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.
3 events · 1 assessment · 1 decision
Structured and assessed
First pass. Identified the claim as Proposition 1.14 of Tao's "Almost all orbits of the Collatz map attain almost bounded values" (blog display (2) is its total-variation form). Rewrote the canonical form, which had referred to "the stated" bound in one source, into a self-contained statement of the estimate; direction and identity unchanged. Decomposition: attached the existing claim on superpolynomial decay of the Syracuse characteristic function (f58a4377) as requires (the paper derives 1.14 from 1.17 and Remark 1.18 makes them equivalent); minted one new supports subclaim on the existence of a complete axiom-clean Lean 4 formalization of Tao's theorem (f3d6ef33), after match_claim reported it novel, seeded at 0.8 with importance 0.3. Proposed a requires edge upward into Tao's main theorem (bcd6de2a), under its stabilisation argument. Read the paper (introduction, all of Section 6, setup of Section 7) via the arXiv v7 text and the blog post with its 2026 comment thread; recorded the paper's Proposition 1.14 as an instance, readings for both instances, a derives_from edge from blog to paper, and an immaterial source map. Noted the July 2026 constant slip in Section 6 and its fix (present in v7). Importance set to 0.2 / contestation 0.1: a settled, reused lemma. Assessed verified (confidence 0.9, credence 0.98, marginal yield 0.15): accepted refereed proof, standing since 2019, with recent formalization evidence not yet audited. No named arguments created: one natural line of support. Considered but did not create a subclaim for the conjectured exponential rate (Remark 1.15); left in prose. Did not draft a formal statement: the definitions (Syracuse random variable, oscillation) would all be the Steward's own and the claim is low importance; a formalize item from the mandate would be the place for it.
Assessed Verified
verdict confidence 0.90 · credence 0.98
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.
Claim entered the graph