The all-α extension of the super-4 theorem

Less than 1 year old · traced to 2026

Let $calpha>2$ be real, and let $c_{n,k}$ and the associated quantities be those defined in the paper. All-α conjecture. The statement of Theorem cref{t:super4} holds true for every real number $calpha>2$ . The conjecture extends the result established in the paper's preceding special case; the supplied text does not state a resolution.

References

Primary source

Heinz H. Bauschke and Yuan Gao, “Boţ-Nguyen Acceleration, Weighted Mean Ergodic Iteration, and the Beta-Binomial Distribution”, arXiv:2604.17084 (2026).

asymptotic-analysis ergodic-iteration

Posted 1 month ago

Progress summary

Refreshed 1 month ago

Claimed solved

A posted argument claims a complete proof for every real $\alpha>2$ , but no independent verification has been found and the paper itself records only the $\alpha=4$ case.

The conjecture asks whether the paper’s super-4 theorem extends from its established special case $\alpha=4$ to every real $\alpha>2$ . Bauschke and Gao state this extension as Conjecture 6.1 in their paper, first posted in April 2026.

Known results

Bauschke and Gao (2026): strong convergence is proved for the special case $\alpha=4$ ; the all- $\alpha$ extension is left as a conjecture.

Posted attempt

A reader-written argument claims a complete proof for every real $\alpha>2$ : applying the paper’s shift-operator result is asserted to establish the missing Lorentz condition, after which the convergence theorem would apply. The attempt has not been independently verified.

Current status (as of August 2026): The $\alpha=4$ case is settled, while a complete all- $\alpha$ proof has only been claimed publicly and remains unverified.

Sources

arxiv.org

ProofThis solution needs a summarySee full solutionHide full solution

Complete proof for every real $\alpha>2$ , including the Lorentz condition and strong convergence. Let $c_{n,k}$ be the source's Boţ–Nguyen coefficient array, extended by $c_{n,k}=0$ outside $0\le k\le n$ . Source equation (43e) states

x_{n+1}=\sum_{k=0}^{n}c_{n,k}T^kx_0.

The source's Theorem 4.4 already proves, for every $\alpha>2$ ,

c_{n,k}\ge0,\qquad \sum_{k=0}^{n}c_{n,k}=1,\qquad c_{n,k}\longrightarrow0\quad\text{for each fixed }k.

Thus the sole missing hypothesis of its Theorem 2.10 is the Lorentz condition

L_n:=\sum_{k=0}^{\infty}|c_{n,k}-c_{n,k+1}|\longrightarrow0.

The key is to apply the source's already-established Fact 1.1 to one universal test operator. Work in the real Hilbert space $\ell^2(\mathbb N_0)$ , let

Se_k=e_{k+1},

and start the iteration at $x_0=e_0$ . The unilateral shift $S$ is a linear isometry with $\operatorname{Fix}S=\{0\}$ , so Fact 1.1 applies for every real $\alpha>2$ and yields

\|x_j-Sx_j\|_2=o(j^{-1}).

Equation (43e) gives

x_{n+1}=\sum_{k=0}^{n}c_{n,k}e_k,

and consequently

x_{n+1}-Sx_{n+1} =c_{n,0}e_0 +\sum_{k=1}^{n}(c_{n,k}-c_{n,k-1})e_k -c_{n,n}e_{n+1}.

This vector has at most $n+2$ nonzero coordinates, and positivity gives the exact identity

\|x_{n+1}-Sx_{n+1}\|_1=c_{n,0}+L_n.

Therefore Cauchy–Schwarz and Fact 1.1 imply

0\le L_n \le\sqrt{n+2}\,\|x_{n+1}-Sx_{n+1}\|_2 =o(n^{-1/2}).

In particular the Lorentz condition holds for every $\alpha>2$ , with the stronger quantitative bound $L_n=o(n^{-1/2})$ .

All four hypotheses of the source's Theorem 2.10 are now satisfied. Hence, for every real Hilbert space, every linear nonexpansive operator $T$ , and every starting point $x_0$ ,

x_{n+1}=\sum_{k=0}^{n}c_{n,k}T^kx_0 \longrightarrow P_{\operatorname{Fix}T}x_0 \qquad\text{in norm}.

Thus every assertion of Theorem 5.5 extends from $\alpha=4$ to the entire conjectured range $\alpha>2$ .

Source: Bauschke and Gao, Boţ–Nguyen Acceleration, Weighted Mean Ergodic Iteration, and the Beta-Binomial Distribution, revised August 13, 2026, Fact 1.1, Theorems 2.10 and 4.4, and Conjecture 6.1.