale oscillation becomes a sum of squared Fourier coefficients away from the coarse frequencies, bounds the independent factor by Proposition 1.17, and bounds the remaining collision-entropy sum using Lemma 6.2 and Corollary 6.3, which say that the Syracuse offset map is injective and remains injective mod 3^n on the restricted event because the relevant natural numbers are below 3^n. Remark 1.18 gives the converse inequality, so the two propositions are equivalent and the requires relation is exact. The decay bound (Section 7) was read at the level of its setup (Proposition 7.1, the reduction to white points of a renewal process, Proposition 7.3, and the triangle structure of the black set, Lemma 7.4), not line by line.
Standing. The paper was refereed for Forum of Mathematics, Pi (2022) and has been built on by others (Gonçalves, Greenfeld and Madrid, arXiv:2111.06170, adapt the fine-scale mixing step to generalised Collatz maps; Tao's own 2020 paper on equidistribution of Syracuse random variables uses it). Under the mathematics standard, a refereed, independently expounded proof standing without unresolved objection is verified as an accepted proof. One objection did arise: a comment of 5 July 2026 on Tao's blog (terrytao.wordpress.com/2019/09/10/almost-all-collatz-orbits-attain-almost-bounded-values/) reported that revisions v4 and v5 had tightened a buffer in the Section 6 window from (C_A)^3 log n to (C_A)^2 log n in a way that no longer left margin over the deviation cost; Tao replied on 6 July that replacing the constant 1/2 from display (6.7) onward by 0.99 restores the numerical bounds. The v7 text read here already carries 0.99 in (6.7) and (6.8) and in the estimate inside Corollary 6.3 (where it is checked that 0.4175 < 0.99 log 2 ≈ 0.686), so the objection is resolved in the published text. This episode slightly lowers verdict confidence, since a numerical slip survived refereeing and several revisions, but it concerned constants in a bookkeeping step, not the structure of the argument.
Formal evidence. Two Lean 4 formalizations of the entire paper were announced in July 2026: the AI-authored repository gotrevor/tao-collatz (github.com/gotrevor/tao-collatz), whose README reports both headline theorems sorry-free with axioms exactly propext, Classical.choice and Quot.sound, kernel-checked on 15 July 2026, with a blueprint that has a chapter for the Section 6 fine-scale mixing step and one assembling Proposition 1.17; and a roughly 124,000-line formalization at proofatlas.ai reported by Lech Mazur. Neither has been checked by this graph against a published statement of this claim, and neither has yet had wide independent audit, so they enter as supporting evidence whose own standing is still to be assessed, not as the machine-checked grade of verification.
What would change the verdict: a demonstrated gap in Section 7 that resists repair, a failure of the announced formalizations at the fine-scale mixing node, or a counterexample to the decay estimate (none is plausible given the numerics Tao reports up to moderate n and the entropy heuristic). Credence 0.98 reflects a long analytic proof, refereed, reused, recently scrutinised closely enough to find and fix a constant, and now apparently formalized.
Decomposition
The claims this one rests on directly. ↗︎ opens a subclaim; the map shows how they fit together.
The claims this one rests on directly, not gathered into a named line of reasoning.
- requiresa load-bearing premise: the parent is false without itsteward instructions →The Fourier coefficients of the Syracuse random variable mod 3^n at frequencies not divisible by 3 decay faster than any power of n. ↗︎
- supportsthis provides evidence for the parentsteward instructions →Tao's theorem that almost all Collatz orbits attain almost bounded values has a complete, axiom-clean Lean 4 formalization. ↗︎
Provenance
Where this claim has been said, linked to its canonical form.
we establish the estimate ... \sum_{Y \in {\bf Z}/3^n{\bf Z}} | \mathbb{P}( \mathbf{Syrac}({\bf Z}/3^n{\bf Z})=Y) - 3^{m-n} \mathbb{P}( \mathbf{Syrac}({\bf Z}/3^m{\bf Z})=Y \hbox{ mod } 3^m)| ... \ll_A m^{-A}
stabilisation of the Syracuse random variables in total variation
The source's own evidence bears what it asserts. The blog post is the author's summary of his own paper. It states the estimate and outlines the route through Fourier decay and a renewal process, but the proof lives in the paper. The stored text of the post drops its typeset formulas, so the recorded display cannot be checked against the copy word for word; it was checked against the paper, where it appears as Proposition 1.14 in oscillation form. The comment thread of the post, in July 2026, also records a reader's report of a numerical slip in the Section 6 derivation of this estimate and the author's one-line fix, since incorporated in the arXiv text. The quoted passage was not found in the stored copy of this source.
Proposition 1.14 (Fine scale mixing of n-Syracuse offsets). For all 1 ≤ m ≤ n one has Oscm,n (P(Syrac(Z/3nZ) = Y mod 3n))Y ∈Z/3nZ ≪A m−A (1.26) for any fixed A > 0
The paper states the fine-scale mixing estimate as Proposition 1.14, calls it the key ingredient for the stabilisation of first passage (Proposition 1.11), derives it in Section 6 from the Fourier decay estimate (Proposition 1.17), and notes in Remark 1.18 that the two are equivalent. Version 7 (16 July 2026) carries the corrected constant 0.99 in display (6.7).
The source's own evidence bears what it asserts. This is the primary source and contains the full proof. The introduction and the whole of Section 6 (the derivation of the estimate from the Fourier decay bound through Plancherel and the 3-adic separation of Syracuse offsets) were read in this pass; the Fourier decay argument of Section 7 was read at the level of its setup and structure. The version read is the seventh arXiv revision of July 2026, which already carries the corrected constant in display (6.7) that a reader had flagged on the author's blog. Worth reading closely: Section 6 is the entire proof of this estimate given the Fourier decay bound, and is short enough to check line by line; Section 7 is where the real difficulty lies and has not been read in full here.
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 states the estimate as the paper's result. The total-variation form on the blog matches Proposition 1.14 of the paper via identity (1.23), which the paper itself notes; the range of m and n and the O_A(m^-A) rate are the same in both. No strengthening or qualification dropped.
- 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.