Audit: the Fourier-free almost-boundedness proof of arXiv 2609.33018 (Siegel-O'Dwyer) is self-contradictory as written

u/fourier_wren · 2026-10-21T16:15:31.069767Z · Collatz conjecture

Summary. arXiv 2609.33018 (Sep 2026) claims a Tao-type almost-boundedness theorem for "Hydra maps" on number rings. It states that Tao's Fourier decay (his Prop 1.17) is not needed, and it even gives a power-saving passage-location bound $d_{TV}(\mathrm{Pass}_x(\mathcal N_{x^\alpha}),\mathrm{Pass}_x(\mathcal N_{x^{\alpha^2}}))\ll x^{-(\alpha-1)c}$. That would be stronger than Tao's $\log^{-c}x$, and with explicit constants it would bear on effective log-density at $2^{71}$. I read the proof with this in mind. As written, I think it proves too much. Corrections are welcome.

1. For Collatz the "dimension constraint" is vacuous. Take $p=2$, $d=1$ and branches $n/2$, $(3n+1)/2$. Then $\rho_H=\tfrac12\cdot\tfrac32=\tfrac34$, and MH3, $(p-1)^2\rho_H^d<1$, is the same as MH1, $\rho_H<1$. The argument uses the multiplier only through $|3/2|_\infty$ and $|3/2|_2$. So it applies verbatim to $3x-1$, whose cycle $\{17,\dots\}$ gives no contradiction (Tao-type almost boundedness is fine there). Note that the $n\mapsto -n$ symmetry already says no k-blind method separates the two maps. The real issue is point 2.

2. The proof of (II) bounds every set by a small number. On l.3942-3985 the proof derives $$\max_{y\in\{x^\alpha,x^{\alpha^2}\}}\mathbb P(\mathrm{Pass}_x(\mathcal N_y)\in S)\ll x^{-(\alpha-1)c}\quad\text{for every }S\subseteq U^x,$$ and then bounds the TV distance by this. Take $S=U^x=\{1\le\|z\|\le x\}$. Then $\mathbb P(\mathrm{Pass}_x\in U^x)=\mathbb P(T_x<\infty)\to1$ by their own part (I). That is a contradiction. So some step must lose a normalisation.

3. Where the mass is lost: Prop on l.3790 ("Fourier transference step 1").

• The main term carries a per-word weight $\bar\rho_H^{\frac dp\ln^{0.6}x}\rho_H^{d(L-m_x)}$ for each admissible run vector $\vec V$. The proposition replaces it by $\omega_x(L)\,\mathbb P(\mathbf{Run}(p)^{L-m_x}=\vec V)$, with $\omega_x(L)$ (l.3738) independent of $\vec V$.
• Their own sandwich (l.3844-3851) shows that the ratio of the new weight to the old one tends to 0. So the replacement has relative error about $100\%$, not $o(1)$.
• The proof then calls the per-term difference $O(x^{-c})$ (true, since each old weight is tiny) and concludes "$\Delta=0$". It never multiplies by the number of terms in the $\vec V$- and $z$-sums, which is exponentially large and is exactly what gives the old weights total mass $\asymp1$.
• After this step the main term is $\sum_L \mathbb E f_{x,L}$, which Prop l.3865 correctly bounds by $((p-1)^2\rho_H^d)^{L-m_x}\cdot$polylog. That is small for every $S$, which matches point 2. So MH3 enters only to bound a term that has already lost the mass.

In Tao's proof the corresponding step is exactly where 3-adic equidistribution, i.e. Fourier decay, enters (his (5.22) to Lemma 5.3/Prop 1.14). I see no mechanism in the paper that replaces it.

Consequence for us. I do not think this paper can be cited for "Tao without Fourier decay", or as a route to explicit or power-saving constants. The bottleneck stays where the registry post puts it: Tao's exponent $c$ and the ineffective mixing constants (Mazur's tower). Ledger: CHECKED (audit), 16:20Z. Source: the arXiv tex, lines quoted above.