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Claim. For $t,\alpha>0$ let $S_t(\alpha)=(\lfloor t\alpha^n\rfloor)_{n\ge1}$, each index used at most once, so that $P(S)$ is the set of sums of distinct elements of the sequence as a multiset; $S$ is complete when $P(S)$ contains all large integers and entirely complete when $P(S)=\mathbb N$. Wouter van Doorn, Completeness of exponentially increasing sequences, arXiv:2602.23394 (v1 2026-02-25, 11 pages), recorded on the card doorn_2026_completeness_exponentially_increasing_sequences, proves: (Proposition 1) for $\alpha\notin[1,2]$ the sequence is not complete for any $t>0$, and for $\alpha=1$ it is (entirely) complete if and only if $1\le t<2$; (Proposition 2) for $\alpha=2$ it is (entirely) complete if and only if $t=1/2^k$ with $k\ge1$; (Proposition 3) for $5^{1/3}\le\alpha<2$ and $t\ge1$ it is not complete; (Proposition 4) for $\varphi\le\alpha<5^{1/3}$, $\varphi=(1+\sqrt5)/2$, it is (entirely) complete if and only if $t<\min(3/\alpha^2,5/\alpha^3)$; (Propositions 5 and 6) for $1<\alpha<\varphi$ it is entirely complete if and only if $t<\min(2/\alpha,3/\alpha^2)$, so that for $1<\alpha\le5^{1/3}$ entire completeness holds exactly when $t<\min(2/\alpha,3/\alpha^2,5/\alpha^3)$. Together with Graham's determination of the complete pairs with $t<1$ (Graham's claim page), this decides completeness for every pair with $\alpha\ge\varphi$. Below $\varphi$ the paper proves completeness regions: (Proposition 7) complete for all $t<4/\alpha$ when $1<\alpha\le5/4$; (Proposition 8, computer-assisted, with the search data in the linked repository) complete for all $t\le3$ when $1.3<\alpha\le1.4$, for all $t\le5$ when $1.2<\alpha\le1.3$, for all $t\le10$ when $1.1<\alpha\le1.2$, and for all $t\le50$ when $1<\alpha\le1.1$; and (Proposition 9) complete whenever $1<\alpha\le1+1/(\lceil t\rceil+2\lceil\sqrt t\rceil)$, a region of infinite area. The paper also describes a computer search that, where it succeeds, certifies completeness on a bounded region of pairs with $1<\alpha<\varphi$ through its Lemma 5, and asserts without proof that any region $t<T$, $\alpha<\varphi-\epsilon$ could in principle be covered this way; the author wrote in the thread on 2026-03-02 that the author cannot prove such a covering exists. If Geneson's claimed counterexample holds (Geneson's claim page), the conjecture fails at one base near $1.25$ for large $t$, so for large $T$ no such search can succeed.

Covers. Every pair $(t,\alpha)$ of Problem 349 with $\alpha\ge\varphi$, with $\alpha\le1$, or with $1<\alpha<\varphi$ and $t<\min(2/\alpha,3/\alpha^2)$, and the pairs in the regions of Propositions 7--9, for the corrected Statement, whose sums and index from $n=1$ are the paper's. On the pairs it covers, the result determines whether the sequence is complete, which is what the problem asks, so the claim's value is answered. It leaves open the pairs with $1<\alpha<\varphi$ and $t\ge\min(2/\alpha,3/\alpha^2)$ outside those regions, which the paper conjectures are all complete; Geneson's later counterexample at a Salem base (Geneson's claim page) refutes that conjecture at one base for large $t$ without contradicting any region proved here.

Claimant and postings. The author announced the results in the site's thread on 2025-09-08 (the first discussion link), with the classification above, the computer-assisted ranges and the infinite-area result, linking a write-up; the paper was posted to the arXiv on 2026-02-25 and the thread link updated on 2026-03-02 (the second discussion link), the author writing that little had changed in the intervening months. The page is dated by the thread announcement.

Standing. Claimed: a preprint with no journal reference or DOI on its arXiv record as of 2026-10-07, no outside review and no site acceptance. The site's remarks credit Graham's paper and do not mention this one; the problem is labeled OPEN. Nat Sothanaphan's note of 2026-03-09 (its claim page) reports an independent verification of the computer-assisted ranges, which the author acknowledged in the thread, and sharpens Proposition 9 by a bounded amount.

Depends on. Graham's claim page, whose Theorems 2 and 3 and determination of the complete pairs with $t<1$ the paper relies on to finish the case $\alpha\ge\varphi$.