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SOP2=SOP3SOP_2 = SOP_3

The classes of SOP_2 and SOP_3 first-order theories coincide. This answers a question of Džamonja and Shelah from 2004.

Result
Proved(see note)
Status
Resolved
AI contribution
AI co-developed
Method
Argument
Field
Model theory
Posed by
Mirna Džamonja, Saharon Shelah
Year posed
2004
Years open
22y
Solved
2026-08-13
Model
ChatGPT 5.6
Vendor
OpenAI
Collaborators
Artem Chernikov
Verification
Unreviewed
Publication
Preprint
Significance
35 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

The new content is SOP2⇒SOP3SOP_2 \Rightarrow SOP_3; the converse implication was known from the start. Dzamonja and Shelah asked whether either implication in SOP3⇒SOP2⇒SOP1SOP_3 \Rightarrow SOP_2 \Rightarrow SOP_1 reverses: Mutchnik answered the second (SOP1=SOP2SOP_1 = SOP_2), and this answers the first, collapsing the bottom of the hierarchy to SOP1=SOP2=SOP3SOP_1 = SOP_2 = SOP_3. The SOPnSOP_n hierarchy for n≥3n \ge 3 remains, as does everything above it.

What the AI did

From the paper: "The proof was found using ChatGPT 5.6 and simplified and streamlined
by the author."

Verification

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 SOP1SOP_1 and SOP2SOP_2 and asked whether either implication SOP3⇒SOP2⇒SOP1SOP_3 \Rightarrow SOP_2 \Rightarrow SOP_1 reverses; Mutchnik's breakthrough proved SOP2=SOP1SOP_2 = SOP_1, and the question SOP2=SOP3SOP_2 = SOP_3 was repeatedly highlighted afterwards, with partial results by Conant, Kaplan-Ramsey-Simon and Mutchnik. This result completes the collapse SOP1=SOP2=SOP3SOP_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

Submitted by LucidManta102 on

Changelog5 changes
  • Rasmus Lindahlchanged Short name from “SOP_2 = SOP_3” to “$SOP_2 = SOP_3$”
  • Rasmus Lindahlset Collaborators to “Artem Chernikov”, also Verification note, Field detail
  • Rasmus Lindahlapproved this entry
  • Rasmus Lindahlchanged Field from “Combinatorics” to “Logic & foundations”, also Name, Significance, Significance note, What was actually shown, Age note
  • LucidManta102submitted this entry

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