{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2025:QLR6IVZVMUZRUCTSE5LW6F5ZNE","short_pith_number":"pith:QLR6IVZV","canonical_record":{"source":{"id":"2506.07694","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","primary_cat":"math.AP","submitted_at":"2025-06-09T12:29:30Z","cross_cats_sorted":[],"title_canon_sha256":"ebac486a525f552dab506f8724ba6daad50629851deb2a5f40dff6fe94725f36","abstract_canon_sha256":"ad6be7a9d41b2a7de78b13a237b61b7ef520dc72118af6ba9c94bf05d81b6e58"},"schema_version":"1.0"},"canonical_sha256":"82e3e4573565331a0a7227576f17b969201ce09b3b555146a75304cb81172f62","source":{"kind":"arxiv","id":"2506.07694","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2506.07694","created_at":"2026-07-05T11:18:31Z"},{"alias_kind":"arxiv_version","alias_value":"2506.07694v1","created_at":"2026-07-05T11:18:31Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2506.07694","created_at":"2026-07-05T11:18:31Z"},{"alias_kind":"pith_short_12","alias_value":"QLR6IVZVMUZR","created_at":"2026-07-05T11:18:31Z"},{"alias_kind":"pith_short_16","alias_value":"QLR6IVZVMUZRUCTS","created_at":"2026-07-05T11:18:31Z"},{"alias_kind":"pith_short_8","alias_value":"QLR6IVZV","created_at":"2026-07-05T11:18:31Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2025:QLR6IVZVMUZRUCTSE5LW6F5ZNE","target":"record","payload":{"canonical_record":{"source":{"id":"2506.07694","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","primary_cat":"math.AP","submitted_at":"2025-06-09T12:29:30Z","cross_cats_sorted":[],"title_canon_sha256":"ebac486a525f552dab506f8724ba6daad50629851deb2a5f40dff6fe94725f36","abstract_canon_sha256":"ad6be7a9d41b2a7de78b13a237b61b7ef520dc72118af6ba9c94bf05d81b6e58"},"schema_version":"1.0"},"canonical_sha256":"82e3e4573565331a0a7227576f17b969201ce09b3b555146a75304cb81172f62","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T11:18:31.639492Z","signature_b64":"Ebr+0SjzoMJybPa+oVKOK23jSOrPQMikTpCr5KYQBmy0e33WFJEclQJ2uCS0WP4AP+wiPpun8jDV3e7hLxJGDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"82e3e4573565331a0a7227576f17b969201ce09b3b555146a75304cb81172f62","last_reissued_at":"2026-07-05T11:18:31.639027Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T11:18:31.639027Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2506.07694","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T11:18:31Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"thpbbSSObIPdzzXa4cengT0BGht+L1gkesxm4eCvLWqdQUPcdiZh4Eoe7DgB/SmiyJwI+cIJalqWDtSOGu/TBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-08T04:03:47.845588Z"},"content_sha256":"6e12749ee4b8f9cef932a440f5af3a18f138b71da1c2e8d9cca5389714999c48","schema_version":"1.0","event_id":"sha256:6e12749ee4b8f9cef932a440f5af3a18f138b71da1c2e8d9cca5389714999c48"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2025:QLR6IVZVMUZRUCTSE5LW6F5ZNE","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Fractional Sobolev spaces and fractional $p$-Laplace equations on locally finite graphs","license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Mengjie Zhang, Yong Lin, Yunyan Yang","submitted_at":"2025-06-09T12:29:30Z","abstract_excerpt":"Graph-based analysis holds both theoretical and applied significance, attracting considerable attention from researchers and yielding abundant results in recent years. However, research on fractional problems remains limited, with most of established results restricted to lattice graphs. In this paper, fractional Sobolev spaces are constructed on general graphs that are connected, locally finite and stochastically complete. Under certain assumptions, these spaces exhibit completeness, reflexivity, and other properties. Moreover, we propose a fractional $p$-Laplace operator, and study the exist"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2506.07694","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/2506.07694/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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T11:18:31Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"qzXZkJcyZixwwlHmyI4Q6Qc+68/CdHxD9oQw69vhpWtDkQ8BgZ3arEMBTR9/4gRliKhVECaWIpTa1Mb26px0Aw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-08T04:03:47.846071Z"},"content_sha256":"735b9e2a0691dc27473b3288f9315024b89e79c3eb1fed50e3100f9b90abe35e","schema_version":"1.0","event_id":"sha256:735b9e2a0691dc27473b3288f9315024b89e79c3eb1fed50e3100f9b90abe35e"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/QLR6IVZVMUZRUCTSE5LW6F5ZNE/bundle.json","state_url":"https://pith.science/pith/QLR6IVZVMUZRUCTSE5LW6F5ZNE/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/QLR6IVZVMUZRUCTSE5LW6F5ZNE/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-08-08T04:03:47Z","links":{"resolver":"https://pith.science/pith/QLR6IVZVMUZRUCTSE5LW6F5ZNE","bundle":"https://pith.science/pith/QLR6IVZVMUZRUCTSE5LW6F5ZNE/bundle.json","state":"https://pith.science/pith/QLR6IVZVMUZRUCTSE5LW6F5ZNE/state.json","well_known_bundle":"https://pith.science/.well-known/pith/QLR6IVZVMUZRUCTSE5LW6F5ZNE/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2025:QLR6IVZVMUZRUCTSE5LW6F5ZNE","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"ad6be7a9d41b2a7de78b13a237b61b7ef520dc72118af6ba9c94bf05d81b6e58","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","primary_cat":"math.AP","submitted_at":"2025-06-09T12:29:30Z","title_canon_sha256":"ebac486a525f552dab506f8724ba6daad50629851deb2a5f40dff6fe94725f36"},"schema_version":"1.0","source":{"id":"2506.07694","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2506.07694","created_at":"2026-07-05T11:18:31Z"},{"alias_kind":"arxiv_version","alias_value":"2506.07694v1","created_at":"2026-07-05T11:18:31Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2506.07694","created_at":"2026-07-05T11:18:31Z"},{"alias_kind":"pith_short_12","alias_value":"QLR6IVZVMUZR","created_at":"2026-07-05T11:18:31Z"},{"alias_kind":"pith_short_16","alias_value":"QLR6IVZVMUZRUCTS","created_at":"2026-07-05T11:18:31Z"},{"alias_kind":"pith_short_8","alias_value":"QLR6IVZV","created_at":"2026-07-05T11:18:31Z"}],"graph_snapshots":[{"event_id":"sha256:735b9e2a0691dc27473b3288f9315024b89e79c3eb1fed50e3100f9b90abe35e","target":"graph","created_at":"2026-07-05T11:18:31Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2506.07694/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Graph-based analysis holds both theoretical and applied significance, attracting considerable attention from researchers and yielding abundant results in recent years. However, research on fractional problems remains limited, with most of established results restricted to lattice graphs. In this paper, fractional Sobolev spaces are constructed on general graphs that are connected, locally finite and stochastically complete. Under certain assumptions, these spaces exhibit completeness, reflexivity, and other properties. Moreover, we propose a fractional $p$-Laplace operator, and study the exist","authors_text":"Mengjie Zhang, Yong Lin, Yunyan Yang","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","primary_cat":"math.AP","submitted_at":"2025-06-09T12:29:30Z","title":"Fractional Sobolev spaces and fractional $p$-Laplace equations on locally finite graphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2506.07694","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:6e12749ee4b8f9cef932a440f5af3a18f138b71da1c2e8d9cca5389714999c48","target":"record","created_at":"2026-07-05T11:18:31Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"ad6be7a9d41b2a7de78b13a237b61b7ef520dc72118af6ba9c94bf05d81b6e58","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","primary_cat":"math.AP","submitted_at":"2025-06-09T12:29:30Z","title_canon_sha256":"ebac486a525f552dab506f8724ba6daad50629851deb2a5f40dff6fe94725f36"},"schema_version":"1.0","source":{"id":"2506.07694","kind":"arxiv","version":1}},"canonical_sha256":"82e3e4573565331a0a7227576f17b969201ce09b3b555146a75304cb81172f62","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"82e3e4573565331a0a7227576f17b969201ce09b3b555146a75304cb81172f62","first_computed_at":"2026-07-05T11:18:31.639027Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T11:18:31.639027Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Ebr+0SjzoMJybPa+oVKOK23jSOrPQMikTpCr5KYQBmy0e33WFJEclQJ2uCS0WP4AP+wiPpun8jDV3e7hLxJGDQ==","signature_status":"signed_v1","signed_at":"2026-07-05T11:18:31.639492Z","signed_message":"canonical_sha256_bytes"},"source_id":"2506.07694","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:6e12749ee4b8f9cef932a440f5af3a18f138b71da1c2e8d9cca5389714999c48","sha256:735b9e2a0691dc27473b3288f9315024b89e79c3eb1fed50e3100f9b90abe35e"],"state_sha256":"d4d1646cf221b70d4972508a40a47807eadf8e6a884aeb074ed0625c3599a3f1"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"UlAiTChm3Wn+CT80lvl49btCCyKoIaWU98ikgK9H26BzF+b/O7ae3FtjuZa3o8JrYT0EnWw38O7vvL5Pz/ynBQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-08T04:03:47.850947Z","bundle_sha256":"b40b0047e5a172360dd039deee8edd7c79022bbcc7671ef30b63d6be2bbcbfe4"}}