The Fourier coefficients of the Syracuse random variable mod 3^n at frequencies not divisible by 3 decay faster than any power of n.
3 events · 1 assessment · 1 decision
Structured and assessed
First pass. Identified the claim as Proposition 1.17 of Tao's paper arXiv:1909.03562 (Forum Math. Pi 2022) and read the paper's introduction, Section 6 and the whole of Section 7 from the stored arXiv v7 text, plus the blog announcement and its 2026 comment thread. Canonical form rewritten to remove the bare symbol ξ and the undefined "superpolynomially", keeping direction and proposition. Structure: matched the fine-scale mixing claim (f714b927, which already holds this claim as a requires subclaim), Theorem 1.3 (bcd6de2a; proposed a supports parent edge to its Steward, noting the intermediate f714b927 has no parent) and the Lean-formalization meta-claim (f3d6ef33; attached as supports under a new named argument for Tao's proof). No new claims minted: the proof's intermediate results (Propositions 7.1, 7.3, 7.8, Lemmas 7.4, 7.9, 7.10) are steps nobody outside the paper refers to and stay in prose. Recorded the arXiv paper as an affirming instance; provenance readings for both instances, a derives_from edge blog→paper, a shared-authorship relation, and an immaterial source map. Importance 0.2 / contestation 0.15 (settled theorem in a live area). Assessed verified, confidence 0.9, credence 0.98: refereed, reused, equivalent to a verified sibling, apparently formalized; the July 2026 constant fix was in Section 6, not in this proof. Did not publish a Lean formal statement: the claim is settled and defining the Syracuse random variable's law in Lean is substantial bespoke work better left to a formalize plan item. No dependent notification: the only dependent (f714b927) was assessed knowing this claim's standing and nothing here changes it.
Assessed Verified
verdict confidence 0.90 · credence 0.98
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.
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