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We prove the following:\n  (1) Every oriented path with at least three arcs and exactly one non-leaf source or sink vertex is tournament anti-Sidorenko.\n  (2) An oriented path is tournament anti-Sidorenko if the distance between any leaf vertex and any source or sink vertex is at least two and the distance between any pair of non-leaf source or sink vertices is a "},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":true,"formal_links_present":false},"canonical_record":{"source":{"id":"2605.14138","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2026-05-13T21:44:25Z","cross_cats_sorted":[],"title_canon_sha256":"c978ff09e75bccabbfd738469fedfd8afbf2fa17bf1ccc7d5dec1d729f2e7e09","abstract_canon_sha256":"0d5a5bdda7f034e4494d2a062413e9bff9c7288f36ed94996f7b347c39677cd7"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:39:11.721029Z","signature_b64":"3+u04DUpa6hTkw3F+aBOK4Em40XCEgx7rVJrtvFId8RgXrKvCgB01sP3sgKKItpRHWsD273wj5RxAMjaS2lUCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"bb0289a9c9be8237e1e2b69f16472e9a2e4f4e7bbe50ffd6ed243b3d8a477632","last_reissued_at":"2026-05-17T23:39:11.720455Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:39:11.720455Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On Tournament Anti-Sidorenko Orientations of Trees","license":"http://creativecommons.org/licenses/by/4.0/","headline":"Oriented paths with exactly one non-leaf source or sink are tournament anti-Sidorenko.","cross_cats":[],"primary_cat":"math.CO","authors_text":"Felix Christian Clemen, Hao Chen, Jonathan A. 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An oriented path is tournament anti-Sidorenko if the distance between any leaf vertex and any source or sink vertex is at least two and the distance between any pair of non-leaf source or sink vertices is a multiple of four. Every spider with exactly three legs admits a tournament anti-Sidorenko orientation.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The proofs for the stated distance conditions and structural properties of the oriented paths and spiders hold without gaps or unstated restrictions on the graphs considered.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Specific orientations of paths with distance conditions and three-legged spiders are tournament anti-Sidorenko, proving conjectures and yielding new families.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Oriented paths with exactly one non-leaf source or sink are tournament anti-Sidorenko.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"e425063248ef346e7843a4c7e4a3ad5e6eaf74445beaeadad3593d732c0649b0"},"source":{"id":"2605.14138","kind":"arxiv","version":1},"verdict":{"id":"23160e29-80d9-4091-b7e1-eeb59c8e59f7","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-15T02:16:07.669349Z","strongest_claim":"Every oriented path with at least three arcs and exactly one non-leaf source or sink vertex is tournament anti-Sidorenko. An oriented path is tournament anti-Sidorenko if the distance between any leaf vertex and any source or sink vertex is at least two and the distance between any pair of non-leaf source or sink vertices is a multiple of four. Every spider with exactly three legs admits a tournament anti-Sidorenko orientation.","one_line_summary":"Specific orientations of paths with distance conditions and three-legged spiders are tournament anti-Sidorenko, proving conjectures and yielding new families.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The proofs for the stated distance conditions and structural properties of the oriented paths and spiders hold without gaps or unstated restrictions on the graphs considered.","pith_extraction_headline":"Oriented paths with exactly one non-leaf source or sink are tournament anti-Sidorenko."},"references":{"count":30,"sample":[{"doi":"","year":2025,"title":"A. Basit, B. Granet, D. Horsley, A. K¨ undgen, and K. Staden. 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