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Claim. Let $A$ be the set of integers whose base-$3$ digits are all $0$ or $1$ and $B$ the set of integers whose base-$4$ digits are all $0$ or $1$. Then the lower density of $A+B$ is $0$: for every $\epsilon>0$ there are arbitrarily large $x$ with $\lvert (A+B)\cap[1,x]\rvert<\epsilon x$. This answers the question of Problem 125 in the negative. The Lean statement proved is erdos_125.variants.positive_lower_density, the negation of 0 < (A + B).lowerDensity, together with lower_density_zero : (A + B).lowerDensity = 0, in the formal-conjectures file at the linked commit.

Argument, in outline. The lemma names of the Lean texts, and the summaries on the thread, indicate a Dirichlet approximation aligning the scales $3^k$ and $4^m$, a gap in $A+B$ forced at each aligned scale, and a sparse sequence of scales at which the count $\lvert (A+B)\cap[1,x]\rvert$ shrinks by a constant factor each time, so that it falls below $\epsilon x$ infinitely often. Thomas Bloom's reconstruction on the thread (post 5114, 2026-03-30) suggests that the argument generalizes to any bases $d_1,\dots,d_r$ with $\sum 1/(d_i-1)<1$. Nat Sothanaphan's reply (post 5119, the same day, crediting GPT-5.4 Thinking) sketches that generalization for $r\geq 2$ bases, repetition allowed, through the simultaneous Dirichlet approximation theorem. The proof has not been reconstructed in this corpus.

The same claimant's first step, DeepMind's earlier result, showed that $A+B$ has no positive density; the present result improves it and does not rest on it logically.

Standing. The claimant is DeepMind, which the site's commentary credits with both results; the result was posted on 2026-03-30 by George Tsoukalas, who reports that a DeepMind prover agent found the Lean proof autonomously. The system is named here as the thread and the community file's header name it, a DeepMind prover agent. The site's curator, Thomas Bloom, marks the problem DISPROVED (LEAN), records in the commentary that DeepMind proved the lower density is $0$, and reconstructed the argument on the thread; Giuseppe Melfi, an author of the problem's literature, confirmed on the thread (post 5169, 2026-04-01) that the result rules out the remaining scenario with positive lower density. That curator acceptance is the reviewed evidence. No manuscript or refereed publication exists; the claimant's thread post 5110 summarizes the argument. Two Lean texts hold the proof: the pinned formal-conjectures copy and Boris Alexeev's repository copy, whose header declares itself a formalization of this result with informal author a DeepMind prover agent and formal authors the agent and George Tsoukalas. Neither has been built or audited in this corpus, so the page lists no formalized evidence; the Lean qualification in the site's label refers to these developments. A generated Lean proof of the same statement by a different argument, in DeepMind's AlphaProof Nexus results repository, names no person and has its own page, a generated Lean proof by AlphaProof Nexus. Whether $A+B$ has positive upper density is a separate question that the formal-conjectures statement file, at its commit of 2026-09-18, keeps open.