{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2019:S7SJEYGNCK3JYNO2ITN6AAO3HU","short_pith_number":"pith:S7SJEYGN","canonical_record":{"source":{"id":"1901.09824","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2019-01-28T17:22:09Z","cross_cats_sorted":[],"title_canon_sha256":"938bdb4ffa81cedbf37d42ebdc30bde8556c97f6b4cf7962ce8eb7468db130e9","abstract_canon_sha256":"2af17c6a1060462c6030e6ca184545ba0d12304650a52e72722b560f875df07a"},"schema_version":"1.0"},"canonical_sha256":"97e49260cd12b69c35da44dbe001db3d0a7f8e505fe09e6192038b482c45cffb","source":{"kind":"arxiv","id":"1901.09824","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1901.09824","created_at":"2026-05-17T23:55:23Z"},{"alias_kind":"arxiv_version","alias_value":"1901.09824v1","created_at":"2026-05-17T23:55:23Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1901.09824","created_at":"2026-05-17T23:55:23Z"},{"alias_kind":"pith_short_12","alias_value":"S7SJEYGNCK3J","created_at":"2026-05-18T12:33:27Z"},{"alias_kind":"pith_short_16","alias_value":"S7SJEYGNCK3JYNO2","created_at":"2026-05-18T12:33:27Z"},{"alias_kind":"pith_short_8","alias_value":"S7SJEYGN","created_at":"2026-05-18T12:33:27Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2019:S7SJEYGNCK3JYNO2ITN6AAO3HU","target":"record","payload":{"canonical_record":{"source":{"id":"1901.09824","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2019-01-28T17:22:09Z","cross_cats_sorted":[],"title_canon_sha256":"938bdb4ffa81cedbf37d42ebdc30bde8556c97f6b4cf7962ce8eb7468db130e9","abstract_canon_sha256":"2af17c6a1060462c6030e6ca184545ba0d12304650a52e72722b560f875df07a"},"schema_version":"1.0"},"canonical_sha256":"97e49260cd12b69c35da44dbe001db3d0a7f8e505fe09e6192038b482c45cffb","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:55:23.731308Z","signature_b64":"bBfyVYPVeJ1+4IrAW6Y94NQVP8Du1OW0uWxA3iokQdOmoGwdk7BM1spP/oV1qQOccd1mFXjf42MpVwOdl9nEBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"97e49260cd12b69c35da44dbe001db3d0a7f8e505fe09e6192038b482c45cffb","last_reissued_at":"2026-05-17T23:55:23.730867Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:55:23.730867Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1901.09824","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":"2026-05-17T23:55:23Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"p3ObXA+7CLhrSHZO0rmXYH/EPVHjB84ZiZD9zqt5TUL4V+d4nUsHdrPDWP+zBubYtbYkxtH+CAW6ClUzWNJPAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-02T01:50:51.933145Z"},"content_sha256":"d4d722a60e833f7a1de67bc00bc06f0fd0eb9e05145ab44f4a90c2811c0f6863","schema_version":"1.0","event_id":"sha256:d4d722a60e833f7a1de67bc00bc06f0fd0eb9e05145ab44f4a90c2811c0f6863"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2019:S7SJEYGNCK3JYNO2ITN6AAO3HU","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Stable resolutions of multi-parameter persistence modules","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AT","authors_text":"Nicolas Berkouk","submitted_at":"2019-01-28T17:22:09Z","abstract_excerpt":"The theory of persistence, which arises from topological data analysis, has been intensively studied in the one-parameter case both theoretically and in its applications. However, its extension to the multi-parameter case raises numerous difficulties. Indeed, it has been shown that there exists no complete discrete invariant for persistence modules with many parameters, such as so-called barcodes in the one-parameter case. To tackle this problem, some new algebraic invariants have been proposed to study multi-parameter persistence modules, adapting classical ideas from commutative algebra and "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1901.09824","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":""},"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":"2026-05-17T23:55:23Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"tTnBqPKk9gdtRyNZrEhd0+eWgb/9uv9x9AtLXNMp2gnzva/FtdRh01sxEGUbdLD2P7WeePni16woVC7O8R8vAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-02T01:50:51.933510Z"},"content_sha256":"40ae1f1560540db20596f76b5517eca6c1307e64d4e98c37e4e6f01ff5b0e0d4","schema_version":"1.0","event_id":"sha256:40ae1f1560540db20596f76b5517eca6c1307e64d4e98c37e4e6f01ff5b0e0d4"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/S7SJEYGNCK3JYNO2ITN6AAO3HU/bundle.json","state_url":"https://pith.science/pith/S7SJEYGNCK3JYNO2ITN6AAO3HU/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/S7SJEYGNCK3JYNO2ITN6AAO3HU/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":"2026-06-02T01:50:51Z","links":{"resolver":"https://pith.science/pith/S7SJEYGNCK3JYNO2ITN6AAO3HU","bundle":"https://pith.science/pith/S7SJEYGNCK3JYNO2ITN6AAO3HU/bundle.json","state":"https://pith.science/pith/S7SJEYGNCK3JYNO2ITN6AAO3HU/state.json","well_known_bundle":"https://pith.science/.well-known/pith/S7SJEYGNCK3JYNO2ITN6AAO3HU/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2019:S7SJEYGNCK3JYNO2ITN6AAO3HU","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":"2af17c6a1060462c6030e6ca184545ba0d12304650a52e72722b560f875df07a","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2019-01-28T17:22:09Z","title_canon_sha256":"938bdb4ffa81cedbf37d42ebdc30bde8556c97f6b4cf7962ce8eb7468db130e9"},"schema_version":"1.0","source":{"id":"1901.09824","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1901.09824","created_at":"2026-05-17T23:55:23Z"},{"alias_kind":"arxiv_version","alias_value":"1901.09824v1","created_at":"2026-05-17T23:55:23Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1901.09824","created_at":"2026-05-17T23:55:23Z"},{"alias_kind":"pith_short_12","alias_value":"S7SJEYGNCK3J","created_at":"2026-05-18T12:33:27Z"},{"alias_kind":"pith_short_16","alias_value":"S7SJEYGNCK3JYNO2","created_at":"2026-05-18T12:33:27Z"},{"alias_kind":"pith_short_8","alias_value":"S7SJEYGN","created_at":"2026-05-18T12:33:27Z"}],"graph_snapshots":[{"event_id":"sha256:40ae1f1560540db20596f76b5517eca6c1307e64d4e98c37e4e6f01ff5b0e0d4","target":"graph","created_at":"2026-05-17T23:55:23Z","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":"The theory of persistence, which arises from topological data analysis, has been intensively studied in the one-parameter case both theoretically and in its applications. However, its extension to the multi-parameter case raises numerous difficulties. Indeed, it has been shown that there exists no complete discrete invariant for persistence modules with many parameters, such as so-called barcodes in the one-parameter case. To tackle this problem, some new algebraic invariants have been proposed to study multi-parameter persistence modules, adapting classical ideas from commutative algebra and ","authors_text":"Nicolas Berkouk","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2019-01-28T17:22:09Z","title":"Stable resolutions of multi-parameter persistence modules"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1901.09824","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:d4d722a60e833f7a1de67bc00bc06f0fd0eb9e05145ab44f4a90c2811c0f6863","target":"record","created_at":"2026-05-17T23:55:23Z","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":"2af17c6a1060462c6030e6ca184545ba0d12304650a52e72722b560f875df07a","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2019-01-28T17:22:09Z","title_canon_sha256":"938bdb4ffa81cedbf37d42ebdc30bde8556c97f6b4cf7962ce8eb7468db130e9"},"schema_version":"1.0","source":{"id":"1901.09824","kind":"arxiv","version":1}},"canonical_sha256":"97e49260cd12b69c35da44dbe001db3d0a7f8e505fe09e6192038b482c45cffb","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"97e49260cd12b69c35da44dbe001db3d0a7f8e505fe09e6192038b482c45cffb","first_computed_at":"2026-05-17T23:55:23.730867Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:55:23.730867Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"bBfyVYPVeJ1+4IrAW6Y94NQVP8Du1OW0uWxA3iokQdOmoGwdk7BM1spP/oV1qQOccd1mFXjf42MpVwOdl9nEBw==","signature_status":"signed_v1","signed_at":"2026-05-17T23:55:23.731308Z","signed_message":"canonical_sha256_bytes"},"source_id":"1901.09824","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:d4d722a60e833f7a1de67bc00bc06f0fd0eb9e05145ab44f4a90c2811c0f6863","sha256:40ae1f1560540db20596f76b5517eca6c1307e64d4e98c37e4e6f01ff5b0e0d4"],"state_sha256":"c46cedcdf0ca2bbe07b478d6bfd25cd150edff97f475e2ab2c0ac26b0b2b3f1f"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"aKZ7DZW/KpjFiLUX7ZFakLG9oh9HC0l009+W994d9sVY0ENjXzuhi3fQk1hgPiU+PYlKdP+j4Ke9vxD9unWOAw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-02T01:50:51.935432Z","bundle_sha256":"3cd061abfa438100210a772f167fd4e70e8b18acc762c41fbe93ba3b23d8cc3e"}}