{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:FCIGV3NPCDOIIP4D774SOI575S","short_pith_number":"pith:FCIGV3NP","canonical_record":{"source":{"id":"1511.05411","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GN","submitted_at":"2015-11-17T14:11:41Z","cross_cats_sorted":["math.DS"],"title_canon_sha256":"861e25e921ecb5bd446522bc838603b7275853b0e0c28fdb0f1c453d9fb9834c","abstract_canon_sha256":"a1de6be064a2ab1d1119eb515f928320c6e65f073964ffb7fce839f553a8a35d"},"schema_version":"1.0"},"canonical_sha256":"28906aedaf10dc843f83fff92723bfecad1a0d1bc511c9a15b6d28a24cd73bb8","source":{"kind":"arxiv","id":"1511.05411","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1511.05411","created_at":"2026-05-18T00:17:55Z"},{"alias_kind":"arxiv_version","alias_value":"1511.05411v2","created_at":"2026-05-18T00:17:55Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1511.05411","created_at":"2026-05-18T00:17:55Z"},{"alias_kind":"pith_short_12","alias_value":"FCIGV3NPCDOI","created_at":"2026-05-18T12:29:19Z"},{"alias_kind":"pith_short_16","alias_value":"FCIGV3NPCDOIIP4D","created_at":"2026-05-18T12:29:19Z"},{"alias_kind":"pith_short_8","alias_value":"FCIGV3NP","created_at":"2026-05-18T12:29:19Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:FCIGV3NPCDOIIP4D774SOI575S","target":"record","payload":{"canonical_record":{"source":{"id":"1511.05411","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GN","submitted_at":"2015-11-17T14:11:41Z","cross_cats_sorted":["math.DS"],"title_canon_sha256":"861e25e921ecb5bd446522bc838603b7275853b0e0c28fdb0f1c453d9fb9834c","abstract_canon_sha256":"a1de6be064a2ab1d1119eb515f928320c6e65f073964ffb7fce839f553a8a35d"},"schema_version":"1.0"},"canonical_sha256":"28906aedaf10dc843f83fff92723bfecad1a0d1bc511c9a15b6d28a24cd73bb8","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:17:55.480509Z","signature_b64":"RyA+tRxLHmq13VHn8F4xw65rcIONjGIxgYTBCJigPuY1z7cZEu7wyKcXXx6qwZqe1vjaYj7/2m43WYM/B1pODQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"28906aedaf10dc843f83fff92723bfecad1a0d1bc511c9a15b6d28a24cd73bb8","last_reissued_at":"2026-05-18T00:17:55.479834Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:17:55.479834Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1511.05411","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:17:55Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"4npU83Dd4h3iWKaSDevKl2z4KdW7xS5OrHguIZkieroBNSVeAtdPjj5psqUCrhQgJCoLc6uM6HRaGA7MQXeyAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-20T04:35:12.719924Z"},"content_sha256":"8906e3ad7434eb61400a538213f152f436ed2ff65f542aa34b85613bdd4af936","schema_version":"1.0","event_id":"sha256:8906e3ad7434eb61400a538213f152f436ed2ff65f542aa34b85613bdd4af936"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:FCIGV3NPCDOIIP4D774SOI575S","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Space-filling curves of self-similar sets (II): Edge-to-trail substitution rule","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DS"],"primary_cat":"math.GN","authors_text":"Hui Rao, Shu-Qin Zhang, Xi-rong Dai","submitted_at":"2015-11-17T14:11:41Z","abstract_excerpt":"It is well-known that the constructions of space-filling curves depend on certain substitution rules. For a given self-similar set, finding such rules is somehow mysterious, and it is the main concern of the present paper. Our first idea is to introduce the notion of skeleton for a self-similar set. Then, from a skeleton, we construct several graphs, define edge-to-trail substitution rules, and explore conditions ensuring the rules lead to space-filling curves. Thirdly, we summarize the classical constructions of the space-filling curves into two classes: the traveling-trail class and the posi"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1511.05411","kind":"arxiv","version":2},"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":"2026-05-18T00:17:55Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"O2f72lD4tDprIly0rPWviFggh2CRZ7ye3HNLgHgEx0CyhrAZosBcYbDAtAnl7wBv2FZAEaq2R/kq/FXj7InmCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-20T04:35:12.720267Z"},"content_sha256":"d33664dfd8b7bdac894efa33c77f45893eaad0feb918b73991e3e01f07062695","schema_version":"1.0","event_id":"sha256:d33664dfd8b7bdac894efa33c77f45893eaad0feb918b73991e3e01f07062695"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/FCIGV3NPCDOIIP4D774SOI575S/bundle.json","state_url":"https://pith.science/pith/FCIGV3NPCDOIIP4D774SOI575S/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/FCIGV3NPCDOIIP4D774SOI575S/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":"2026-06-20T04:35:12Z","links":{"resolver":"https://pith.science/pith/FCIGV3NPCDOIIP4D774SOI575S","bundle":"https://pith.science/pith/FCIGV3NPCDOIIP4D774SOI575S/bundle.json","state":"https://pith.science/pith/FCIGV3NPCDOIIP4D774SOI575S/state.json","well_known_bundle":"https://pith.science/.well-known/pith/FCIGV3NPCDOIIP4D774SOI575S/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:FCIGV3NPCDOIIP4D774SOI575S","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":"a1de6be064a2ab1d1119eb515f928320c6e65f073964ffb7fce839f553a8a35d","cross_cats_sorted":["math.DS"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GN","submitted_at":"2015-11-17T14:11:41Z","title_canon_sha256":"861e25e921ecb5bd446522bc838603b7275853b0e0c28fdb0f1c453d9fb9834c"},"schema_version":"1.0","source":{"id":"1511.05411","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1511.05411","created_at":"2026-05-18T00:17:55Z"},{"alias_kind":"arxiv_version","alias_value":"1511.05411v2","created_at":"2026-05-18T00:17:55Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1511.05411","created_at":"2026-05-18T00:17:55Z"},{"alias_kind":"pith_short_12","alias_value":"FCIGV3NPCDOI","created_at":"2026-05-18T12:29:19Z"},{"alias_kind":"pith_short_16","alias_value":"FCIGV3NPCDOIIP4D","created_at":"2026-05-18T12:29:19Z"},{"alias_kind":"pith_short_8","alias_value":"FCIGV3NP","created_at":"2026-05-18T12:29:19Z"}],"graph_snapshots":[{"event_id":"sha256:d33664dfd8b7bdac894efa33c77f45893eaad0feb918b73991e3e01f07062695","target":"graph","created_at":"2026-05-18T00:17:55Z","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":"It is well-known that the constructions of space-filling curves depend on certain substitution rules. For a given self-similar set, finding such rules is somehow mysterious, and it is the main concern of the present paper. Our first idea is to introduce the notion of skeleton for a self-similar set. Then, from a skeleton, we construct several graphs, define edge-to-trail substitution rules, and explore conditions ensuring the rules lead to space-filling curves. Thirdly, we summarize the classical constructions of the space-filling curves into two classes: the traveling-trail class and the posi","authors_text":"Hui Rao, Shu-Qin Zhang, Xi-rong Dai","cross_cats":["math.DS"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GN","submitted_at":"2015-11-17T14:11:41Z","title":"Space-filling curves of self-similar sets (II): Edge-to-trail substitution rule"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1511.05411","kind":"arxiv","version":2},"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":"sha256:8906e3ad7434eb61400a538213f152f436ed2ff65f542aa34b85613bdd4af936","target":"record","created_at":"2026-05-18T00:17:55Z","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":"a1de6be064a2ab1d1119eb515f928320c6e65f073964ffb7fce839f553a8a35d","cross_cats_sorted":["math.DS"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GN","submitted_at":"2015-11-17T14:11:41Z","title_canon_sha256":"861e25e921ecb5bd446522bc838603b7275853b0e0c28fdb0f1c453d9fb9834c"},"schema_version":"1.0","source":{"id":"1511.05411","kind":"arxiv","version":2}},"canonical_sha256":"28906aedaf10dc843f83fff92723bfecad1a0d1bc511c9a15b6d28a24cd73bb8","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"28906aedaf10dc843f83fff92723bfecad1a0d1bc511c9a15b6d28a24cd73bb8","first_computed_at":"2026-05-18T00:17:55.479834Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:17:55.479834Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"RyA+tRxLHmq13VHn8F4xw65rcIONjGIxgYTBCJigPuY1z7cZEu7wyKcXXx6qwZqe1vjaYj7/2m43WYM/B1pODQ==","signature_status":"signed_v1","signed_at":"2026-05-18T00:17:55.480509Z","signed_message":"canonical_sha256_bytes"},"source_id":"1511.05411","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:8906e3ad7434eb61400a538213f152f436ed2ff65f542aa34b85613bdd4af936","sha256:d33664dfd8b7bdac894efa33c77f45893eaad0feb918b73991e3e01f07062695"],"state_sha256":"ed29da73e5b5fd38b4dd1b706b2974278e5868b085bbb7db655cd7913d30ba79"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"WPJTGmxuh0FLDvwW9wBiMsGOm/S/fRqXt7P/7NAD2nyS14ErCHwjZP/BKEO6T4tsMVizwGp6Cg5H8rF689a4Cg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-20T04:35:12.722190Z","bundle_sha256":"64cfeb09f9ff806da30739b86117dc1fbff325f9aaccd44c7bc4db0ef291f44d"}}