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Claim. Let $\gamma$ be the Salem number with minimal polynomial $x^{18}-x^{12}-x^{11}-x^{10}-x^9-x^8-x^7-x^6+1$, for which $6/5<\gamma<13/10$. There are $\alpha,\beta>0$ with $\beta/\alpha\ne r\gamma^k$ for all $r\in\mathbb Q$ and $k\in\mathbb Z$, so that $\alpha/\beta$ is irrational and neither sequence is a tail of the other, such that every term of $(\lfloor\alpha\gamma^n\rfloor)_{n\ge0}$ and $(\lfloor\beta\gamma^n\rfloor)_{n\ge0}$ is even; hence the interleaved sequence, repeated occurrences retained, is not complete. This is Corollary 12 (p. 12) of Jesse Geneson, Deletion thresholds and exponential examples for complete sequences, arXiv:2609.25107v1, 20 September 2026, 14 pages; the result page Corollary 12 states it and the source card geneson_2026_deletion_thresholds_exponential_examples_complete_sequences records the preprint.

Covers. The second question of Problem 354, "What if $2$ is replaced by some $\gamma\in(1,2)$?", under the reading "for every $\gamma\in(1,2)$", answered no at one base. The paper says that this answers the variable-base extension of the two-sequence question negatively and does not resolve the base-$2$ question; it says nothing under the reading "for some $\gamma$", which Kitamura's claim page addresses. The construction is existential and gives no explicit $\alpha$ or $\beta$.

Argument. Theorem 9 (p. 11) gives arbitrarily large $t>0$ with $\lfloor t\gamma^n\rfloor$ even for every $n\ge0$ at this base, through Dubickas's theorem on the fractional parts of $t\gamma^n$ (the paper's Lemma 10) with a sign adjustment (Proposition 11); that theorem is also the paper's counterexample to Graham's conjecture for bases below the golden ratio (Problem 349). Corollary 12 chooses two such coefficients whose ratio is not a rational multiple of a power of $\gamma$. The library card records the two-page proof, which has not been independently reviewed.

Claimant and posting. Jesse Geneson, the preprint's sole author; its only version on 2026-09-28 was v1 of 20 September 2026, with no journal reference or DOI. The paper's closing declaration states that the proofs were found with the assistance of Codex with GPT-6 Astra Ultra, that the author directed separate writing and auditing teams, read and edited the proofs and requested revisions, and that the author takes responsibility for the content; the source's own disclosure.

Standing. Claimed: a preprint, not refereed, with no review, no site mention and no catalog record found. The reading of the second question is unfixed by the site and by the original wordings, so this claim settles the question only under one of its two readings and leaves the first question, base $2$, untouched.

Depends on. Nothing in this wiki.