ChatGPT 5.6
From the paper: "The proof was found using ChatGPT 5.6 and simplified and streamlined by the author."
Provider: OpenAI · Prompt public: unknown · Independence: unknown
Problem detail · source-aware
$SOP_2 = SOP_3$
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
Precise statement
The classes of SOP_2 and SOP_3 first-order theories coincide. This answers a question of Džamonja and Shelah from 2004.
The source statement is reproduced for indexing with attribution. Mathematical correctness requires domain-expert or mechanical review. VibeMath has not independently audited statement fidelity, correctness, priority, or novelty.
What AI did
Verification boundary
Checked by this site on 14 August 2026 against the paper's LaTeX source. The paper is real - five pages, math.LO, posted 13 August - and its author, Artem Chernikov, is a leading model theorist in exactly this area. The AI disclosure is verbatim as the entry quotes it. The history checks out in the paper's own introduction: Dzamonja and Shelah introduced the tree configurations $SOP_1$ and $SOP_2$ and asked whether either implication $SOP_3 \Rightarrow SOP_2 \Rightarrow SOP_1$ reverses; Mutchnik's breakthrough proved $SOP_2 = SOP_1$, and the question $SOP_2 = SOP_3$ was repeatedly highlighted afterwards, with partial results by Conant, Kaplan-Ramsey-Simon and Mutchnik. This result completes the collapse $SOP_1 = SOP_2 = SOP_3$. The acknowledgements record comments by Itay Kaplan and Scott Mutchnik on a preliminary version - expert eyes, but comments on a draft are not independent verification, and the manuscript is one day old and unrefereed, so the tier is Unreviewed. The proof itself - five pages of tree-indiscernible manipulation - was not checked here; it needs a model theorist.
Correctness: unknown · statement fidelity: unaudited · peer review: none
Timeline
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SOP_2 = SOP_3
The new content is $SOP_2 \Rightarrow SOP_3$; the converse implication was known from the start. Dzamonja and Shelah asked whether either implication in $SOP_3 \Rightarrow SOP_2 \Rightarrow SOP_1$ reverses: Mutchnik answered the second ($SOP_1 = SOP_2$), and this answers the first, collapsing the bottom of the hierarchy to $SOP_1 = SOP_2 = SOP_3$. The $SOP_n$ hierarchy for $n \ge 3$ remains, as does everything above it.
Known method families
argument (source-reported)
Source-reported tools: argument.
Independent: unknown · difference confidence: 0
What remains uncertain
VibeMath has not independently audited the mathematical statement, proof, or novelty claim.
- VibeMath has not independently verified the mathematical claim.
- AI-attempt independence and training-data exposure are unknown unless explicitly documented.
- VibeMath has not independently audited the mathematical statement, proof, or novelty claim.