SOP₂ and SOP₃ theories shown to coincide
The classes of and 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
What was found
Džamonja and Shelah introduced the tree configurations and in 2004 and asked whether either implication in the chain from to to reverses. Mutchnik answered the second, proving equals , and the surviving half organised much of the subsequent literature, with partial results by Conant, by Kaplan, Ramsey and Simon, and by Mutchnik. This five-page paper proves that implies , 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 hierarchy for 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 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A preprint one day old at entry, not peer-reviewed and not formalized.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.
Also recorded at
Nothing mechanically. It is a parallel listing of the same result, carrying its own verification label rather than an independent check.
Nobody outside the lab has checked this yet.
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History of this entry
- CorrectedRecorded the vibemathed record for this result as a registration. The vibemathed link it replaces was classified as commentary, which a registry listing is not.
- 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). SOP₂ and SOP₃ 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 = {SOP₂ and SOP₃ 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}
}