{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2020:VQWOODRCUA7XW46E264LDWZRFR","short_pith_number":"pith:VQWOODRC","canonical_record":{"source":{"id":"2001.05477","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2020-01-15T18:45:06Z","cross_cats_sorted":["math-ph","math.MP"],"title_canon_sha256":"31e1e3643d1e36bcf4924bd9802e41d2ae6edff2677abf153a5664e8e3af3991","abstract_canon_sha256":"3381531d1fb3046baf7c1aeb4e1b970bc0ec96e1eb952736185c8ebeebb08c48"},"schema_version":"1.0"},"canonical_sha256":"ac2ce70e22a03f7b73c4d7b8b1db312c72502601aeb536b57ad47abfe78c6438","source":{"kind":"arxiv","id":"2001.05477","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2001.05477","created_at":"2026-07-05T08:19:06Z"},{"alias_kind":"arxiv_version","alias_value":"2001.05477v2","created_at":"2026-07-05T08:19:06Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2001.05477","created_at":"2026-07-05T08:19:06Z"},{"alias_kind":"pith_short_12","alias_value":"VQWOODRCUA7X","created_at":"2026-07-05T08:19:06Z"},{"alias_kind":"pith_short_16","alias_value":"VQWOODRCUA7XW46E","created_at":"2026-07-05T08:19:06Z"},{"alias_kind":"pith_short_8","alias_value":"VQWOODRC","created_at":"2026-07-05T08:19:06Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2020:VQWOODRCUA7XW46E264LDWZRFR","target":"record","payload":{"canonical_record":{"source":{"id":"2001.05477","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2020-01-15T18:45:06Z","cross_cats_sorted":["math-ph","math.MP"],"title_canon_sha256":"31e1e3643d1e36bcf4924bd9802e41d2ae6edff2677abf153a5664e8e3af3991","abstract_canon_sha256":"3381531d1fb3046baf7c1aeb4e1b970bc0ec96e1eb952736185c8ebeebb08c48"},"schema_version":"1.0"},"canonical_sha256":"ac2ce70e22a03f7b73c4d7b8b1db312c72502601aeb536b57ad47abfe78c6438","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T08:19:06.274767Z","signature_b64":"begi4LKWznx9OwuXtDl5NQ1axCXK6FpkvUlAYTuyxuLGGbEW4as6hpb/IShEgUaheVgoAu/nd/WEyfbzM0BHDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ac2ce70e22a03f7b73c4d7b8b1db312c72502601aeb536b57ad47abfe78c6438","last_reissued_at":"2026-07-05T08:19:06.274295Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T08:19:06.274295Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2001.05477","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T08:19:06Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"SqhYX4sxb/+xib4ys8UUP9rrFzJ/sUYfGVSNiZfCC5bI+YfJMvnjshj+5h5o3uvgswUQb/aqRnBOziq50aWWDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-06T13:01:35.810830Z"},"content_sha256":"f2dc8bc64033d1f0d0de6a385b1db56c9cf3063816177a2806b1315266b4bbad","schema_version":"1.0","event_id":"sha256:f2dc8bc64033d1f0d0de6a385b1db56c9cf3063816177a2806b1315266b4bbad"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2020:VQWOODRCUA7XW46E264LDWZRFR","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Blow-up criteria below scaling for defocusing energy-supercritical NLS and quantitative global scattering bounds","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.MP"],"primary_cat":"math.AP","authors_text":"Aynur Bulut","submitted_at":"2020-01-15T18:45:06Z","abstract_excerpt":"We establish quantitative blow-up criteria below the scaling threshold for radially symmetric solutions to the defocusing nonlinear Schr\\\"odinger equation with nonlinearity $|u|^6u$. This provides to our knowledge the first generic results distinguishing potential blow-up solutions of the defocusing equation from many of the known examples of blow-up in the focusing case. Our main tool is a quantitative version of a result showing that uniform bounds on $L^2$-based critical Sobolev norms imply scattering estimates.\n  As another application of our techniques, we establish a variant which allows"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2001.05477","kind":"arxiv","version":2},"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/2001.05477/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":"2026-07-05T08:19:06Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"jlfNiW30l43RH2Y661C36RPBhqlExGdQPsGBlTNESWfE52nbDkcj1EhLSXFWpDxu5GUrW3+f9kEJLtQKbkunAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-06T13:01:35.811447Z"},"content_sha256":"de11233551a9b0619afab675e8397dfb066723001e19b2f86064cc1b648904d8","schema_version":"1.0","event_id":"sha256:de11233551a9b0619afab675e8397dfb066723001e19b2f86064cc1b648904d8"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/VQWOODRCUA7XW46E264LDWZRFR/bundle.json","state_url":"https://pith.science/pith/VQWOODRCUA7XW46E264LDWZRFR/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/VQWOODRCUA7XW46E264LDWZRFR/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":"2026-08-06T13:01:35Z","links":{"resolver":"https://pith.science/pith/VQWOODRCUA7XW46E264LDWZRFR","bundle":"https://pith.science/pith/VQWOODRCUA7XW46E264LDWZRFR/bundle.json","state":"https://pith.science/pith/VQWOODRCUA7XW46E264LDWZRFR/state.json","well_known_bundle":"https://pith.science/.well-known/pith/VQWOODRCUA7XW46E264LDWZRFR/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2020:VQWOODRCUA7XW46E264LDWZRFR","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":"3381531d1fb3046baf7c1aeb4e1b970bc0ec96e1eb952736185c8ebeebb08c48","cross_cats_sorted":["math-ph","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2020-01-15T18:45:06Z","title_canon_sha256":"31e1e3643d1e36bcf4924bd9802e41d2ae6edff2677abf153a5664e8e3af3991"},"schema_version":"1.0","source":{"id":"2001.05477","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2001.05477","created_at":"2026-07-05T08:19:06Z"},{"alias_kind":"arxiv_version","alias_value":"2001.05477v2","created_at":"2026-07-05T08:19:06Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2001.05477","created_at":"2026-07-05T08:19:06Z"},{"alias_kind":"pith_short_12","alias_value":"VQWOODRCUA7X","created_at":"2026-07-05T08:19:06Z"},{"alias_kind":"pith_short_16","alias_value":"VQWOODRCUA7XW46E","created_at":"2026-07-05T08:19:06Z"},{"alias_kind":"pith_short_8","alias_value":"VQWOODRC","created_at":"2026-07-05T08:19:06Z"}],"graph_snapshots":[{"event_id":"sha256:de11233551a9b0619afab675e8397dfb066723001e19b2f86064cc1b648904d8","target":"graph","created_at":"2026-07-05T08:19:06Z","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/2001.05477/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We establish quantitative blow-up criteria below the scaling threshold for radially symmetric solutions to the defocusing nonlinear Schr\\\"odinger equation with nonlinearity $|u|^6u$. This provides to our knowledge the first generic results distinguishing potential blow-up solutions of the defocusing equation from many of the known examples of blow-up in the focusing case. Our main tool is a quantitative version of a result showing that uniform bounds on $L^2$-based critical Sobolev norms imply scattering estimates.\n  As another application of our techniques, we establish a variant which allows","authors_text":"Aynur Bulut","cross_cats":["math-ph","math.MP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2020-01-15T18:45:06Z","title":"Blow-up criteria below scaling for defocusing energy-supercritical NLS and quantitative global scattering bounds"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2001.05477","kind":"arxiv","version":2},"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:f2dc8bc64033d1f0d0de6a385b1db56c9cf3063816177a2806b1315266b4bbad","target":"record","created_at":"2026-07-05T08:19:06Z","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":"3381531d1fb3046baf7c1aeb4e1b970bc0ec96e1eb952736185c8ebeebb08c48","cross_cats_sorted":["math-ph","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2020-01-15T18:45:06Z","title_canon_sha256":"31e1e3643d1e36bcf4924bd9802e41d2ae6edff2677abf153a5664e8e3af3991"},"schema_version":"1.0","source":{"id":"2001.05477","kind":"arxiv","version":2}},"canonical_sha256":"ac2ce70e22a03f7b73c4d7b8b1db312c72502601aeb536b57ad47abfe78c6438","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"ac2ce70e22a03f7b73c4d7b8b1db312c72502601aeb536b57ad47abfe78c6438","first_computed_at":"2026-07-05T08:19:06.274295Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T08:19:06.274295Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"begi4LKWznx9OwuXtDl5NQ1axCXK6FpkvUlAYTuyxuLGGbEW4as6hpb/IShEgUaheVgoAu/nd/WEyfbzM0BHDg==","signature_status":"signed_v1","signed_at":"2026-07-05T08:19:06.274767Z","signed_message":"canonical_sha256_bytes"},"source_id":"2001.05477","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:f2dc8bc64033d1f0d0de6a385b1db56c9cf3063816177a2806b1315266b4bbad","sha256:de11233551a9b0619afab675e8397dfb066723001e19b2f86064cc1b648904d8"],"state_sha256":"d94999105e727bb71158df7ce06209f7d2f0ab10c8c7edca1ccaa0b8392d9935"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"N9/SAZOeU5kERfZ8CQ2N1+nQQpiDKuiVCrK0n0zKuInIcrV567TLeHrwBRt+V/snvbuCHHe2+M5fyvpNQeA3BA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-06T13:01:35.816413Z","bundle_sha256":"6e419b179aa7cfdcb81547ca7b52146afd559feb21ca1594c29e68616c64609d"}}