{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2014:PJK4VRE53TKTLNX63QWF6SGF5N","short_pith_number":"pith:PJK4VRE5","canonical_record":{"source":{"id":"1409.0132","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2014-08-30T16:38:30Z","cross_cats_sorted":[],"title_canon_sha256":"f9ff4f0b1a978a505a3a0391adebef6595c51bc3c2ddcfacad333a92119e3dd7","abstract_canon_sha256":"088f349ce6ada2f9eb5f7f319132d3429e72c2e32d33d488c95965eac3ccb5a7"},"schema_version":"1.0"},"canonical_sha256":"7a55cac49ddcd535b6fedc2c5f48c5eb77d93059fc45579afb29af0d1cf6ff5c","source":{"kind":"arxiv","id":"1409.0132","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1409.0132","created_at":"2026-05-18T01:32:18Z"},{"alias_kind":"arxiv_version","alias_value":"1409.0132v3","created_at":"2026-05-18T01:32:18Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1409.0132","created_at":"2026-05-18T01:32:18Z"},{"alias_kind":"pith_short_12","alias_value":"PJK4VRE53TKT","created_at":"2026-05-18T12:28:43Z"},{"alias_kind":"pith_short_16","alias_value":"PJK4VRE53TKTLNX6","created_at":"2026-05-18T12:28:43Z"},{"alias_kind":"pith_short_8","alias_value":"PJK4VRE5","created_at":"2026-05-18T12:28:43Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2014:PJK4VRE53TKTLNX63QWF6SGF5N","target":"record","payload":{"canonical_record":{"source":{"id":"1409.0132","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2014-08-30T16:38:30Z","cross_cats_sorted":[],"title_canon_sha256":"f9ff4f0b1a978a505a3a0391adebef6595c51bc3c2ddcfacad333a92119e3dd7","abstract_canon_sha256":"088f349ce6ada2f9eb5f7f319132d3429e72c2e32d33d488c95965eac3ccb5a7"},"schema_version":"1.0"},"canonical_sha256":"7a55cac49ddcd535b6fedc2c5f48c5eb77d93059fc45579afb29af0d1cf6ff5c","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:32:18.984236Z","signature_b64":"lthcaxPaoSLu3aVit4l+YU4zbponj3xQ7B/8Mi/S69odBiZODLE5IoZyjZQXEQG/d4TLVRuOSsSb1+PN1d8KAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7a55cac49ddcd535b6fedc2c5f48c5eb77d93059fc45579afb29af0d1cf6ff5c","last_reissued_at":"2026-05-18T01:32:18.983526Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:32:18.983526Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1409.0132","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:32:18Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Gy2x9SwxgA4bfgxJEgCJGCN88ph6UhOgqqZC1dCLDoRJkkzazlDN1l60l8RX48cK55XTOAMfHYmyBRDkD1rkBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-24T23:12:22.634178Z"},"content_sha256":"2e823137848743079180eb3ef4148ae1225adf0ed31eff7913805b8148a98f13","schema_version":"1.0","event_id":"sha256:2e823137848743079180eb3ef4148ae1225adf0ed31eff7913805b8148a98f13"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2014:PJK4VRE53TKTLNX63QWF6SGF5N","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"The Sub-Index of Critical Points of Distance Functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Barbara Herzog, Frederick Wilhelm","submitted_at":"2014-08-30T16:38:30Z","abstract_excerpt":"We define a new notion---the sub-index of a critical point of a distance function. We show how sub-index affects the homotopy type of sublevel sets of distance functions."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1409.0132","kind":"arxiv","version":3},"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":"2026-05-18T01:32:18Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"RhMKXZbHJqM6maWzKaYQ5WrDbbtdjidR3Nv85rIVuihuWOgc/rEPIVfiQPn1Fg4+bOPT+46hUPkjdH41eglICQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-24T23:12:22.634526Z"},"content_sha256":"9d0744dac874997b4af69aa3914120fcab915c0eba8663046fa97cc19ec99e3c","schema_version":"1.0","event_id":"sha256:9d0744dac874997b4af69aa3914120fcab915c0eba8663046fa97cc19ec99e3c"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/PJK4VRE53TKTLNX63QWF6SGF5N/bundle.json","state_url":"https://pith.science/pith/PJK4VRE53TKTLNX63QWF6SGF5N/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/PJK4VRE53TKTLNX63QWF6SGF5N/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":"2026-06-24T23:12:22Z","links":{"resolver":"https://pith.science/pith/PJK4VRE53TKTLNX63QWF6SGF5N","bundle":"https://pith.science/pith/PJK4VRE53TKTLNX63QWF6SGF5N/bundle.json","state":"https://pith.science/pith/PJK4VRE53TKTLNX63QWF6SGF5N/state.json","well_known_bundle":"https://pith.science/.well-known/pith/PJK4VRE53TKTLNX63QWF6SGF5N/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:PJK4VRE53TKTLNX63QWF6SGF5N","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":"088f349ce6ada2f9eb5f7f319132d3429e72c2e32d33d488c95965eac3ccb5a7","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2014-08-30T16:38:30Z","title_canon_sha256":"f9ff4f0b1a978a505a3a0391adebef6595c51bc3c2ddcfacad333a92119e3dd7"},"schema_version":"1.0","source":{"id":"1409.0132","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1409.0132","created_at":"2026-05-18T01:32:18Z"},{"alias_kind":"arxiv_version","alias_value":"1409.0132v3","created_at":"2026-05-18T01:32:18Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1409.0132","created_at":"2026-05-18T01:32:18Z"},{"alias_kind":"pith_short_12","alias_value":"PJK4VRE53TKT","created_at":"2026-05-18T12:28:43Z"},{"alias_kind":"pith_short_16","alias_value":"PJK4VRE53TKTLNX6","created_at":"2026-05-18T12:28:43Z"},{"alias_kind":"pith_short_8","alias_value":"PJK4VRE5","created_at":"2026-05-18T12:28:43Z"}],"graph_snapshots":[{"event_id":"sha256:9d0744dac874997b4af69aa3914120fcab915c0eba8663046fa97cc19ec99e3c","target":"graph","created_at":"2026-05-18T01:32:18Z","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":"We define a new notion---the sub-index of a critical point of a distance function. We show how sub-index affects the homotopy type of sublevel sets of distance functions.","authors_text":"Barbara Herzog, Frederick Wilhelm","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2014-08-30T16:38:30Z","title":"The Sub-Index of Critical Points of Distance Functions"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1409.0132","kind":"arxiv","version":3},"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":"sha256:2e823137848743079180eb3ef4148ae1225adf0ed31eff7913805b8148a98f13","target":"record","created_at":"2026-05-18T01:32:18Z","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":"088f349ce6ada2f9eb5f7f319132d3429e72c2e32d33d488c95965eac3ccb5a7","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2014-08-30T16:38:30Z","title_canon_sha256":"f9ff4f0b1a978a505a3a0391adebef6595c51bc3c2ddcfacad333a92119e3dd7"},"schema_version":"1.0","source":{"id":"1409.0132","kind":"arxiv","version":3}},"canonical_sha256":"7a55cac49ddcd535b6fedc2c5f48c5eb77d93059fc45579afb29af0d1cf6ff5c","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"7a55cac49ddcd535b6fedc2c5f48c5eb77d93059fc45579afb29af0d1cf6ff5c","first_computed_at":"2026-05-18T01:32:18.983526Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:32:18.983526Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"lthcaxPaoSLu3aVit4l+YU4zbponj3xQ7B/8Mi/S69odBiZODLE5IoZyjZQXEQG/d4TLVRuOSsSb1+PN1d8KAw==","signature_status":"signed_v1","signed_at":"2026-05-18T01:32:18.984236Z","signed_message":"canonical_sha256_bytes"},"source_id":"1409.0132","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:2e823137848743079180eb3ef4148ae1225adf0ed31eff7913805b8148a98f13","sha256:9d0744dac874997b4af69aa3914120fcab915c0eba8663046fa97cc19ec99e3c"],"state_sha256":"80517539f3dfd1425c9864148b9b92b25d25081009fa7d96a297bbdb0ec07ca4"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"i16jYzJuQoHeqx29Aa/2+CzwcPT8F3hc2mCvR7qzVEfIz6aNb6QpJt4qKl33ScOyc0jSsO3oCP6FM/X7F8XQCQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-24T23:12:22.636416Z","bundle_sha256":"a408a1dcaf22aaf68f796e4dbb7f05f1d02f4bc2ae44dfe3f4f9f9bd55c7dc66"}}