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Claim. Write $D_2(G)$ for the fewest edges whose deletion makes a finite simple graph $G$ bipartite, and $e(G)$ for its number of edges. Let $G$ be a triangle-free graph on $N\ge1$ vertices.

1. If $2e(G)/N^2\le0.2486$ or $2e(G)/N^2\ge0.3197$, then $D_2(G)\le N^2/25$. In particular, for every $n\ge1$, every triangle-free graph on $5n$ vertices whose edge density $2e(G)/(5n)^2$ lies in either range can be made bipartite by deleting at most $n^2$ edges.
2. In general $D_2(G)\le2N^2/47$; on $N=5n$ vertices this is $D_2(G)\le(50/47)n^2$.

Covers. The question of Problem 23, answered yes at every order for the triangle-free graphs of item 1, whose edge density $2e(G)/N^2$ is at most $0.2486$ or at least $0.3197$, the two boundary densities included. Not covered: the graphs with $0.2486<2e(G)/N^2<0.3197$, at every order. Blow-ups preserve edge density, so the transfer cannot reach this middle band, and the problem remains open. Item 2 answers the question for no graph, since $50/47>1$. For $1\le n\le40$, Ferudun's pending claim asserts the bound $n^2$ at every density; its middle-band certificate is not used here.

Credit. Both items carry Theorem 2(a)–(c) of József Balogh, Felix Christian Clemen and Bernard Lidický, Max Cuts in Triangle-free Graphs (Extended Abstracts EuroComb 2021, pp. 509–514), to every order. That theorem gives the bound $N^2/23.5$, and the bound $N^2/25$ when $e(G)\ge0.3197\binom N2$ or $e(G)\le0.2486\binom N2$, for all orders above an unspecified threshold. The transfer to every order is theirs too: Section 2.3 of the same paper blows a smaller triangle-free graph up to a large order and compares the two by Theorem 7 of Erdős, Győri and Simonovits. Alper Ferudun's preprint arXiv:2606.28041v1 of 26 June 2026, the paper of Ferudun's pending claim, proves the identity $D_2(G[t])=t^2D_2(G)$ for the blow-up $G[t]$ that replaces each vertex by $t$ independent copies as its Lemma 2.1, and states the same two tail bounds at every order, both boundary densities included, as its Corollary 4.2. That corollary's proof blows up only the strict tails and sends the two boundary densities to the preprint's positive-error band certificate. The tail statements are therefore not new. Plasma AI's part is the direct treatment of the two boundary densities, at the sparse end by a blow-up with one isolated vertex adjoined before letting $t\to\infty$, and the statement of clause (a) at every order, as item 2.

Standing. Tier 0: the transfer is author-recorded by Plasma AI, with no independent review or Lean proof, and it is a written argument with no evidence program. Theorem 2 of Balogh, Clemen and Lidický is a published input used at its statement. Its held source is a six-page extended abstract whose proof is a sketch, with detailed computations omitted in the high-density section; that proof has not been reconstructed or reviewed here. Catalog acceptance is pending.

Depends on. The density-tail lead (tier 0), through the homogeneous transfer page it links, with its external input Balogh, Clemen and Lidický's Theorem 2.