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Source. Lemma 5, p. 4, of Wouter van Doorn, Completeness of exponentially increasing sequences, arXiv:2602.23394v1 (25 February 2026), the version named on the source card. A preprint.

Read depth. Claims checked: the statement was read clause by clause on the page images of the print; the proof was read for structure only. Nothing here is independently reviewed.

Statement

Setting (p. 1). For positive reals $t$ and $\alpha$, $S_t(\alpha)=(s_1,s_2,\ldots)$ with $s_n=\lfloor t\alpha^n\rfloor$, indexed from $n=1$. For a sequence or multiset $S$ of positive integers, $P(S)$ is the set of integers that are sums of distinct elements of $S$; $S$ is complete when $\mathbb N\setminus P(S)$ is finite and entirely complete when $P(S)=\mathbb N$. Throughout, $\varphi=(1+\sqrt5)/2$.

Lemma 5 (p. 4). Let $t\ge1$ and $1<\alpha<\varphi$, and suppose positive integers $r$ and $X$ exist with $m\in P(\{s_1,\ldots,s_r\})$ for every $m$ with $X\le m<X+s_{r+1}$. Then $S_t(\alpha)$ is complete.

Proof pointer

Adjoining $s_{r+1}$ extends the run of representable integers to $[X,X+2s_{r+1})$, which contains $[X,X+s_{r+2})$ because $s_{r+2}\le2s_{r+1}$ by Lemma 4; induction then represents every $m\ge X$ (p. 4).

Dependencies

Lemma 4.

Bears on

• Problem 349: the finite certificate of completeness below $\varphi$; the paper applies it in Proposition 7, Proposition 8 and Proposition 9, and its computer search (Section 4) certifies a region only where the search finds $r$ and $X$. The lemma decides no pair until such $r$ and $X$ are exhibited.