{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2006:22MC7E55PDNCHIT7W2X3T6PT7C","short_pith_number":"pith:22MC7E55","canonical_record":{"source":{"id":"math/0603059","kind":"arxiv","version":3},"metadata":{"license":"","primary_cat":"math.GR","submitted_at":"2006-03-02T20:55:15Z","cross_cats_sorted":[],"title_canon_sha256":"25e21e7f9f64edadc9fe37e2b8570227943b15f2f1919ccf3fc1debbae5c4971","abstract_canon_sha256":"36773ae82f256d5c980564b98e5ff31105942f7caec2a5eb0dbd0b81acc22e4d"},"schema_version":"1.0"},"canonical_sha256":"d6982f93bd78da23a27fb6afb9f9f3f893a97189fc1fc18ed1dd9e84507ae433","source":{"kind":"arxiv","id":"math/0603059","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/0603059","created_at":"2026-05-18T04:42:22Z"},{"alias_kind":"arxiv_version","alias_value":"math/0603059v3","created_at":"2026-05-18T04:42:22Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0603059","created_at":"2026-05-18T04:42:22Z"},{"alias_kind":"pith_short_12","alias_value":"22MC7E55PDNC","created_at":"2026-05-18T12:25:53Z"},{"alias_kind":"pith_short_16","alias_value":"22MC7E55PDNCHIT7","created_at":"2026-05-18T12:25:53Z"},{"alias_kind":"pith_short_8","alias_value":"22MC7E55","created_at":"2026-05-18T12:25:53Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2006:22MC7E55PDNCHIT7W2X3T6PT7C","target":"record","payload":{"canonical_record":{"source":{"id":"math/0603059","kind":"arxiv","version":3},"metadata":{"license":"","primary_cat":"math.GR","submitted_at":"2006-03-02T20:55:15Z","cross_cats_sorted":[],"title_canon_sha256":"25e21e7f9f64edadc9fe37e2b8570227943b15f2f1919ccf3fc1debbae5c4971","abstract_canon_sha256":"36773ae82f256d5c980564b98e5ff31105942f7caec2a5eb0dbd0b81acc22e4d"},"schema_version":"1.0"},"canonical_sha256":"d6982f93bd78da23a27fb6afb9f9f3f893a97189fc1fc18ed1dd9e84507ae433","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:42:22.399641Z","signature_b64":"2zQCp/z73JkhvvkTpQH4R2L+hz/TPLbOoNSu68mRAu4hA91waQLmdgi0PJ0SXBT/5wGXL12KgJnfxy9uo7AYDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d6982f93bd78da23a27fb6afb9f9f3f893a97189fc1fc18ed1dd9e84507ae433","last_reissued_at":"2026-05-18T04:42:22.399023Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:42:22.399023Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"math/0603059","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":"2026-05-18T04:42:22Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"TDvAaeITAC2edYPK3AXY7808/CqKDd9zHPUP4hB4JTlUOvYQCzXyBn7ykyaVHAUcU3hJ+0XIv8ITpNwlfGEyCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-11T17:43:24.255322Z"},"content_sha256":"d59c8527ccf53d1248e859e714e76b28964618a827f68fb2e7d4462e915f311b","schema_version":"1.0","event_id":"sha256:d59c8527ccf53d1248e859e714e76b28964618a827f68fb2e7d4462e915f311b"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2006:22MC7E55PDNCHIT7W2X3T6PT7C","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Filling functions","license":"","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"T.R.Riley","submitted_at":"2006-03-02T20:55:15Z","abstract_excerpt":"Filling functions are asymptotic invariants of finitely presentable groups; the seminal work on the subject is by M.Gromov. They record features of combinatorial homotopy discs (van Kampen diagrams) filling loops in Cayley 2-complexes. Examples are the Dehn (or isoperimetric) function, the filling length function and the intrinsic diameter (or isodiametric) function. We discuss filling functions from geometric, combinatorial and computational points of view, we survey their interrelationships, and we sketch their roles in the studies of nilpotent groups, hyperbolic groups and asymptotic cones."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0603059","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":"2026-05-18T04:42:22Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"k67ND9Ch99U8A0rCrLDxGeuMVsBKUxwsybv3SE5jnA8gZNA2hMKh8eKmntYgqQTgVlXr3ZZroBtm9AxXGFI/Dw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-11T17:43:24.255778Z"},"content_sha256":"33e7649f4367a967d4fec7c65e0cba76299624a69d69377221d1fefebcf36764","schema_version":"1.0","event_id":"sha256:33e7649f4367a967d4fec7c65e0cba76299624a69d69377221d1fefebcf36764"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/22MC7E55PDNCHIT7W2X3T6PT7C/bundle.json","state_url":"https://pith.science/pith/22MC7E55PDNCHIT7W2X3T6PT7C/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/22MC7E55PDNCHIT7W2X3T6PT7C/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":"2026-08-11T17:43:24Z","links":{"resolver":"https://pith.science/pith/22MC7E55PDNCHIT7W2X3T6PT7C","bundle":"https://pith.science/pith/22MC7E55PDNCHIT7W2X3T6PT7C/bundle.json","state":"https://pith.science/pith/22MC7E55PDNCHIT7W2X3T6PT7C/state.json","well_known_bundle":"https://pith.science/.well-known/pith/22MC7E55PDNCHIT7W2X3T6PT7C/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2006:22MC7E55PDNCHIT7W2X3T6PT7C","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":"36773ae82f256d5c980564b98e5ff31105942f7caec2a5eb0dbd0b81acc22e4d","cross_cats_sorted":[],"license":"","primary_cat":"math.GR","submitted_at":"2006-03-02T20:55:15Z","title_canon_sha256":"25e21e7f9f64edadc9fe37e2b8570227943b15f2f1919ccf3fc1debbae5c4971"},"schema_version":"1.0","source":{"id":"math/0603059","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/0603059","created_at":"2026-05-18T04:42:22Z"},{"alias_kind":"arxiv_version","alias_value":"math/0603059v3","created_at":"2026-05-18T04:42:22Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0603059","created_at":"2026-05-18T04:42:22Z"},{"alias_kind":"pith_short_12","alias_value":"22MC7E55PDNC","created_at":"2026-05-18T12:25:53Z"},{"alias_kind":"pith_short_16","alias_value":"22MC7E55PDNCHIT7","created_at":"2026-05-18T12:25:53Z"},{"alias_kind":"pith_short_8","alias_value":"22MC7E55","created_at":"2026-05-18T12:25:53Z"}],"graph_snapshots":[{"event_id":"sha256:33e7649f4367a967d4fec7c65e0cba76299624a69d69377221d1fefebcf36764","target":"graph","created_at":"2026-05-18T04:42:22Z","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":"Filling functions are asymptotic invariants of finitely presentable groups; the seminal work on the subject is by M.Gromov. They record features of combinatorial homotopy discs (van Kampen diagrams) filling loops in Cayley 2-complexes. Examples are the Dehn (or isoperimetric) function, the filling length function and the intrinsic diameter (or isodiametric) function. We discuss filling functions from geometric, combinatorial and computational points of view, we survey their interrelationships, and we sketch their roles in the studies of nilpotent groups, hyperbolic groups and asymptotic cones.","authors_text":"T.R.Riley","cross_cats":[],"headline":"","license":"","primary_cat":"math.GR","submitted_at":"2006-03-02T20:55:15Z","title":"Filling functions"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0603059","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:d59c8527ccf53d1248e859e714e76b28964618a827f68fb2e7d4462e915f311b","target":"record","created_at":"2026-05-18T04:42:22Z","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":"36773ae82f256d5c980564b98e5ff31105942f7caec2a5eb0dbd0b81acc22e4d","cross_cats_sorted":[],"license":"","primary_cat":"math.GR","submitted_at":"2006-03-02T20:55:15Z","title_canon_sha256":"25e21e7f9f64edadc9fe37e2b8570227943b15f2f1919ccf3fc1debbae5c4971"},"schema_version":"1.0","source":{"id":"math/0603059","kind":"arxiv","version":3}},"canonical_sha256":"d6982f93bd78da23a27fb6afb9f9f3f893a97189fc1fc18ed1dd9e84507ae433","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"d6982f93bd78da23a27fb6afb9f9f3f893a97189fc1fc18ed1dd9e84507ae433","first_computed_at":"2026-05-18T04:42:22.399023Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:42:22.399023Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"2zQCp/z73JkhvvkTpQH4R2L+hz/TPLbOoNSu68mRAu4hA91waQLmdgi0PJ0SXBT/5wGXL12KgJnfxy9uo7AYDw==","signature_status":"signed_v1","signed_at":"2026-05-18T04:42:22.399641Z","signed_message":"canonical_sha256_bytes"},"source_id":"math/0603059","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:d59c8527ccf53d1248e859e714e76b28964618a827f68fb2e7d4462e915f311b","sha256:33e7649f4367a967d4fec7c65e0cba76299624a69d69377221d1fefebcf36764"],"state_sha256":"1f34fc0967347799567e69658603e810d8185f7fc661e1aacfd9df3c0e3473fa"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"tb34aP3tbZ5/Uq7d8ioNHSK8dZaAyz5zxuZSlt83rXel1bFQ2XDJW6J/TF240rs/Bh3iZhoIxuz/n3o6djAvDw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-11T17:43:24.286592Z","bundle_sha256":"0bd80297910dd46bbfe621a865bce2d9ce89e1aad1fbd9cc3ea306f700a860ce"}}