{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2021:W5ZVJ4DGWQVPMKRXOCICDMS5TD","short_pith_number":"pith:W5ZVJ4DG","canonical_record":{"source":{"id":"2109.08721","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"astro-ph.CO","submitted_at":"2021-09-17T19:04:49Z","cross_cats_sorted":["math.AT"],"title_canon_sha256":"e5e7716528ffd68febc1a2be054e2085419d3f75e1f9789e8547dada50978834","abstract_canon_sha256":"56f43eb3a85440e2ac62c6e062114ae0bd9f8769fd543475698e7bd42d3c648d"},"schema_version":"1.0"},"canonical_sha256":"b77354f066b42af62a37709021b25d98f76bf74e594191acef291870f203643a","source":{"kind":"arxiv","id":"2109.08721","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2109.08721","created_at":"2026-07-05T03:15:19Z"},{"alias_kind":"arxiv_version","alias_value":"2109.08721v1","created_at":"2026-07-05T03:15:19Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2109.08721","created_at":"2026-07-05T03:15:19Z"},{"alias_kind":"pith_short_12","alias_value":"W5ZVJ4DGWQVP","created_at":"2026-07-05T03:15:19Z"},{"alias_kind":"pith_short_16","alias_value":"W5ZVJ4DGWQVPMKRX","created_at":"2026-07-05T03:15:19Z"},{"alias_kind":"pith_short_8","alias_value":"W5ZVJ4DG","created_at":"2026-07-05T03:15:19Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2021:W5ZVJ4DGWQVPMKRXOCICDMS5TD","target":"record","payload":{"canonical_record":{"source":{"id":"2109.08721","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"astro-ph.CO","submitted_at":"2021-09-17T19:04:49Z","cross_cats_sorted":["math.AT"],"title_canon_sha256":"e5e7716528ffd68febc1a2be054e2085419d3f75e1f9789e8547dada50978834","abstract_canon_sha256":"56f43eb3a85440e2ac62c6e062114ae0bd9f8769fd543475698e7bd42d3c648d"},"schema_version":"1.0"},"canonical_sha256":"b77354f066b42af62a37709021b25d98f76bf74e594191acef291870f203643a","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T03:15:19.640269Z","signature_b64":"tsjiTGPpufz3XNrK9fqOY7Dt0wozFHHDKRjgIrogUnGls5YlZmagGvv37jXnLoa49/m4JnZDuEOrYu4ars/tDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b77354f066b42af62a37709021b25d98f76bf74e594191acef291870f203643a","last_reissued_at":"2026-07-05T03:15:19.639865Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T03:15:19.639865Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2109.08721","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":"2026-07-05T03:15:19Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"sE5RY9PrGVtM3UPTZ+87oNnTIGwTjHNhHUMneiyMx9NnN8P6yTI6CWtr9ZnCAw6BiQlP9ojC6I8DtdzeEpJPCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-18T10:39:48.113876Z"},"content_sha256":"0fb1b49aaf9c4aba3d0a471170236aaf2a41cc172470f462eae50c0fa6abdd43","schema_version":"1.0","event_id":"sha256:0fb1b49aaf9c4aba3d0a471170236aaf2a41cc172470f462eae50c0fa6abdd43"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2021:W5ZVJ4DGWQVPMKRXOCICDMS5TD","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Topology and geometry of Gaussian random fields II: on critical points, excursion sets, and persistent homology","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.AT"],"primary_cat":"astro-ph.CO","authors_text":"Pratyush Pranav","submitted_at":"2021-09-17T19:04:49Z","abstract_excerpt":"This paper is second in the series, following Pranav et al. (2019), focused on the characterization of geometric and topological properties of 3D Gaussian random fields. We focus on the formalism of persistent homology, the mainstay of Topological Data Analysis (TDA), in the context of excursion set formalism. We also focus on the structure of critical points of stochastic fields, and their relationship with formation and evolution of structures in the universe.\n  The topological background is accompanied by an investigation of Gaussian field simulations based on the LCDM spectrum, as well as "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2109.08721","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/2109.08721/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":"2026-07-05T03:15:19Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"kGnu/cbbxHBcCqH06Wl8SMS0pqAAxTZVPiLdAcNO3abWkjyl5vcoMJRQXMkcn+OrA82vg4uYnK6w8U41lbBvAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-18T10:39:48.114256Z"},"content_sha256":"d88e1f3765abb197e6247ca0e481de350bbf11e4c0266e42971d23ecbc551316","schema_version":"1.0","event_id":"sha256:d88e1f3765abb197e6247ca0e481de350bbf11e4c0266e42971d23ecbc551316"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/W5ZVJ4DGWQVPMKRXOCICDMS5TD/bundle.json","state_url":"https://pith.science/pith/W5ZVJ4DGWQVPMKRXOCICDMS5TD/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/W5ZVJ4DGWQVPMKRXOCICDMS5TD/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":"2026-08-18T10:39:48Z","links":{"resolver":"https://pith.science/pith/W5ZVJ4DGWQVPMKRXOCICDMS5TD","bundle":"https://pith.science/pith/W5ZVJ4DGWQVPMKRXOCICDMS5TD/bundle.json","state":"https://pith.science/pith/W5ZVJ4DGWQVPMKRXOCICDMS5TD/state.json","well_known_bundle":"https://pith.science/.well-known/pith/W5ZVJ4DGWQVPMKRXOCICDMS5TD/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2021:W5ZVJ4DGWQVPMKRXOCICDMS5TD","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":"56f43eb3a85440e2ac62c6e062114ae0bd9f8769fd543475698e7bd42d3c648d","cross_cats_sorted":["math.AT"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"astro-ph.CO","submitted_at":"2021-09-17T19:04:49Z","title_canon_sha256":"e5e7716528ffd68febc1a2be054e2085419d3f75e1f9789e8547dada50978834"},"schema_version":"1.0","source":{"id":"2109.08721","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2109.08721","created_at":"2026-07-05T03:15:19Z"},{"alias_kind":"arxiv_version","alias_value":"2109.08721v1","created_at":"2026-07-05T03:15:19Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2109.08721","created_at":"2026-07-05T03:15:19Z"},{"alias_kind":"pith_short_12","alias_value":"W5ZVJ4DGWQVP","created_at":"2026-07-05T03:15:19Z"},{"alias_kind":"pith_short_16","alias_value":"W5ZVJ4DGWQVPMKRX","created_at":"2026-07-05T03:15:19Z"},{"alias_kind":"pith_short_8","alias_value":"W5ZVJ4DG","created_at":"2026-07-05T03:15:19Z"}],"graph_snapshots":[{"event_id":"sha256:d88e1f3765abb197e6247ca0e481de350bbf11e4c0266e42971d23ecbc551316","target":"graph","created_at":"2026-07-05T03:15:19Z","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/2109.08721/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"This paper is second in the series, following Pranav et al. (2019), focused on the characterization of geometric and topological properties of 3D Gaussian random fields. We focus on the formalism of persistent homology, the mainstay of Topological Data Analysis (TDA), in the context of excursion set formalism. We also focus on the structure of critical points of stochastic fields, and their relationship with formation and evolution of structures in the universe.\n  The topological background is accompanied by an investigation of Gaussian field simulations based on the LCDM spectrum, as well as ","authors_text":"Pratyush Pranav","cross_cats":["math.AT"],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"astro-ph.CO","submitted_at":"2021-09-17T19:04:49Z","title":"Topology and geometry of Gaussian random fields II: on critical points, excursion sets, and persistent homology"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2109.08721","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":"sha256:0fb1b49aaf9c4aba3d0a471170236aaf2a41cc172470f462eae50c0fa6abdd43","target":"record","created_at":"2026-07-05T03:15:19Z","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":"56f43eb3a85440e2ac62c6e062114ae0bd9f8769fd543475698e7bd42d3c648d","cross_cats_sorted":["math.AT"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"astro-ph.CO","submitted_at":"2021-09-17T19:04:49Z","title_canon_sha256":"e5e7716528ffd68febc1a2be054e2085419d3f75e1f9789e8547dada50978834"},"schema_version":"1.0","source":{"id":"2109.08721","kind":"arxiv","version":1}},"canonical_sha256":"b77354f066b42af62a37709021b25d98f76bf74e594191acef291870f203643a","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"b77354f066b42af62a37709021b25d98f76bf74e594191acef291870f203643a","first_computed_at":"2026-07-05T03:15:19.639865Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T03:15:19.639865Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"tsjiTGPpufz3XNrK9fqOY7Dt0wozFHHDKRjgIrogUnGls5YlZmagGvv37jXnLoa49/m4JnZDuEOrYu4ars/tDA==","signature_status":"signed_v1","signed_at":"2026-07-05T03:15:19.640269Z","signed_message":"canonical_sha256_bytes"},"source_id":"2109.08721","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:0fb1b49aaf9c4aba3d0a471170236aaf2a41cc172470f462eae50c0fa6abdd43","sha256:d88e1f3765abb197e6247ca0e481de350bbf11e4c0266e42971d23ecbc551316"],"state_sha256":"cb84767209a21f22d434eac77d97b959387cd6839228622c08aefba7ac03b75b"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Msq8sv0R+fv6owHzmMCtSOAri63ks0l33F1jYsSbH9Q/FUdwepsd8Xatl/MnuqQsympCgpfig5eX/JSIjLVTDA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-18T10:39:48.116618Z","bundle_sha256":"c49da71c865211522e309dc7e2f197b1822d1987b3f51438b4aa42db87c5694c"}}