{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:GMNPU3FCQPHVOXU3BXKCQANDHN","short_pith_number":"pith:GMNPU3FC","canonical_record":{"source":{"id":"2607.04575","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AP","submitted_at":"2026-07-06T01:03:03Z","cross_cats_sorted":[],"title_canon_sha256":"136ccca29e1d2c1b1c805442de41adbb5f62bcfe0d90ca00e39ba65415b85b95","abstract_canon_sha256":"49143a6111eeb9f2e5f2f61d832875bc37fca4331c0ef74385734d0013204c0b"},"schema_version":"1.0"},"canonical_sha256":"331afa6ca283cf575e9b0dd42801a33b7399ff9866a2381f57885621ec13ac1b","source":{"kind":"arxiv","id":"2607.04575","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2607.04575","created_at":"2026-07-07T02:19:25Z"},{"alias_kind":"arxiv_version","alias_value":"2607.04575v1","created_at":"2026-07-07T02:19:25Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.04575","created_at":"2026-07-07T02:19:25Z"},{"alias_kind":"pith_short_12","alias_value":"GMNPU3FCQPHV","created_at":"2026-07-07T02:19:25Z"},{"alias_kind":"pith_short_16","alias_value":"GMNPU3FCQPHVOXU3","created_at":"2026-07-07T02:19:25Z"},{"alias_kind":"pith_short_8","alias_value":"GMNPU3FC","created_at":"2026-07-07T02:19:25Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:GMNPU3FCQPHVOXU3BXKCQANDHN","target":"record","payload":{"canonical_record":{"source":{"id":"2607.04575","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AP","submitted_at":"2026-07-06T01:03:03Z","cross_cats_sorted":[],"title_canon_sha256":"136ccca29e1d2c1b1c805442de41adbb5f62bcfe0d90ca00e39ba65415b85b95","abstract_canon_sha256":"49143a6111eeb9f2e5f2f61d832875bc37fca4331c0ef74385734d0013204c0b"},"schema_version":"1.0"},"canonical_sha256":"331afa6ca283cf575e9b0dd42801a33b7399ff9866a2381f57885621ec13ac1b","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-07T02:19:25.541925Z","signature_b64":"dpwIrcMXU7fPD+JTYR9xaNiqRou7w9v9FHFXNEMt4nSU63mx2tVltIqlR1HfdA9akvfz4o9ZT2d/MBout8XpAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"331afa6ca283cf575e9b0dd42801a33b7399ff9866a2381f57885621ec13ac1b","last_reissued_at":"2026-07-07T02:19:25.541295Z","signature_status":"signed_v1","first_computed_at":"2026-07-07T02:19:25.541295Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2607.04575","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":"2026-07-07T02:19:25Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"xMXFEcbPz90joXLtFacqcvQwInEiqWv5EVcn8m7ZRkzs5jA2ZkhODETVe2jNhtWUUzE1Y8gMpuyblRPPervZBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-21T09:34:28.035364Z"},"content_sha256":"bb038cafa62c8d195addf9ec07034002aea1c56c7f4adbfd3476cabd47ba42ef","schema_version":"1.0","event_id":"sha256:bb038cafa62c8d195addf9ec07034002aea1c56c7f4adbfd3476cabd47ba42ef"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:GMNPU3FCQPHVOXU3BXKCQANDHN","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Global existence of small data solutions to 3-D semilinear Euler-Poisson-Darboux equations","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Huicheng Yin, QianQian Li","submitted_at":"2026-07-06T01:03:03Z","abstract_excerpt":"There is an interesting open question: for $n$-D ($n\\ge 1$) semilinear Euler-Poisson-Darboux equation $\\partial_t^2u-\\Delta u+\\frac{\\mu}{t}\\partial_tu=|u|^p$, where $t\\ge 1$, $p>1$ and $\\mu>0$, the global small data weak solution $u$ will exist when $p>p_{crit}(n,\\mu)=\\max\\{p_s(n+\\mu), p_f(n)\\}$ with the Strauss exponent $p_{s}(n+\\mu)=\\frac{n+\\mu+1+\\sqrt{(n+\\mu)^2+10(n+\\mu)-7}}{2(n+\\mu-1)}$ and the Fujita exponent $p_f(n)=1+\\frac{2}{n}$. The blowup of weak solution $u$ has been shown when $1<p\\le p_{crit}(n,\\mu)$ meanwhile this open question has been solved for $n=1,2$. In the present paper, w"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.04575","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/2607.04575/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":"2026-07-07T02:19:25Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"2BdS9V+6K8k5z5clPJV7z5sT7L1+QvOzMGt6jaUxEHhwU5IHK8DEJW9FgJBQQjhUgoxPJ24vugdDixWoZKk9Aw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-21T09:34:28.035857Z"},"content_sha256":"4337142faa09674e3869f1bac6930a51bbdf4cb92fd9a153a095f128ee50d20f","schema_version":"1.0","event_id":"sha256:4337142faa09674e3869f1bac6930a51bbdf4cb92fd9a153a095f128ee50d20f"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/GMNPU3FCQPHVOXU3BXKCQANDHN/bundle.json","state_url":"https://pith.science/pith/GMNPU3FCQPHVOXU3BXKCQANDHN/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/GMNPU3FCQPHVOXU3BXKCQANDHN/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":"2026-08-21T09:34:28Z","links":{"resolver":"https://pith.science/pith/GMNPU3FCQPHVOXU3BXKCQANDHN","bundle":"https://pith.science/pith/GMNPU3FCQPHVOXU3BXKCQANDHN/bundle.json","state":"https://pith.science/pith/GMNPU3FCQPHVOXU3BXKCQANDHN/state.json","well_known_bundle":"https://pith.science/.well-known/pith/GMNPU3FCQPHVOXU3BXKCQANDHN/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:GMNPU3FCQPHVOXU3BXKCQANDHN","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":"49143a6111eeb9f2e5f2f61d832875bc37fca4331c0ef74385734d0013204c0b","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AP","submitted_at":"2026-07-06T01:03:03Z","title_canon_sha256":"136ccca29e1d2c1b1c805442de41adbb5f62bcfe0d90ca00e39ba65415b85b95"},"schema_version":"1.0","source":{"id":"2607.04575","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2607.04575","created_at":"2026-07-07T02:19:25Z"},{"alias_kind":"arxiv_version","alias_value":"2607.04575v1","created_at":"2026-07-07T02:19:25Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.04575","created_at":"2026-07-07T02:19:25Z"},{"alias_kind":"pith_short_12","alias_value":"GMNPU3FCQPHV","created_at":"2026-07-07T02:19:25Z"},{"alias_kind":"pith_short_16","alias_value":"GMNPU3FCQPHVOXU3","created_at":"2026-07-07T02:19:25Z"},{"alias_kind":"pith_short_8","alias_value":"GMNPU3FC","created_at":"2026-07-07T02:19:25Z"}],"graph_snapshots":[{"event_id":"sha256:4337142faa09674e3869f1bac6930a51bbdf4cb92fd9a153a095f128ee50d20f","target":"graph","created_at":"2026-07-07T02:19:25Z","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/2607.04575/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"There is an interesting open question: for $n$-D ($n\\ge 1$) semilinear Euler-Poisson-Darboux equation $\\partial_t^2u-\\Delta u+\\frac{\\mu}{t}\\partial_tu=|u|^p$, where $t\\ge 1$, $p>1$ and $\\mu>0$, the global small data weak solution $u$ will exist when $p>p_{crit}(n,\\mu)=\\max\\{p_s(n+\\mu), p_f(n)\\}$ with the Strauss exponent $p_{s}(n+\\mu)=\\frac{n+\\mu+1+\\sqrt{(n+\\mu)^2+10(n+\\mu)-7}}{2(n+\\mu-1)}$ and the Fujita exponent $p_f(n)=1+\\frac{2}{n}$. The blowup of weak solution $u$ has been shown when $1<p\\le p_{crit}(n,\\mu)$ meanwhile this open question has been solved for $n=1,2$. In the present paper, w","authors_text":"Huicheng Yin, QianQian Li","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AP","submitted_at":"2026-07-06T01:03:03Z","title":"Global existence of small data solutions to 3-D semilinear Euler-Poisson-Darboux equations"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.04575","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:bb038cafa62c8d195addf9ec07034002aea1c56c7f4adbfd3476cabd47ba42ef","target":"record","created_at":"2026-07-07T02:19:25Z","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":"49143a6111eeb9f2e5f2f61d832875bc37fca4331c0ef74385734d0013204c0b","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AP","submitted_at":"2026-07-06T01:03:03Z","title_canon_sha256":"136ccca29e1d2c1b1c805442de41adbb5f62bcfe0d90ca00e39ba65415b85b95"},"schema_version":"1.0","source":{"id":"2607.04575","kind":"arxiv","version":1}},"canonical_sha256":"331afa6ca283cf575e9b0dd42801a33b7399ff9866a2381f57885621ec13ac1b","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"331afa6ca283cf575e9b0dd42801a33b7399ff9866a2381f57885621ec13ac1b","first_computed_at":"2026-07-07T02:19:25.541295Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-07T02:19:25.541295Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"dpwIrcMXU7fPD+JTYR9xaNiqRou7w9v9FHFXNEMt4nSU63mx2tVltIqlR1HfdA9akvfz4o9ZT2d/MBout8XpAw==","signature_status":"signed_v1","signed_at":"2026-07-07T02:19:25.541925Z","signed_message":"canonical_sha256_bytes"},"source_id":"2607.04575","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:bb038cafa62c8d195addf9ec07034002aea1c56c7f4adbfd3476cabd47ba42ef","sha256:4337142faa09674e3869f1bac6930a51bbdf4cb92fd9a153a095f128ee50d20f"],"state_sha256":"ba76b5112e6ecb9e65bd579f6dff2f9b0913c6d49cea4d1fb06f9af19ac44854"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"6a2tJCLN8K+8Wh2SBG53wUdLdpbFQN0KG/iUKteGujY6C77ySHdwbltWs7sJJpFP154uWPuqd84Da+TZRFSnBw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-21T09:34:28.040574Z","bundle_sha256":"f5c7d9cb9fec1310e0673cc0a0cb03241920ba177974d9dbec058251e06fe97a"}}