{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2024:SDTWO4P2DLFCZCSJMJ7TNDJLGN","short_pith_number":"pith:SDTWO4P2","canonical_record":{"source":{"id":"2412.16066","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2024-12-20T17:09:55Z","cross_cats_sorted":[],"title_canon_sha256":"6221496e5e05eadbef23fba49827f1a1d8d24d1a60ed958ed06d6f59f0aad744","abstract_canon_sha256":"daf7168c401b995234d0c26dfc825ade297fa863324a0c3ff6357ba83bb1253d"},"schema_version":"1.0"},"canonical_sha256":"90e76771fa1aca2c8a49627f368d2b334828074c4a8e6b4faccfb6d4bc7436c9","source":{"kind":"arxiv","id":"2412.16066","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2412.16066","created_at":"2026-07-05T09:52:34Z"},{"alias_kind":"arxiv_version","alias_value":"2412.16066v1","created_at":"2026-07-05T09:52:34Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2412.16066","created_at":"2026-07-05T09:52:34Z"},{"alias_kind":"pith_short_12","alias_value":"SDTWO4P2DLFC","created_at":"2026-07-05T09:52:34Z"},{"alias_kind":"pith_short_16","alias_value":"SDTWO4P2DLFCZCSJ","created_at":"2026-07-05T09:52:34Z"},{"alias_kind":"pith_short_8","alias_value":"SDTWO4P2","created_at":"2026-07-05T09:52:34Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2024:SDTWO4P2DLFCZCSJMJ7TNDJLGN","target":"record","payload":{"canonical_record":{"source":{"id":"2412.16066","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2024-12-20T17:09:55Z","cross_cats_sorted":[],"title_canon_sha256":"6221496e5e05eadbef23fba49827f1a1d8d24d1a60ed958ed06d6f59f0aad744","abstract_canon_sha256":"daf7168c401b995234d0c26dfc825ade297fa863324a0c3ff6357ba83bb1253d"},"schema_version":"1.0"},"canonical_sha256":"90e76771fa1aca2c8a49627f368d2b334828074c4a8e6b4faccfb6d4bc7436c9","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T09:52:34.001905Z","signature_b64":"L5QCGlua7OlU8seZRZOdWaFveic2k9m+p08ZTjtzkc0i7jHBqII1PhM++0HBO6g+qIC7gG0GbdQCmufiz822Dw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"90e76771fa1aca2c8a49627f368d2b334828074c4a8e6b4faccfb6d4bc7436c9","last_reissued_at":"2026-07-05T09:52:34.001389Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T09:52:34.001389Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2412.16066","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":"2026-07-05T09:52:34Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"2qQtWc9iN3uxAwzUMH0wldV00iUqrMQ8rQVW8JiolOlloxafG7F2F/03ZdbGuwth5ctA6RY60piWc0eaJRJdAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-07T19:33:16.640379Z"},"content_sha256":"58bcc051346f9e0f926b0cd111ac6e43591bdfb59d85edfa15c1ebfd38a3634a","schema_version":"1.0","event_id":"sha256:58bcc051346f9e0f926b0cd111ac6e43591bdfb59d85edfa15c1ebfd38a3634a"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2024:SDTWO4P2DLFCZCSJMJ7TNDJLGN","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Generic regularity of free boundaries in the obstacle problem for the fractional Laplacian","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Matteo Carducci, Roberto Colombo","submitted_at":"2024-12-20T17:09:55Z","abstract_excerpt":"We establish generic regularity results of free boundaries for solutions of the obstacle problem for the fractional Laplacian $(-\\Delta)^s$. We prove that, for almost every obstacle, the free boundary contains only regular points up to dimension $3$, for every $s\\in(0,1)$. To do so, we extend some results on the fine structure of the free boundary to the case $s\\in (0,1)$ and general non-zero obstacle, including a blow-up analysis at points with frequency $2m+2s$, and we prove new explicit uniform frequency gaps for solutions of the fractional obstacle problem."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2412.16066","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/2412.16066/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":"2026-07-05T09:52:34Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"7V7MVMCQowCdiVlU+Goy7EYenHSez/xUwDP22jap7J6awnAqOF7xgSHSWWTc0VrPdHPvtc/vYRbanwYxuj0zAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-07T19:33:16.640935Z"},"content_sha256":"832f5c058dc3a337187efddfd2202bcc7195ea5a7722ee3f2bb398b4ab954c65","schema_version":"1.0","event_id":"sha256:832f5c058dc3a337187efddfd2202bcc7195ea5a7722ee3f2bb398b4ab954c65"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/SDTWO4P2DLFCZCSJMJ7TNDJLGN/bundle.json","state_url":"https://pith.science/pith/SDTWO4P2DLFCZCSJMJ7TNDJLGN/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/SDTWO4P2DLFCZCSJMJ7TNDJLGN/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":"2026-08-07T19:33:16Z","links":{"resolver":"https://pith.science/pith/SDTWO4P2DLFCZCSJMJ7TNDJLGN","bundle":"https://pith.science/pith/SDTWO4P2DLFCZCSJMJ7TNDJLGN/bundle.json","state":"https://pith.science/pith/SDTWO4P2DLFCZCSJMJ7TNDJLGN/state.json","well_known_bundle":"https://pith.science/.well-known/pith/SDTWO4P2DLFCZCSJMJ7TNDJLGN/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2024:SDTWO4P2DLFCZCSJMJ7TNDJLGN","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":"daf7168c401b995234d0c26dfc825ade297fa863324a0c3ff6357ba83bb1253d","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2024-12-20T17:09:55Z","title_canon_sha256":"6221496e5e05eadbef23fba49827f1a1d8d24d1a60ed958ed06d6f59f0aad744"},"schema_version":"1.0","source":{"id":"2412.16066","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2412.16066","created_at":"2026-07-05T09:52:34Z"},{"alias_kind":"arxiv_version","alias_value":"2412.16066v1","created_at":"2026-07-05T09:52:34Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2412.16066","created_at":"2026-07-05T09:52:34Z"},{"alias_kind":"pith_short_12","alias_value":"SDTWO4P2DLFC","created_at":"2026-07-05T09:52:34Z"},{"alias_kind":"pith_short_16","alias_value":"SDTWO4P2DLFCZCSJ","created_at":"2026-07-05T09:52:34Z"},{"alias_kind":"pith_short_8","alias_value":"SDTWO4P2","created_at":"2026-07-05T09:52:34Z"}],"graph_snapshots":[{"event_id":"sha256:832f5c058dc3a337187efddfd2202bcc7195ea5a7722ee3f2bb398b4ab954c65","target":"graph","created_at":"2026-07-05T09:52:34Z","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/2412.16066/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We establish generic regularity results of free boundaries for solutions of the obstacle problem for the fractional Laplacian $(-\\Delta)^s$. We prove that, for almost every obstacle, the free boundary contains only regular points up to dimension $3$, for every $s\\in(0,1)$. To do so, we extend some results on the fine structure of the free boundary to the case $s\\in (0,1)$ and general non-zero obstacle, including a blow-up analysis at points with frequency $2m+2s$, and we prove new explicit uniform frequency gaps for solutions of the fractional obstacle problem.","authors_text":"Matteo Carducci, Roberto Colombo","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2024-12-20T17:09:55Z","title":"Generic regularity of free boundaries in the obstacle problem for the fractional Laplacian"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2412.16066","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":"sha256:58bcc051346f9e0f926b0cd111ac6e43591bdfb59d85edfa15c1ebfd38a3634a","target":"record","created_at":"2026-07-05T09:52:34Z","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":"daf7168c401b995234d0c26dfc825ade297fa863324a0c3ff6357ba83bb1253d","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2024-12-20T17:09:55Z","title_canon_sha256":"6221496e5e05eadbef23fba49827f1a1d8d24d1a60ed958ed06d6f59f0aad744"},"schema_version":"1.0","source":{"id":"2412.16066","kind":"arxiv","version":1}},"canonical_sha256":"90e76771fa1aca2c8a49627f368d2b334828074c4a8e6b4faccfb6d4bc7436c9","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"90e76771fa1aca2c8a49627f368d2b334828074c4a8e6b4faccfb6d4bc7436c9","first_computed_at":"2026-07-05T09:52:34.001389Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T09:52:34.001389Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"L5QCGlua7OlU8seZRZOdWaFveic2k9m+p08ZTjtzkc0i7jHBqII1PhM++0HBO6g+qIC7gG0GbdQCmufiz822Dw==","signature_status":"signed_v1","signed_at":"2026-07-05T09:52:34.001905Z","signed_message":"canonical_sha256_bytes"},"source_id":"2412.16066","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:58bcc051346f9e0f926b0cd111ac6e43591bdfb59d85edfa15c1ebfd38a3634a","sha256:832f5c058dc3a337187efddfd2202bcc7195ea5a7722ee3f2bb398b4ab954c65"],"state_sha256":"7cbd2104f8d2e6a048ef778118f535910e47e7ee28b211953d2fa6233545d7f1"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"1Xd6xdWxSKwg34q9DdTIrvJ7lHaeb+bOYbsrw4jPg0Wn/wZrNy4M6M84QpXsFik7eh1gesDt3CutRzjJfNGyAA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-07T19:33:16.644942Z","bundle_sha256":"55b09ab1e0037d469de96e9d22bf6af57134c2b4bf1adf9ea161c618c580ed64"}}