$SOP_2 = SOP_3$

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.

Exact FrontierDelta

Prior state unknown→proved

Scope and record

Occurred: Aug 13, 2026

Delta type: SOURCE CLAIM

Assumptions: VibeMathed verification: unreviewed. Publication: preprint. AI contribution: ai-co-developed. Imported under CC BY 4.0.

Canonical aliases: $SOP_2 = SOP_3$

Confidence: Not scored

Registry verification: unreviewed · preprint · resolved

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Attribution

VibeMathed
registry · event recorded by

Artem Chernikov
human · human collaborator

ChatGPT 5.6
model · ai model contributor · OpenAI

Evidence vector

SourceReplayReproducedFormal proofStatement auditExternal checkExpert reviewPeer review

Named evidence

VibeMathed record: $SOP_2 = SOP_3$

specialist registry · passed

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.

Source ↗

$SOP_2 = SOP_3$

preprint · pending

Primary source named by the VibeMathed record. Presence in this import is not an independent mathematical verification.

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Lineage and corrections

This event attributed to Artem Chernikov

This event attributed to ChatGPT 5.6

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