{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2021:CQWZHSOVU6CIJKU4ID2I7GXUK2","short_pith_number":"pith:CQWZHSOV","canonical_record":{"source":{"id":"2112.12165","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.CG","submitted_at":"2021-12-22T19:00:41Z","cross_cats_sorted":["cs.LG","math.AT"],"title_canon_sha256":"2b3449fd080c17711eb383ab5306d8355d60a3538de11b62f316e97a8c4bcac0","abstract_canon_sha256":"c2d9faee219f5e316df4e85eb57b67caadaeeff3c6d5da52db760ea383bdec9c"},"schema_version":"1.0"},"canonical_sha256":"142d93c9d5a78484aa9c40f48f9af4568e7c7b577bf1f74a90d5f07881c7bde6","source":{"kind":"arxiv","id":"2112.12165","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2112.12165","created_at":"2026-07-05T04:07:07Z"},{"alias_kind":"arxiv_version","alias_value":"2112.12165v2","created_at":"2026-07-05T04:07:07Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2112.12165","created_at":"2026-07-05T04:07:07Z"},{"alias_kind":"pith_short_12","alias_value":"CQWZHSOVU6CI","created_at":"2026-07-05T04:07:07Z"},{"alias_kind":"pith_short_16","alias_value":"CQWZHSOVU6CIJKU4","created_at":"2026-07-05T04:07:07Z"},{"alias_kind":"pith_short_8","alias_value":"CQWZHSOV","created_at":"2026-07-05T04:07:07Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2021:CQWZHSOVU6CIJKU4ID2I7GXUK2","target":"record","payload":{"canonical_record":{"source":{"id":"2112.12165","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.CG","submitted_at":"2021-12-22T19:00:41Z","cross_cats_sorted":["cs.LG","math.AT"],"title_canon_sha256":"2b3449fd080c17711eb383ab5306d8355d60a3538de11b62f316e97a8c4bcac0","abstract_canon_sha256":"c2d9faee219f5e316df4e85eb57b67caadaeeff3c6d5da52db760ea383bdec9c"},"schema_version":"1.0"},"canonical_sha256":"142d93c9d5a78484aa9c40f48f9af4568e7c7b577bf1f74a90d5f07881c7bde6","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T04:07:07.072784Z","signature_b64":"Bfdc0YcVkLdd/1TC9pMgSsfxsJaZRlaSlH4+ceTdRLAb9oYXIDqgHsQ5IQ8bVWyMhQPBjMtEdZEDj3cRoKkzDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"142d93c9d5a78484aa9c40f48f9af4568e7c7b577bf1f74a90d5f07881c7bde6","last_reissued_at":"2026-07-05T04:07:07.072291Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T04:07:07.072291Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2112.12165","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":"2026-07-05T04:07:07Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"N7INBeNzUVBBsM3lf1NgkkdxENogGMUFUiZK9bEEmt1f87eXdOPx/ewjyyroK4SqYKxTv/N83xYxZQKN0xESDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-03T16:31:55.358151Z"},"content_sha256":"6408ade681a476a96a103f681336977cdfd7e47ab61d3511d972c78a3e59b894","schema_version":"1.0","event_id":"sha256:6408ade681a476a96a103f681336977cdfd7e47ab61d3511d972c78a3e59b894"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2021:CQWZHSOVU6CIJKU4ID2I7GXUK2","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"The Universal $\\ell^p$-Metric on Merge Trees","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["cs.LG","math.AT"],"primary_cat":"cs.CG","authors_text":"Justin Curry, Michael Lesnick, Robert Cardona, Tung Lam","submitted_at":"2021-12-22T19:00:41Z","abstract_excerpt":"Adapting a definition given by Bjerkevik and Lesnick for multiparameter persistence modules, we introduce an $\\ell^p$-type extension of the interleaving distance on merge trees. We show that our distance is a metric, and that it upper-bounds the $p$-Wasserstein distance between the associated barcodes. For each $p\\in[1,\\infty]$, we prove that this distance is stable with respect to cellular sublevel filtrations and that it is the universal (i.e., largest) distance satisfying this stability property. In the $p=\\infty$ case, this gives a novel proof of universality for the interleaving distance "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2112.12165","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/2112.12165/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":"2026-07-05T04:07:07Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"vHZEfZ/T6+0g+meRY+JxOYHZ46ElDV/F0TNQVwbEn8PHbxXK8TlaWzQD6TrLFCx3T166rFVKKWKSHVkvAFdXCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-03T16:31:55.358874Z"},"content_sha256":"04c9f1ee5c48fe2862d1dfbd00b7eb730f232769e113921ef8f1928b257a7ad5","schema_version":"1.0","event_id":"sha256:04c9f1ee5c48fe2862d1dfbd00b7eb730f232769e113921ef8f1928b257a7ad5"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/CQWZHSOVU6CIJKU4ID2I7GXUK2/bundle.json","state_url":"https://pith.science/pith/CQWZHSOVU6CIJKU4ID2I7GXUK2/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/CQWZHSOVU6CIJKU4ID2I7GXUK2/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":"2026-08-03T16:31:55Z","links":{"resolver":"https://pith.science/pith/CQWZHSOVU6CIJKU4ID2I7GXUK2","bundle":"https://pith.science/pith/CQWZHSOVU6CIJKU4ID2I7GXUK2/bundle.json","state":"https://pith.science/pith/CQWZHSOVU6CIJKU4ID2I7GXUK2/state.json","well_known_bundle":"https://pith.science/.well-known/pith/CQWZHSOVU6CIJKU4ID2I7GXUK2/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2021:CQWZHSOVU6CIJKU4ID2I7GXUK2","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":"c2d9faee219f5e316df4e85eb57b67caadaeeff3c6d5da52db760ea383bdec9c","cross_cats_sorted":["cs.LG","math.AT"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.CG","submitted_at":"2021-12-22T19:00:41Z","title_canon_sha256":"2b3449fd080c17711eb383ab5306d8355d60a3538de11b62f316e97a8c4bcac0"},"schema_version":"1.0","source":{"id":"2112.12165","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2112.12165","created_at":"2026-07-05T04:07:07Z"},{"alias_kind":"arxiv_version","alias_value":"2112.12165v2","created_at":"2026-07-05T04:07:07Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2112.12165","created_at":"2026-07-05T04:07:07Z"},{"alias_kind":"pith_short_12","alias_value":"CQWZHSOVU6CI","created_at":"2026-07-05T04:07:07Z"},{"alias_kind":"pith_short_16","alias_value":"CQWZHSOVU6CIJKU4","created_at":"2026-07-05T04:07:07Z"},{"alias_kind":"pith_short_8","alias_value":"CQWZHSOV","created_at":"2026-07-05T04:07:07Z"}],"graph_snapshots":[{"event_id":"sha256:04c9f1ee5c48fe2862d1dfbd00b7eb730f232769e113921ef8f1928b257a7ad5","target":"graph","created_at":"2026-07-05T04:07:07Z","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/2112.12165/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Adapting a definition given by Bjerkevik and Lesnick for multiparameter persistence modules, we introduce an $\\ell^p$-type extension of the interleaving distance on merge trees. We show that our distance is a metric, and that it upper-bounds the $p$-Wasserstein distance between the associated barcodes. For each $p\\in[1,\\infty]$, we prove that this distance is stable with respect to cellular sublevel filtrations and that it is the universal (i.e., largest) distance satisfying this stability property. In the $p=\\infty$ case, this gives a novel proof of universality for the interleaving distance ","authors_text":"Justin Curry, Michael Lesnick, Robert Cardona, Tung Lam","cross_cats":["cs.LG","math.AT"],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.CG","submitted_at":"2021-12-22T19:00:41Z","title":"The Universal $\\ell^p$-Metric on Merge Trees"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2112.12165","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":"sha256:6408ade681a476a96a103f681336977cdfd7e47ab61d3511d972c78a3e59b894","target":"record","created_at":"2026-07-05T04:07:07Z","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":"c2d9faee219f5e316df4e85eb57b67caadaeeff3c6d5da52db760ea383bdec9c","cross_cats_sorted":["cs.LG","math.AT"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.CG","submitted_at":"2021-12-22T19:00:41Z","title_canon_sha256":"2b3449fd080c17711eb383ab5306d8355d60a3538de11b62f316e97a8c4bcac0"},"schema_version":"1.0","source":{"id":"2112.12165","kind":"arxiv","version":2}},"canonical_sha256":"142d93c9d5a78484aa9c40f48f9af4568e7c7b577bf1f74a90d5f07881c7bde6","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"142d93c9d5a78484aa9c40f48f9af4568e7c7b577bf1f74a90d5f07881c7bde6","first_computed_at":"2026-07-05T04:07:07.072291Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T04:07:07.072291Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Bfdc0YcVkLdd/1TC9pMgSsfxsJaZRlaSlH4+ceTdRLAb9oYXIDqgHsQ5IQ8bVWyMhQPBjMtEdZEDj3cRoKkzDA==","signature_status":"signed_v1","signed_at":"2026-07-05T04:07:07.072784Z","signed_message":"canonical_sha256_bytes"},"source_id":"2112.12165","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:6408ade681a476a96a103f681336977cdfd7e47ab61d3511d972c78a3e59b894","sha256:04c9f1ee5c48fe2862d1dfbd00b7eb730f232769e113921ef8f1928b257a7ad5"],"state_sha256":"1c2c312305bea3fd7c2d8db90e04403a8bdcd3c271cfc6eb5dfbe1122fd193c1"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"0oekMraWdzN0ROQd279bxA+DIXiSMdzwJnok/S4BE+wGR1wqrrSwAwNZ+8ywREEZY3zQXcCk6ukYxsaG7KaJAQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-03T16:31:55.362750Z","bundle_sha256":"5ab1e9af5b3157aeca770ccf5f862212d7bb58d6d3fe57560ba6e8e01c23c7e6"}}