{"paper":{"title":"Vaught's Conjecture for Unions of Products of Rooted Trees","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.LO","authors_text":"Milo\\v{s} S. Kurili\\'c","submitted_at":"2026-06-10T12:40:25Z","abstract_excerpt":"Let ${\\mathcal C} ^{\\rm rt}$ be the class of rooted trees and $\\langle {\\mathcal C} ^{\\rm rt}\\rangle _{\\dot{\\cup }\\Pi}$ its minimal closure under isomorphism, finite direct products and finite disjoint unions. Posets from that closure are isomorphic to ${\\mathbb X}= \\dot{\\bigcup} _{i<n}\\prod _{j<m_i}{\\mathbb X}_i^j$, where ${\\mathbb X}_i^j$ are rooted trees. Defining ${\\mathcal T}=\\mathop{\\rm Th} ({\\mathbb X})$, ${\\mathcal T} _i ^j=\\mathop{\\rm Th}({\\mathbb X}_i^j)$, for $i<n$ and $j<m_i$, and $\\kappa = \\prod _{i<n}\\prod _{j<m_i}I({\\mathcal T} _i^j)$, we have\n  (a) Vaught's conjecture is true f"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.12014","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2606.12014/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}