{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2011:XBMMGOQNYOKYL2CYPXML6GYQCA","short_pith_number":"pith:XBMMGOQN","canonical_record":{"source":{"id":"1106.5305","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CG","submitted_at":"2011-06-27T06:05:20Z","cross_cats_sorted":["math.AT"],"title_canon_sha256":"f018920cec1ff39279adcf7db9e6654e9f942bc7c01369a37207f577fe41c5b1","abstract_canon_sha256":"48862a9f5b7f933914d4dca33659f42cc10bce7c5894647dcb9e1a5758586c85"},"schema_version":"1.0"},"canonical_sha256":"b858c33a0dc39585e8587dd8bf1b101016a71fa980d62eecfb7e8c048d85b59f","source":{"kind":"arxiv","id":"1106.5305","version":4},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1106.5305","created_at":"2026-05-18T02:03:56Z"},{"alias_kind":"arxiv_version","alias_value":"1106.5305v4","created_at":"2026-05-18T02:03:56Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1106.5305","created_at":"2026-05-18T02:03:56Z"},{"alias_kind":"pith_short_12","alias_value":"XBMMGOQNYOKY","created_at":"2026-05-18T12:26:44Z"},{"alias_kind":"pith_short_16","alias_value":"XBMMGOQNYOKYL2CY","created_at":"2026-05-18T12:26:44Z"},{"alias_kind":"pith_short_8","alias_value":"XBMMGOQN","created_at":"2026-05-18T12:26:44Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2011:XBMMGOQNYOKYL2CYPXML6GYQCA","target":"record","payload":{"canonical_record":{"source":{"id":"1106.5305","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CG","submitted_at":"2011-06-27T06:05:20Z","cross_cats_sorted":["math.AT"],"title_canon_sha256":"f018920cec1ff39279adcf7db9e6654e9f942bc7c01369a37207f577fe41c5b1","abstract_canon_sha256":"48862a9f5b7f933914d4dca33659f42cc10bce7c5894647dcb9e1a5758586c85"},"schema_version":"1.0"},"canonical_sha256":"b858c33a0dc39585e8587dd8bf1b101016a71fa980d62eecfb7e8c048d85b59f","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:03:56.016616Z","signature_b64":"cqp9BhiopNVGmtYYsRkuxtcMzY/4495am06MWcp7keildfQZGk864XmqWA5Tqk1g9A1XAUVEW3UXZVfhVoK4Dw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b858c33a0dc39585e8587dd8bf1b101016a71fa980d62eecfb7e8c048d85b59f","last_reissued_at":"2026-05-18T02:03:56.015997Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:03:56.015997Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1106.5305","source_version":4,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:03:56Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"AaC4RCOqh7FYlqcvTS7Py3jCy8rbNosMhoMP56rrJ13AXx2K8mbwA9a3nILbyR+sSOjqMt9Ps7cyRimK3fFsDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-25T19:39:19.220749Z"},"content_sha256":"6e6ed09ca3edd3108c99ee0b845c6291a80b0acfa5d3d98af5f22bbeaf934254","schema_version":"1.0","event_id":"sha256:6e6ed09ca3edd3108c99ee0b845c6291a80b0acfa5d3d98af5f22bbeaf934254"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2011:XBMMGOQNYOKYL2CYPXML6GYQCA","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"The Theory of the Interleaving Distance on Multidimensional Persistence Modules","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AT"],"primary_cat":"cs.CG","authors_text":"Michael Lesnick","submitted_at":"2011-06-27T06:05:20Z","abstract_excerpt":"In 2009, Chazal et al. introduced $\\epsilon$-interleavings of persistence modules. $\\epsilon$-interleavings induce a pseudometric $d_I$ on (isomorphism classes of) persistence modules, the interleaving distance. The definitions of $\\epsilon$-interleavings and $d_I$ generalize readily to multidimensional persistence modules. In this paper, we develop the theory of multidimensional interleavings, with a view towards applications to topological data analysis. We present four main results. First, we show that on 1-D persistence modules, $d_I$ is equal to the bottleneck distance $d_B$. This result,"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1106.5305","kind":"arxiv","version":4},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"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":"2026-05-18T02:03:56Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"7Xs4/V0wJrcSeX3WYMbswZ7FIJVUTiueuhQYrrjEqD6NTWtcOP8EQ7/mRyP3EZl52kVZnCQOajVIb3En+pDxCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-25T19:39:19.221120Z"},"content_sha256":"eb3e6c9c2a32c4cd5bd8bd1fd3109f8338cdbf98c8733d39b15294ef7d87101e","schema_version":"1.0","event_id":"sha256:eb3e6c9c2a32c4cd5bd8bd1fd3109f8338cdbf98c8733d39b15294ef7d87101e"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/XBMMGOQNYOKYL2CYPXML6GYQCA/bundle.json","state_url":"https://pith.science/pith/XBMMGOQNYOKYL2CYPXML6GYQCA/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/XBMMGOQNYOKYL2CYPXML6GYQCA/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":"2026-05-25T19:39:19Z","links":{"resolver":"https://pith.science/pith/XBMMGOQNYOKYL2CYPXML6GYQCA","bundle":"https://pith.science/pith/XBMMGOQNYOKYL2CYPXML6GYQCA/bundle.json","state":"https://pith.science/pith/XBMMGOQNYOKYL2CYPXML6GYQCA/state.json","well_known_bundle":"https://pith.science/.well-known/pith/XBMMGOQNYOKYL2CYPXML6GYQCA/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:XBMMGOQNYOKYL2CYPXML6GYQCA","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":"48862a9f5b7f933914d4dca33659f42cc10bce7c5894647dcb9e1a5758586c85","cross_cats_sorted":["math.AT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CG","submitted_at":"2011-06-27T06:05:20Z","title_canon_sha256":"f018920cec1ff39279adcf7db9e6654e9f942bc7c01369a37207f577fe41c5b1"},"schema_version":"1.0","source":{"id":"1106.5305","kind":"arxiv","version":4}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1106.5305","created_at":"2026-05-18T02:03:56Z"},{"alias_kind":"arxiv_version","alias_value":"1106.5305v4","created_at":"2026-05-18T02:03:56Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1106.5305","created_at":"2026-05-18T02:03:56Z"},{"alias_kind":"pith_short_12","alias_value":"XBMMGOQNYOKY","created_at":"2026-05-18T12:26:44Z"},{"alias_kind":"pith_short_16","alias_value":"XBMMGOQNYOKYL2CY","created_at":"2026-05-18T12:26:44Z"},{"alias_kind":"pith_short_8","alias_value":"XBMMGOQN","created_at":"2026-05-18T12:26:44Z"}],"graph_snapshots":[{"event_id":"sha256:eb3e6c9c2a32c4cd5bd8bd1fd3109f8338cdbf98c8733d39b15294ef7d87101e","target":"graph","created_at":"2026-05-18T02:03:56Z","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"},"paper":{"abstract_excerpt":"In 2009, Chazal et al. introduced $\\epsilon$-interleavings of persistence modules. $\\epsilon$-interleavings induce a pseudometric $d_I$ on (isomorphism classes of) persistence modules, the interleaving distance. The definitions of $\\epsilon$-interleavings and $d_I$ generalize readily to multidimensional persistence modules. In this paper, we develop the theory of multidimensional interleavings, with a view towards applications to topological data analysis. We present four main results. First, we show that on 1-D persistence modules, $d_I$ is equal to the bottleneck distance $d_B$. This result,","authors_text":"Michael Lesnick","cross_cats":["math.AT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CG","submitted_at":"2011-06-27T06:05:20Z","title":"The Theory of the Interleaving Distance on Multidimensional Persistence Modules"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1106.5305","kind":"arxiv","version":4},"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":"sha256:6e6ed09ca3edd3108c99ee0b845c6291a80b0acfa5d3d98af5f22bbeaf934254","target":"record","created_at":"2026-05-18T02:03:56Z","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":"48862a9f5b7f933914d4dca33659f42cc10bce7c5894647dcb9e1a5758586c85","cross_cats_sorted":["math.AT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CG","submitted_at":"2011-06-27T06:05:20Z","title_canon_sha256":"f018920cec1ff39279adcf7db9e6654e9f942bc7c01369a37207f577fe41c5b1"},"schema_version":"1.0","source":{"id":"1106.5305","kind":"arxiv","version":4}},"canonical_sha256":"b858c33a0dc39585e8587dd8bf1b101016a71fa980d62eecfb7e8c048d85b59f","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"b858c33a0dc39585e8587dd8bf1b101016a71fa980d62eecfb7e8c048d85b59f","first_computed_at":"2026-05-18T02:03:56.015997Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:03:56.015997Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"cqp9BhiopNVGmtYYsRkuxtcMzY/4495am06MWcp7keildfQZGk864XmqWA5Tqk1g9A1XAUVEW3UXZVfhVoK4Dw==","signature_status":"signed_v1","signed_at":"2026-05-18T02:03:56.016616Z","signed_message":"canonical_sha256_bytes"},"source_id":"1106.5305","source_kind":"arxiv","source_version":4}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:6e6ed09ca3edd3108c99ee0b845c6291a80b0acfa5d3d98af5f22bbeaf934254","sha256:eb3e6c9c2a32c4cd5bd8bd1fd3109f8338cdbf98c8733d39b15294ef7d87101e"],"state_sha256":"b5db4a2a9ee8c914197af99b8caaf9c9b0c3301546aac1f6933f3c99d5b757af"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"kK9NJoKtygRRE7TyYwc1xVzdY7OWjVN8Ndq5j79g+jiNbB+wBkBsoVcLHRQqfHwVNzeTKLJ+7Qs8Z1/VrD7eCg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-25T19:39:19.223826Z","bundle_sha256":"1b90b86698f411452b7ef6ac28eb3f5c0bf5326e3327cbc7143509fc861852f8"}}