{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2024:I7GIYYT34F7PRHM2TMU3MNQ5IO","short_pith_number":"pith:I7GIYYT3","canonical_record":{"source":{"id":"2402.04076","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2024-01-17T09:19:28Z","cross_cats_sorted":[],"title_canon_sha256":"b4a154a01c3f6ede5fe18f4e412449dd736167d766236a5757b14978e5dca61f","abstract_canon_sha256":"02d4e635c7138218f0154a362afe97b3a59bb0d2950eb74ecdfecaa1061732b0"},"schema_version":"1.0"},"canonical_sha256":"47cc8c627be17ef89d9a9b29b6361d439a4be06d4ff1506cb17de996400c9388","source":{"kind":"arxiv","id":"2402.04076","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2402.04076","created_at":"2026-07-05T10:02:02Z"},{"alias_kind":"arxiv_version","alias_value":"2402.04076v3","created_at":"2026-07-05T10:02:02Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2402.04076","created_at":"2026-07-05T10:02:02Z"},{"alias_kind":"pith_short_12","alias_value":"I7GIYYT34F7P","created_at":"2026-07-05T10:02:02Z"},{"alias_kind":"pith_short_16","alias_value":"I7GIYYT34F7PRHM2","created_at":"2026-07-05T10:02:02Z"},{"alias_kind":"pith_short_8","alias_value":"I7GIYYT3","created_at":"2026-07-05T10:02:02Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2024:I7GIYYT34F7PRHM2TMU3MNQ5IO","target":"record","payload":{"canonical_record":{"source":{"id":"2402.04076","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2024-01-17T09:19:28Z","cross_cats_sorted":[],"title_canon_sha256":"b4a154a01c3f6ede5fe18f4e412449dd736167d766236a5757b14978e5dca61f","abstract_canon_sha256":"02d4e635c7138218f0154a362afe97b3a59bb0d2950eb74ecdfecaa1061732b0"},"schema_version":"1.0"},"canonical_sha256":"47cc8c627be17ef89d9a9b29b6361d439a4be06d4ff1506cb17de996400c9388","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T10:02:02.030598Z","signature_b64":"i5nXm9r3k+Ysu1n/E64VUiYLK1Af8xUif5w9eZn5kmjGdldhFTC56oSTQr6H2VfKOKj93wNdQ4NfnZDwRFtbCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"47cc8c627be17ef89d9a9b29b6361d439a4be06d4ff1506cb17de996400c9388","last_reissued_at":"2026-07-05T10:02:02.030183Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T10:02:02.030183Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2402.04076","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T10:02:02Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Dmw1PL3GwrywoYGN7HcSzBVTqEawCJVMg3ZRtP4l84ay3QT9vX8oAWIEWrq4YEykDegCk5L7l9VrnLRBfVsZBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-13T11:30:46.251735Z"},"content_sha256":"151935b9dd3ac2e9543195d6d6cac273f44facaaa840308101af000ce5a6ca8e","schema_version":"1.0","event_id":"sha256:151935b9dd3ac2e9543195d6d6cac273f44facaaa840308101af000ce5a6ca8e"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2024:I7GIYYT34F7PRHM2TMU3MNQ5IO","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Fractional Sobolev spaces on Riemannian manifolds","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Enric Florit-Simon, Joaquim Serra, Michele Caselli","submitted_at":"2024-01-17T09:19:28Z","abstract_excerpt":"This article studies the canonical Hilbert energy $H^{s/2}(M)$ on a Riemannian manifold for $s\\in(0,2)$, with particular focus on the case of closed manifolds. Several equivalent definitions for this energy and the fractional Laplacian on a manifold are given, and they are shown to be identical up to explicit multiplicative constants. Moreover, the precise behavior of the kernel associated with the singular integral definition of the fractional Laplacian is obtained through an in-depth study of the heat kernel on a Riemannian manifold. Furthermore, a monotonicity formula for stationary points "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2402.04076","kind":"arxiv","version":3},"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/2402.04076/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":"2026-07-05T10:02:02Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"1Fx4nDOQ7B/4tmFCdZ57YwPve1P7R7uTEvt3GZr3xlLRYAgyvf/6ZILMnW/umK12QC1bpN+Z2TAi3AyOeZkpAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-13T11:30:46.252463Z"},"content_sha256":"2e8b18bb25b3044a36ff3e3219083f2099e301977adfc0580489017514bb1b02","schema_version":"1.0","event_id":"sha256:2e8b18bb25b3044a36ff3e3219083f2099e301977adfc0580489017514bb1b02"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/I7GIYYT34F7PRHM2TMU3MNQ5IO/bundle.json","state_url":"https://pith.science/pith/I7GIYYT34F7PRHM2TMU3MNQ5IO/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/I7GIYYT34F7PRHM2TMU3MNQ5IO/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":"2026-08-13T11:30:46Z","links":{"resolver":"https://pith.science/pith/I7GIYYT34F7PRHM2TMU3MNQ5IO","bundle":"https://pith.science/pith/I7GIYYT34F7PRHM2TMU3MNQ5IO/bundle.json","state":"https://pith.science/pith/I7GIYYT34F7PRHM2TMU3MNQ5IO/state.json","well_known_bundle":"https://pith.science/.well-known/pith/I7GIYYT34F7PRHM2TMU3MNQ5IO/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2024:I7GIYYT34F7PRHM2TMU3MNQ5IO","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":"02d4e635c7138218f0154a362afe97b3a59bb0d2950eb74ecdfecaa1061732b0","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2024-01-17T09:19:28Z","title_canon_sha256":"b4a154a01c3f6ede5fe18f4e412449dd736167d766236a5757b14978e5dca61f"},"schema_version":"1.0","source":{"id":"2402.04076","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2402.04076","created_at":"2026-07-05T10:02:02Z"},{"alias_kind":"arxiv_version","alias_value":"2402.04076v3","created_at":"2026-07-05T10:02:02Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2402.04076","created_at":"2026-07-05T10:02:02Z"},{"alias_kind":"pith_short_12","alias_value":"I7GIYYT34F7P","created_at":"2026-07-05T10:02:02Z"},{"alias_kind":"pith_short_16","alias_value":"I7GIYYT34F7PRHM2","created_at":"2026-07-05T10:02:02Z"},{"alias_kind":"pith_short_8","alias_value":"I7GIYYT3","created_at":"2026-07-05T10:02:02Z"}],"graph_snapshots":[{"event_id":"sha256:2e8b18bb25b3044a36ff3e3219083f2099e301977adfc0580489017514bb1b02","target":"graph","created_at":"2026-07-05T10:02:02Z","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/2402.04076/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"This article studies the canonical Hilbert energy $H^{s/2}(M)$ on a Riemannian manifold for $s\\in(0,2)$, with particular focus on the case of closed manifolds. Several equivalent definitions for this energy and the fractional Laplacian on a manifold are given, and they are shown to be identical up to explicit multiplicative constants. Moreover, the precise behavior of the kernel associated with the singular integral definition of the fractional Laplacian is obtained through an in-depth study of the heat kernel on a Riemannian manifold. Furthermore, a monotonicity formula for stationary points ","authors_text":"Enric Florit-Simon, Joaquim Serra, Michele Caselli","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2024-01-17T09:19:28Z","title":"Fractional Sobolev spaces on Riemannian manifolds"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2402.04076","kind":"arxiv","version":3},"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":"sha256:151935b9dd3ac2e9543195d6d6cac273f44facaaa840308101af000ce5a6ca8e","target":"record","created_at":"2026-07-05T10:02:02Z","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":"02d4e635c7138218f0154a362afe97b3a59bb0d2950eb74ecdfecaa1061732b0","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2024-01-17T09:19:28Z","title_canon_sha256":"b4a154a01c3f6ede5fe18f4e412449dd736167d766236a5757b14978e5dca61f"},"schema_version":"1.0","source":{"id":"2402.04076","kind":"arxiv","version":3}},"canonical_sha256":"47cc8c627be17ef89d9a9b29b6361d439a4be06d4ff1506cb17de996400c9388","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"47cc8c627be17ef89d9a9b29b6361d439a4be06d4ff1506cb17de996400c9388","first_computed_at":"2026-07-05T10:02:02.030183Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T10:02:02.030183Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"i5nXm9r3k+Ysu1n/E64VUiYLK1Af8xUif5w9eZn5kmjGdldhFTC56oSTQr6H2VfKOKj93wNdQ4NfnZDwRFtbCQ==","signature_status":"signed_v1","signed_at":"2026-07-05T10:02:02.030598Z","signed_message":"canonical_sha256_bytes"},"source_id":"2402.04076","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:151935b9dd3ac2e9543195d6d6cac273f44facaaa840308101af000ce5a6ca8e","sha256:2e8b18bb25b3044a36ff3e3219083f2099e301977adfc0580489017514bb1b02"],"state_sha256":"e82ddc1043dcf2f532bd67431d1e8252e3db7aca4aacc4454edd32add103d3ce"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"k4BurwjUGA4VKKIeiNNdO50zkoJn5eODsPbwUcECSZuOa9/BFiuo40suDicu5LGjC2ucnn6N5poBGPpILnOxAA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-13T11:30:46.259277Z","bundle_sha256":"61c332874a08c503216d8ef1bb593aec39c596e4ce22e261760f2d0ecb63ead5"}}