SOP_2 and SOP_3 theories shown to coincide
The classes of SOP_2 and SOP_3 first-order theories are the same, answering a 2004 question of Džamonja and Shelah and collapsing the bottom of the SOP hierarchy to a single class.
- Lab
- Independent
- Model
- ChatGPT 5.6
- Field
- Mathematics
- Date
- 2026-08-13
- Human collaborators
- Artem Chernikov
- Problem posed
- 2004 · open 22 yrs
Sources
Original work
Independent commentary
What was found
Džamonja and Shelah introduced the tree configurations SOP_1 and SOP_2 in 2004 and asked whether either implication in the chain from SOP_3 to SOP_2 to SOP_1 reverses. Mutchnik answered the second, proving SOP_1 equals SOP_2, and the surviving half organised much of the subsequent NSOP_1 literature, with partial results by Conant, by Kaplan, Ramsey and Simon, and by Mutchnik. This five-page paper proves that SOP_2 implies SOP_3, the converse having been known from the start, which collapses the bottom of the hierarchy to a single class and carries consequences for Keisler-order maximality. The SOP_n hierarchy for n at least three is untouched, as is everything above it.
Novelty check
The question is one of the two 2004 Džamonja–Shelah questions that organised two decades of classification theory, and the paper's own introduction traces it as open through Mutchnik's celebrated resolution of the other half and the partial results that followed. It is repeatedly highlighted in the NSOP_1 literature as the surviving open case. No prior proof appears. The result is new.
Caveats and known objections
A preprint one day old at entry, not peer-reviewed and not formalized. The acknowledgements record comments from Itay Kaplan and Scott Mutchnik on a preliminary version, which is expert attention on a draft rather than independent verification, so the entry stays at author verified. Autonomy graded collaborative: the entire disclosure is one sentence, that the proof was found using ChatGPT 5.6 and simplified and streamlined by the author, which is ambiguous between the model supplying the key insight and the author using it as a tool. The weaker defensible reading applies.
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Entry history (1 event)
- AddedEntered the registry graded Author verified and Collaborative.
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Graded author verified for verification and collaborative for autonomy. What these mean.
Cite this entry
whataifound.org. (2026). SOP_2 and SOP_3 theories shown to coincide. whataifound.org: A Registry of AI Scientific and Mathematical Discoveries. https://whataifound.org/finding/2026-08-13-sop2-sop3BibTeX
@misc{whataifound-independent-2026-sop3,
title = {SOP_2 and SOP_3 theories shown to coincide},
author = {{whataifound.org}},
year = {2026},
howpublished = {whataifound.org: A Registry of AI Scientific and Mathematical Discoveries},
note = {Result by Independent. Verification: Author verified. Autonomy: Collaborative.},
url = {https://whataifound.org/finding/2026-08-13-sop2-sop3}
}