Claim. Let $\gamma=\sqrt\varphi\approx1.27202$, $\varphi$ the golden ratio. For every $u>0$ there is $L\in\mathbb Z$ such that every integer $z\ge L$ is $\sum_{i\in t}\lfloor\gamma^iu\rfloor$ for some finite $t\subset\mathbb N$. Hence the catalog statement erdos_354.parts.ii of formal-conjectures,

∃ γ ∈ Set.Ioo (1 : ℝ) 2, ∀ᵉ (α > 0) (β > 0),
  Irrational (α / β) → IsAddCompleteNatSeq' (FloorMultiples.interleave α β γ)

holds with answer(True): the $\alpha$-sequence alone represents every large integer, its representation maps to the even positions of the interleaving, and the hypothesis that $\alpha/\beta$ is irrational is discarded. The result page erdos_354_part_ii_solved and the source card kitamura_2026_lean_proof_erdos_problem_354_ii record the theorem and the repository.

Covers. The second question of Problem 354, "What if $2$ is replaced by some $\gamma\in(1,2)$?", under the reading "for some $\gamma$", which is the catalog's existential form; it says nothing under the reading "for every $\gamma$", which Geneson's claim page answers no, and nothing about base $2$. The author's README says the result does not solve the first question and is stronger than the catalog's part (ii) because the second sequence and the irrational ratio are not needed at this base.

Argument. The even indices give $c_n=\lfloor u\varphi^n\rfloor$ with $c_{n+2}=c_n+c_{n+1}+\delta_n$, $\delta_n\in\{0,1\}$, and $\delta_n=1$ infinitely often, since otherwise the fractional parts would satisfy the exact Fibonacci recurrence from some point on, whose bounded nonnegative solutions vanish, making two consecutive $u\varphi^n$ integers and $\varphi$ rational; $D$ disjoint carry-one triples represent an interval of $D+1$ consecutive integers by choosing in each triple either $c_n+c_{n+1}$ or $c_{n+2}$; the odd indices, $\lfloor(u\gamma)\varphi^n\rfloor$, are eventually positive with each term at most twice its predecessor, and adding them one at a time extends the represented interval without gaps past every threshold.

Claimant and postings. Kenta Kitamura (GitHub KitaKen1), who gives that name in the erdosproblems.com thread comment of 5 September 2026 announcing the proof, published the repository the same day (the pinned revision) and opened formal-conjectures PR #5286 to mark part (ii) solved. The README states that the formalization was developed with assistance from ChatGPT and OpenAI Codex, using GPT-6 (Astra), the author's own disclosure. The README reports kernel checks under Lean 4.33.1 against a pinned formal-conjectures commit and under Lean 4.34.0-rc2 with Mathlib alone, no sorry, admit, custom axiom, native_decide or unsafe, and #print axioms giving propext, Classical.choice and Quot.sound; this corpus has built none of it.

Standing. Claimed, with no acceptance evidence: PR #5286 was approved by a catalog contributor (GitHub Deicyde) on 2026-09-18 and was still open and unmerged on 2026-10-07 (labels awaiting-author and solution found). A catalog pull-request approval checks that a formal statement and its proof replay in the catalog's build; it is not a named reviewer's examination of the argument, so it is not reviewed evidence, and formalized evidence needs this corpus's own build and audit of the file. The catalog's own audit issue #6542 of 2026-09-24 argues that parts.ii quantifies $\gamma$ existentially and so is provable at one base rather than asking the variable-base question; the site's commentary does not mention the proof, and no review or refereed source exists. The reading of the second question is unfixed by the site and by both of Graham's and Erdős–Graham's wordings, so this claim settles the question only under one of its two readings. The thread's own comment raises the concern that the result may already follow from Hegyvári's 1989 paper, which nobody in the thread verified.

Depends on. Nothing in this wiki.