{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2026:WDNG6EXDJUXILMWFR7SKD6YUNW","short_pith_number":"pith:WDNG6EXD","schema_version":"1.0","canonical_sha256":"b0da6f12e34d2e85b2c58fe4a1fb146db343dfd07b25e503257889f0b54fa208","source":{"kind":"arxiv","id":"2606.02203","version":1},"attestation_state":"computed","paper":{"title":"Stability of nontrivial graph pairs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Xiaomeng Wang, Xing Gao","submitted_at":"2026-06-01T12:59:54Z","abstract_excerpt":"A graph pair $(\\Gamma, \\Sigma)$ is called stable if every automorphism of the direct product $\\Gamma\\times\\Sigma$ is induced componentwise by automorphisms of $\\Gamma$ and $\\Sigma$. A graph is twin-free if no two distinct vertices share the same neighbourhood in the graph. Two graphs $\\Gamma$ and $\\Sigma$ are coprime with respect to the direct product if there is no graph $\\Delta$ of order greater than $1$ such that $\\Gamma\\cong\\Gamma'\\times\\Delta$ and $\\Sigma\\cong\\Sigma'\\times\\Delta$ for some graphs $\\Gamma'$ and $\\Sigma'$. A graph pair $(\\Gamma,\\Sigma)$ is nontrivial if $\\Gamma$ and $\\Sigma$"},"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":false,"formal_links_present":false},"canonical_record":{"source":{"id":"2606.02203","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2026-06-01T12:59:54Z","cross_cats_sorted":[],"title_canon_sha256":"b46b913dc04e794eb5dda4e2241b27fff52b9bb2eea483c161745217d46cc416","abstract_canon_sha256":"2cbea88559cecd36d6fba07e0723defcd2ba4b18568f96d1778554b07269c118"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-02T03:04:53.047419Z","signature_b64":"XxVXizHgOaPkMsDs+gHUPiJK9DJ/54tRBE9altQhiTAjqntPZ1uqKU5+3iYdpItbSU0jqUT62AM5CBVZrD2hAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b0da6f12e34d2e85b2c58fe4a1fb146db343dfd07b25e503257889f0b54fa208","last_reissued_at":"2026-06-02T03:04:53.047029Z","signature_status":"signed_v1","first_computed_at":"2026-06-02T03:04:53.047029Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Stability of nontrivial graph pairs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Xiaomeng Wang, Xing Gao","submitted_at":"2026-06-01T12:59:54Z","abstract_excerpt":"A graph pair $(\\Gamma, \\Sigma)$ is called stable if every automorphism of the direct product $\\Gamma\\times\\Sigma$ is induced componentwise by automorphisms of $\\Gamma$ and $\\Sigma$. A graph is twin-free if no two distinct vertices share the same neighbourhood in the graph. Two graphs $\\Gamma$ and $\\Sigma$ are coprime with respect to the direct product if there is no graph $\\Delta$ of order greater than $1$ such that $\\Gamma\\cong\\Gamma'\\times\\Delta$ and $\\Sigma\\cong\\Sigma'\\times\\Delta$ for some graphs $\\Gamma'$ and $\\Sigma'$. A graph pair $(\\Gamma,\\Sigma)$ is nontrivial if $\\Gamma$ and $\\Sigma$"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.02203","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/2606.02203/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"},"aliases":[{"alias_kind":"arxiv","alias_value":"2606.02203","created_at":"2026-06-02T03:04:53.047090+00:00"},{"alias_kind":"arxiv_version","alias_value":"2606.02203v1","created_at":"2026-06-02T03:04:53.047090+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.02203","created_at":"2026-06-02T03:04:53.047090+00:00"},{"alias_kind":"pith_short_12","alias_value":"WDNG6EXDJUXI","created_at":"2026-06-02T03:04:53.047090+00:00"},{"alias_kind":"pith_short_16","alias_value":"WDNG6EXDJUXILMWF","created_at":"2026-06-02T03:04:53.047090+00:00"},{"alias_kind":"pith_short_8","alias_value":"WDNG6EXD","created_at":"2026-06-02T03:04:53.047090+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/WDNG6EXDJUXILMWFR7SKD6YUNW","json":"https://pith.science/pith/WDNG6EXDJUXILMWFR7SKD6YUNW.json","graph_json":"https://pith.science/api/pith-number/WDNG6EXDJUXILMWFR7SKD6YUNW/graph.json","events_json":"https://pith.science/api/pith-number/WDNG6EXDJUXILMWFR7SKD6YUNW/events.json","paper":"https://pith.science/paper/WDNG6EXD"},"agent_actions":{"view_html":"https://pith.science/pith/WDNG6EXDJUXILMWFR7SKD6YUNW","download_json":"https://pith.science/pith/WDNG6EXDJUXILMWFR7SKD6YUNW.json","view_paper":"https://pith.science/paper/WDNG6EXD","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2606.02203&json=true","fetch_graph":"https://pith.science/api/pith-number/WDNG6EXDJUXILMWFR7SKD6YUNW/graph.json","fetch_events":"https://pith.science/api/pith-number/WDNG6EXDJUXILMWFR7SKD6YUNW/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/WDNG6EXDJUXILMWFR7SKD6YUNW/action/timestamp_anchor","attest_storage":"https://pith.science/pith/WDNG6EXDJUXILMWFR7SKD6YUNW/action/storage_attestation","attest_author":"https://pith.science/pith/WDNG6EXDJUXILMWFR7SKD6YUNW/action/author_attestation","sign_citation":"https://pith.science/pith/WDNG6EXDJUXILMWFR7SKD6YUNW/action/citation_signature","submit_replication":"https://pith.science/pith/WDNG6EXDJUXILMWFR7SKD6YUNW/action/replication_record"}},"created_at":"2026-06-02T03:04:53.047090+00:00","updated_at":"2026-06-02T03:04:53.047090+00:00"}