{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2025:TGLSYDY4JJ5SZBWCDZY7UQAODL","short_pith_number":"pith:TGLSYDY4","canonical_record":{"source":{"id":"2503.02551","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2025-03-04T12:22:47Z","cross_cats_sorted":[],"title_canon_sha256":"4a2a04d7380d2257456d945ef193ff1f5308ec757b94217d5bac46e6ac98d0f5","abstract_canon_sha256":"7041a33a1a8ec6a27294a32f6f0a89fd4caf3e1a6accc6d38a133dfee15a0609"},"schema_version":"1.0"},"canonical_sha256":"99972c0f1c4a7b2c86c21e71fa400e1af283cda067059d3829895502d3e86eb4","source":{"kind":"arxiv","id":"2503.02551","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2503.02551","created_at":"2026-07-05T10:24:08Z"},{"alias_kind":"arxiv_version","alias_value":"2503.02551v1","created_at":"2026-07-05T10:24:08Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2503.02551","created_at":"2026-07-05T10:24:08Z"},{"alias_kind":"pith_short_12","alias_value":"TGLSYDY4JJ5S","created_at":"2026-07-05T10:24:08Z"},{"alias_kind":"pith_short_16","alias_value":"TGLSYDY4JJ5SZBWC","created_at":"2026-07-05T10:24:08Z"},{"alias_kind":"pith_short_8","alias_value":"TGLSYDY4","created_at":"2026-07-05T10:24:08Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2025:TGLSYDY4JJ5SZBWCDZY7UQAODL","target":"record","payload":{"canonical_record":{"source":{"id":"2503.02551","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2025-03-04T12:22:47Z","cross_cats_sorted":[],"title_canon_sha256":"4a2a04d7380d2257456d945ef193ff1f5308ec757b94217d5bac46e6ac98d0f5","abstract_canon_sha256":"7041a33a1a8ec6a27294a32f6f0a89fd4caf3e1a6accc6d38a133dfee15a0609"},"schema_version":"1.0"},"canonical_sha256":"99972c0f1c4a7b2c86c21e71fa400e1af283cda067059d3829895502d3e86eb4","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T10:24:08.026485Z","signature_b64":"zcPCN2JjpMdmCdTOYUSSFZ7r3A5DAk30uKLVALOJyWy3VkLlACWMQrfeNPyv0FWxelLm9ZzjSPi1SoD9JZ59Dg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"99972c0f1c4a7b2c86c21e71fa400e1af283cda067059d3829895502d3e86eb4","last_reissued_at":"2026-07-05T10:24:08.025852Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T10:24:08.025852Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2503.02551","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":"2026-07-05T10:24:08Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"haR4eadzps2Kxxw40L+BSHg+ysf8jVueAmEbLrnavTH/XpXsRZi0/Y43CPFgnqsOh3KnyWF2Q7kNfef6WhCCAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-09T18:37:01.743443Z"},"content_sha256":"89f99ed2081704d29e45b69eb343b6ba8ae3654ca3828f6e532f690eb012be16","schema_version":"1.0","event_id":"sha256:89f99ed2081704d29e45b69eb343b6ba8ae3654ca3828f6e532f690eb012be16"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2025:TGLSYDY4JJ5SZBWCDZY7UQAODL","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Uniqueness of solutions to elliptic and parabolic equations on metric graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Fabio Punzo, Giulia Meglioli","submitted_at":"2025-03-04T12:22:47Z","abstract_excerpt":"We investigate uniqueness of solutions to certain classes of elliptic and parabolic equations posed on metric graphs. In particular, we address the linear Schr\\\"odinger equation with a potential, and the heat equation with a variable density. We assume suitable growth conditions on the solutions, which are related to the behaviour at infinity of the potential or of the density."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2503.02551","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/2503.02551/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":"2026-07-05T10:24:08Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"CLEfPPjqYyNYS59KmaU8dgyagVbVvE7BFd9Yo/AyLHr4DeKR+3YxdH2lBCfRTarYCCPslaQtwCRqscO2MYgTAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-09T18:37:01.744177Z"},"content_sha256":"143d6bf0341bf5ead4da0ff21c5f33977febe019106d2158d0021aabf44691c7","schema_version":"1.0","event_id":"sha256:143d6bf0341bf5ead4da0ff21c5f33977febe019106d2158d0021aabf44691c7"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/TGLSYDY4JJ5SZBWCDZY7UQAODL/bundle.json","state_url":"https://pith.science/pith/TGLSYDY4JJ5SZBWCDZY7UQAODL/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/TGLSYDY4JJ5SZBWCDZY7UQAODL/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":"2026-08-09T18:37:01Z","links":{"resolver":"https://pith.science/pith/TGLSYDY4JJ5SZBWCDZY7UQAODL","bundle":"https://pith.science/pith/TGLSYDY4JJ5SZBWCDZY7UQAODL/bundle.json","state":"https://pith.science/pith/TGLSYDY4JJ5SZBWCDZY7UQAODL/state.json","well_known_bundle":"https://pith.science/.well-known/pith/TGLSYDY4JJ5SZBWCDZY7UQAODL/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2025:TGLSYDY4JJ5SZBWCDZY7UQAODL","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":"7041a33a1a8ec6a27294a32f6f0a89fd4caf3e1a6accc6d38a133dfee15a0609","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2025-03-04T12:22:47Z","title_canon_sha256":"4a2a04d7380d2257456d945ef193ff1f5308ec757b94217d5bac46e6ac98d0f5"},"schema_version":"1.0","source":{"id":"2503.02551","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2503.02551","created_at":"2026-07-05T10:24:08Z"},{"alias_kind":"arxiv_version","alias_value":"2503.02551v1","created_at":"2026-07-05T10:24:08Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2503.02551","created_at":"2026-07-05T10:24:08Z"},{"alias_kind":"pith_short_12","alias_value":"TGLSYDY4JJ5S","created_at":"2026-07-05T10:24:08Z"},{"alias_kind":"pith_short_16","alias_value":"TGLSYDY4JJ5SZBWC","created_at":"2026-07-05T10:24:08Z"},{"alias_kind":"pith_short_8","alias_value":"TGLSYDY4","created_at":"2026-07-05T10:24:08Z"}],"graph_snapshots":[{"event_id":"sha256:143d6bf0341bf5ead4da0ff21c5f33977febe019106d2158d0021aabf44691c7","target":"graph","created_at":"2026-07-05T10:24:08Z","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/2503.02551/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We investigate uniqueness of solutions to certain classes of elliptic and parabolic equations posed on metric graphs. In particular, we address the linear Schr\\\"odinger equation with a potential, and the heat equation with a variable density. We assume suitable growth conditions on the solutions, which are related to the behaviour at infinity of the potential or of the density.","authors_text":"Fabio Punzo, Giulia Meglioli","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2025-03-04T12:22:47Z","title":"Uniqueness of solutions to elliptic and parabolic equations on metric graphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2503.02551","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":"sha256:89f99ed2081704d29e45b69eb343b6ba8ae3654ca3828f6e532f690eb012be16","target":"record","created_at":"2026-07-05T10:24:08Z","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":"7041a33a1a8ec6a27294a32f6f0a89fd4caf3e1a6accc6d38a133dfee15a0609","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2025-03-04T12:22:47Z","title_canon_sha256":"4a2a04d7380d2257456d945ef193ff1f5308ec757b94217d5bac46e6ac98d0f5"},"schema_version":"1.0","source":{"id":"2503.02551","kind":"arxiv","version":1}},"canonical_sha256":"99972c0f1c4a7b2c86c21e71fa400e1af283cda067059d3829895502d3e86eb4","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"99972c0f1c4a7b2c86c21e71fa400e1af283cda067059d3829895502d3e86eb4","first_computed_at":"2026-07-05T10:24:08.025852Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T10:24:08.025852Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"zcPCN2JjpMdmCdTOYUSSFZ7r3A5DAk30uKLVALOJyWy3VkLlACWMQrfeNPyv0FWxelLm9ZzjSPi1SoD9JZ59Dg==","signature_status":"signed_v1","signed_at":"2026-07-05T10:24:08.026485Z","signed_message":"canonical_sha256_bytes"},"source_id":"2503.02551","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:89f99ed2081704d29e45b69eb343b6ba8ae3654ca3828f6e532f690eb012be16","sha256:143d6bf0341bf5ead4da0ff21c5f33977febe019106d2158d0021aabf44691c7"],"state_sha256":"1704542c6ac19f196b4cc72cc82155f0fda91937cfc44dbd4fe3d82b5fa7af40"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Iuv/t+hjwAybSxAYoSPy1+dc9avjjIzNpHo8QNkzpyaWQ9EJEk5bOWn+EUdQ9M3fz2ZJTBufi1AKIG18GwsOAw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-09T18:37:01.750968Z","bundle_sha256":"e1fa2b6a2d88b067b3b28f1e9832709bf38c462b70e68fb012e3b11fadb4d605"}}