{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2024:FKQM6CP47K3OYWJ2ZK5WEJA26H","short_pith_number":"pith:FKQM6CP4","canonical_record":{"source":{"id":"2412.15145","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2024-12-19T18:27:07Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"5510cf6bdd0ba74fdb12e5ae73763d60e07ed09b27e10e2fa3caecdb152c8486","abstract_canon_sha256":"394712e55725a3f9e14de0e4f901e65d10c8a60577981006248563c6e29ba646"},"schema_version":"1.0"},"canonical_sha256":"2aa0cf09fcfab6ec593acabb62241af1d1a41a989ff30af93f140d0180a67b80","source":{"kind":"arxiv","id":"2412.15145","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2412.15145","created_at":"2026-07-05T09:51:58Z"},{"alias_kind":"arxiv_version","alias_value":"2412.15145v1","created_at":"2026-07-05T09:51:58Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2412.15145","created_at":"2026-07-05T09:51:58Z"},{"alias_kind":"pith_short_12","alias_value":"FKQM6CP47K3O","created_at":"2026-07-05T09:51:58Z"},{"alias_kind":"pith_short_16","alias_value":"FKQM6CP47K3OYWJ2","created_at":"2026-07-05T09:51:58Z"},{"alias_kind":"pith_short_8","alias_value":"FKQM6CP4","created_at":"2026-07-05T09:51:58Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2024:FKQM6CP47K3OYWJ2ZK5WEJA26H","target":"record","payload":{"canonical_record":{"source":{"id":"2412.15145","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2024-12-19T18:27:07Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"5510cf6bdd0ba74fdb12e5ae73763d60e07ed09b27e10e2fa3caecdb152c8486","abstract_canon_sha256":"394712e55725a3f9e14de0e4f901e65d10c8a60577981006248563c6e29ba646"},"schema_version":"1.0"},"canonical_sha256":"2aa0cf09fcfab6ec593acabb62241af1d1a41a989ff30af93f140d0180a67b80","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T09:51:58.503691Z","signature_b64":"uXCFnDdYBMrWBxa4P9/bBntemBVwzte00N5FXwO0kl/iijKKt3ROo1gBUOfXZPysbiczLc7bTIKKiH6okMHZCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2aa0cf09fcfab6ec593acabb62241af1d1a41a989ff30af93f140d0180a67b80","last_reissued_at":"2026-07-05T09:51:58.503211Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T09:51:58.503211Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2412.15145","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":"2026-07-05T09:51:58Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"tsx6xYBnnOGBgxx/uhmRq0+S9iJpDXxfOWrWnqlWbLTfhF1jgbs81OcrWTcTJb3RX5NihVaVSUVzIag+tzY1Dw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-07T08:07:30.011212Z"},"content_sha256":"b34fc7e6c116c43221e4867cecd4ae04abacb16f1270b6af167df615dc68257e","schema_version":"1.0","event_id":"sha256:b34fc7e6c116c43221e4867cecd4ae04abacb16f1270b6af167df615dc68257e"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2024:FKQM6CP47K3OYWJ2ZK5WEJA26H","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Ancient caloric functions and parabolic frequency on graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.AP","authors_text":"Archana Mohandas, Tang-Kai Lee","submitted_at":"2024-12-19T18:27:07Z","abstract_excerpt":"We study ancient solutions to discrete heat equations on some weighted graphs. On a graph of the form of a product with $\\bb Z,$ we show that there are no non-trivial ancient solutions with polynomial growth. This result is parallel to the case of finite graphs, which is also discussed. Along the way, we prove a backward uniqueness result for solutions with appropriate decaying rate based on a monotonicity formula of parabolic frequency."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2412.15145","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/2412.15145/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":"2026-07-05T09:51:58Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"F1KrWqexk7SMNt5QNCiaJZEcNp+TM9o3g91sAt3IV/II9HqaMOTvhWLu2T1x8QoPtq8cW1bvVC7GDE8FCbnRBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-07T08:07:30.011579Z"},"content_sha256":"d19f50aec2dc6acf2eaae6d858bfb5c74ec9ae206c108b16b87bd3b6dd0521bc","schema_version":"1.0","event_id":"sha256:d19f50aec2dc6acf2eaae6d858bfb5c74ec9ae206c108b16b87bd3b6dd0521bc"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/FKQM6CP47K3OYWJ2ZK5WEJA26H/bundle.json","state_url":"https://pith.science/pith/FKQM6CP47K3OYWJ2ZK5WEJA26H/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/FKQM6CP47K3OYWJ2ZK5WEJA26H/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":"2026-07-07T08:07:30Z","links":{"resolver":"https://pith.science/pith/FKQM6CP47K3OYWJ2ZK5WEJA26H","bundle":"https://pith.science/pith/FKQM6CP47K3OYWJ2ZK5WEJA26H/bundle.json","state":"https://pith.science/pith/FKQM6CP47K3OYWJ2ZK5WEJA26H/state.json","well_known_bundle":"https://pith.science/.well-known/pith/FKQM6CP47K3OYWJ2ZK5WEJA26H/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2024:FKQM6CP47K3OYWJ2ZK5WEJA26H","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":"394712e55725a3f9e14de0e4f901e65d10c8a60577981006248563c6e29ba646","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2024-12-19T18:27:07Z","title_canon_sha256":"5510cf6bdd0ba74fdb12e5ae73763d60e07ed09b27e10e2fa3caecdb152c8486"},"schema_version":"1.0","source":{"id":"2412.15145","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2412.15145","created_at":"2026-07-05T09:51:58Z"},{"alias_kind":"arxiv_version","alias_value":"2412.15145v1","created_at":"2026-07-05T09:51:58Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2412.15145","created_at":"2026-07-05T09:51:58Z"},{"alias_kind":"pith_short_12","alias_value":"FKQM6CP47K3O","created_at":"2026-07-05T09:51:58Z"},{"alias_kind":"pith_short_16","alias_value":"FKQM6CP47K3OYWJ2","created_at":"2026-07-05T09:51:58Z"},{"alias_kind":"pith_short_8","alias_value":"FKQM6CP4","created_at":"2026-07-05T09:51:58Z"}],"graph_snapshots":[{"event_id":"sha256:d19f50aec2dc6acf2eaae6d858bfb5c74ec9ae206c108b16b87bd3b6dd0521bc","target":"graph","created_at":"2026-07-05T09:51:58Z","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/2412.15145/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We study ancient solutions to discrete heat equations on some weighted graphs. On a graph of the form of a product with $\\bb Z,$ we show that there are no non-trivial ancient solutions with polynomial growth. This result is parallel to the case of finite graphs, which is also discussed. Along the way, we prove a backward uniqueness result for solutions with appropriate decaying rate based on a monotonicity formula of parabolic frequency.","authors_text":"Archana Mohandas, Tang-Kai Lee","cross_cats":["math.CO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2024-12-19T18:27:07Z","title":"Ancient caloric functions and parabolic frequency on graphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2412.15145","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:b34fc7e6c116c43221e4867cecd4ae04abacb16f1270b6af167df615dc68257e","target":"record","created_at":"2026-07-05T09:51:58Z","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":"394712e55725a3f9e14de0e4f901e65d10c8a60577981006248563c6e29ba646","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2024-12-19T18:27:07Z","title_canon_sha256":"5510cf6bdd0ba74fdb12e5ae73763d60e07ed09b27e10e2fa3caecdb152c8486"},"schema_version":"1.0","source":{"id":"2412.15145","kind":"arxiv","version":1}},"canonical_sha256":"2aa0cf09fcfab6ec593acabb62241af1d1a41a989ff30af93f140d0180a67b80","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"2aa0cf09fcfab6ec593acabb62241af1d1a41a989ff30af93f140d0180a67b80","first_computed_at":"2026-07-05T09:51:58.503211Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T09:51:58.503211Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"uXCFnDdYBMrWBxa4P9/bBntemBVwzte00N5FXwO0kl/iijKKt3ROo1gBUOfXZPysbiczLc7bTIKKiH6okMHZCg==","signature_status":"signed_v1","signed_at":"2026-07-05T09:51:58.503691Z","signed_message":"canonical_sha256_bytes"},"source_id":"2412.15145","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:b34fc7e6c116c43221e4867cecd4ae04abacb16f1270b6af167df615dc68257e","sha256:d19f50aec2dc6acf2eaae6d858bfb5c74ec9ae206c108b16b87bd3b6dd0521bc"],"state_sha256":"68dc43d1146ea9636b7f5a51815981fb8b922f4eee13a38205e0c35f4c762f5c"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"9GyVwt4SZcfoURa09fBg6f5fUomwy14E5f+3wC9AB9nKymzPozwAlLYHTO9KD3hqaLjXVcbhc5xU3LWbVvkHCA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-07-07T08:07:30.013397Z","bundle_sha256":"970aea465f137850b785fe390d20d2def23f5a1f28276e76da41c4ad7aa2d55e"}}