{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2025:GWT5ACTEZXXQEDTSRCU5FWEE2D","short_pith_number":"pith:GWT5ACTE","canonical_record":{"source":{"id":"2507.15103","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"math.NA","submitted_at":"2025-07-20T19:54:43Z","cross_cats_sorted":["cs.NA"],"title_canon_sha256":"90d2d0b7688be4a9010a22b4ff6204fd46063a86db3b61ddff6f792fff7f16b2","abstract_canon_sha256":"5521736325093e87cff8bc5512e3d41fd3d9c7e097469cb90f4962e92f19922c"},"schema_version":"1.0"},"canonical_sha256":"35a7d00a64cdef020e7288a9d2d884d0e77f26fe03f7714fc7b354174c60f61a","source":{"kind":"arxiv","id":"2507.15103","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2507.15103","created_at":"2026-07-05T11:42:33Z"},{"alias_kind":"arxiv_version","alias_value":"2507.15103v2","created_at":"2026-07-05T11:42:33Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2507.15103","created_at":"2026-07-05T11:42:33Z"},{"alias_kind":"pith_short_12","alias_value":"GWT5ACTEZXXQ","created_at":"2026-07-05T11:42:33Z"},{"alias_kind":"pith_short_16","alias_value":"GWT5ACTEZXXQEDTS","created_at":"2026-07-05T11:42:33Z"},{"alias_kind":"pith_short_8","alias_value":"GWT5ACTE","created_at":"2026-07-05T11:42:33Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2025:GWT5ACTEZXXQEDTSRCU5FWEE2D","target":"record","payload":{"canonical_record":{"source":{"id":"2507.15103","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"math.NA","submitted_at":"2025-07-20T19:54:43Z","cross_cats_sorted":["cs.NA"],"title_canon_sha256":"90d2d0b7688be4a9010a22b4ff6204fd46063a86db3b61ddff6f792fff7f16b2","abstract_canon_sha256":"5521736325093e87cff8bc5512e3d41fd3d9c7e097469cb90f4962e92f19922c"},"schema_version":"1.0"},"canonical_sha256":"35a7d00a64cdef020e7288a9d2d884d0e77f26fe03f7714fc7b354174c60f61a","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T11:42:33.139028Z","signature_b64":"zygG1QkFBun2UqM6gdv92KAuP9t3WXFeTufp4D9Yg2uetU2KJQy01Y2ptsKmbvLnC9LYQ4G02rjB2AGPBB6nBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"35a7d00a64cdef020e7288a9d2d884d0e77f26fe03f7714fc7b354174c60f61a","last_reissued_at":"2026-07-05T11:42:33.138535Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T11:42:33.138535Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2507.15103","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":"2026-07-05T11:42:33Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"93sRy66XSYBTQmsp4sLksUl6eGAR4K3ZEPpAZYF9/Pcke6OvIyTNKQz1F/LGgs8nIgiqlbIbFDGhCxJSd5thDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-13T16:29:12.220821Z"},"content_sha256":"2e675d527379994e263bff6dc1db1bfa53747e81372366a5236a1622c21d772f","schema_version":"1.0","event_id":"sha256:2e675d527379994e263bff6dc1db1bfa53747e81372366a5236a1622c21d772f"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2025:GWT5ACTEZXXQEDTSRCU5FWEE2D","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Analysis of fully discrete Crank-Nicolson finite element methods for a stochastic Keller-Segel chemotaxis system with gradient-type multiplicative noise","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","headline":"","cross_cats":["cs.NA"],"primary_cat":"math.NA","authors_text":"Liet Vo","submitted_at":"2025-07-20T19:54:43Z","abstract_excerpt":"We develop and analyze numerical methods for a stochastic Keller-Segel system perturbed by Stratonovich noise, which models chemotactic behavior under randomly fluctuating environmental conditions. The proposed fully discrete scheme couples a Crank-Nicolson time discretization with a splitting mixed finite element method in space. We rigorously prove the stability of the numerical scheme and establish strong convergence rates of order $O(k^{1/2} + k^{-1/2}h^2)$, where $k$ and $h$ denote the time and spatial step sizes, respectively. Notably, the presence of stochastic forcing leads to an inver"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2507.15103","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/2507.15103/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":"2026-07-05T11:42:33Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"S2OvH5ei4cW7I5x7tJmS7JuiusmQlhyCO/PkMXbIu2aNMDDMvV+Z885okAzDsjtOsKqyEvsnwaw8XC0ojcciCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-13T16:29:12.221213Z"},"content_sha256":"ed5db4c40a79481bc0a15707b59b3ed745bcdde8f11548c4d145b448aeb91986","schema_version":"1.0","event_id":"sha256:ed5db4c40a79481bc0a15707b59b3ed745bcdde8f11548c4d145b448aeb91986"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/GWT5ACTEZXXQEDTSRCU5FWEE2D/bundle.json","state_url":"https://pith.science/pith/GWT5ACTEZXXQEDTSRCU5FWEE2D/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/GWT5ACTEZXXQEDTSRCU5FWEE2D/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":"2026-08-13T16:29:12Z","links":{"resolver":"https://pith.science/pith/GWT5ACTEZXXQEDTSRCU5FWEE2D","bundle":"https://pith.science/pith/GWT5ACTEZXXQEDTSRCU5FWEE2D/bundle.json","state":"https://pith.science/pith/GWT5ACTEZXXQEDTSRCU5FWEE2D/state.json","well_known_bundle":"https://pith.science/.well-known/pith/GWT5ACTEZXXQEDTSRCU5FWEE2D/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2025:GWT5ACTEZXXQEDTSRCU5FWEE2D","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":"5521736325093e87cff8bc5512e3d41fd3d9c7e097469cb90f4962e92f19922c","cross_cats_sorted":["cs.NA"],"license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"math.NA","submitted_at":"2025-07-20T19:54:43Z","title_canon_sha256":"90d2d0b7688be4a9010a22b4ff6204fd46063a86db3b61ddff6f792fff7f16b2"},"schema_version":"1.0","source":{"id":"2507.15103","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2507.15103","created_at":"2026-07-05T11:42:33Z"},{"alias_kind":"arxiv_version","alias_value":"2507.15103v2","created_at":"2026-07-05T11:42:33Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2507.15103","created_at":"2026-07-05T11:42:33Z"},{"alias_kind":"pith_short_12","alias_value":"GWT5ACTEZXXQ","created_at":"2026-07-05T11:42:33Z"},{"alias_kind":"pith_short_16","alias_value":"GWT5ACTEZXXQEDTS","created_at":"2026-07-05T11:42:33Z"},{"alias_kind":"pith_short_8","alias_value":"GWT5ACTE","created_at":"2026-07-05T11:42:33Z"}],"graph_snapshots":[{"event_id":"sha256:ed5db4c40a79481bc0a15707b59b3ed745bcdde8f11548c4d145b448aeb91986","target":"graph","created_at":"2026-07-05T11:42:33Z","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/2507.15103/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We develop and analyze numerical methods for a stochastic Keller-Segel system perturbed by Stratonovich noise, which models chemotactic behavior under randomly fluctuating environmental conditions. The proposed fully discrete scheme couples a Crank-Nicolson time discretization with a splitting mixed finite element method in space. We rigorously prove the stability of the numerical scheme and establish strong convergence rates of order $O(k^{1/2} + k^{-1/2}h^2)$, where $k$ and $h$ denote the time and spatial step sizes, respectively. Notably, the presence of stochastic forcing leads to an inver","authors_text":"Liet Vo","cross_cats":["cs.NA"],"headline":"","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"math.NA","submitted_at":"2025-07-20T19:54:43Z","title":"Analysis of fully discrete Crank-Nicolson finite element methods for a stochastic Keller-Segel chemotaxis system with gradient-type multiplicative noise"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2507.15103","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":"sha256:2e675d527379994e263bff6dc1db1bfa53747e81372366a5236a1622c21d772f","target":"record","created_at":"2026-07-05T11:42:33Z","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":"5521736325093e87cff8bc5512e3d41fd3d9c7e097469cb90f4962e92f19922c","cross_cats_sorted":["cs.NA"],"license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"math.NA","submitted_at":"2025-07-20T19:54:43Z","title_canon_sha256":"90d2d0b7688be4a9010a22b4ff6204fd46063a86db3b61ddff6f792fff7f16b2"},"schema_version":"1.0","source":{"id":"2507.15103","kind":"arxiv","version":2}},"canonical_sha256":"35a7d00a64cdef020e7288a9d2d884d0e77f26fe03f7714fc7b354174c60f61a","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"35a7d00a64cdef020e7288a9d2d884d0e77f26fe03f7714fc7b354174c60f61a","first_computed_at":"2026-07-05T11:42:33.138535Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T11:42:33.138535Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"zygG1QkFBun2UqM6gdv92KAuP9t3WXFeTufp4D9Yg2uetU2KJQy01Y2ptsKmbvLnC9LYQ4G02rjB2AGPBB6nBw==","signature_status":"signed_v1","signed_at":"2026-07-05T11:42:33.139028Z","signed_message":"canonical_sha256_bytes"},"source_id":"2507.15103","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:2e675d527379994e263bff6dc1db1bfa53747e81372366a5236a1622c21d772f","sha256:ed5db4c40a79481bc0a15707b59b3ed745bcdde8f11548c4d145b448aeb91986"],"state_sha256":"5cdcb3045b228efebf9f1c585c4d877a6ba3f2d10d726a83c4c99b7b83c1c9f6"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"prkdgHqoOPTQOIrZxRvcxa3icwIGHEF+XbZgzrhaPK71LpD+GnvTBrpw/iTNENv2HE8nfTvhjLWtOZ3a+g7nAw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-13T16:29:12.226585Z","bundle_sha256":"7b875f0bb0f3c2329536f043a48c0a8deb44a5d359b6c0e6a00926c8dfd3aea4"}}