{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2025:XU5CCUIPDFKFIQ25MNLBAW2TBO","short_pith_number":"pith:XU5CCUIP","canonical_record":{"source":{"id":"2509.09191","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math-ph","submitted_at":"2025-09-11T07:02:48Z","cross_cats_sorted":["math.MP"],"title_canon_sha256":"d7ec2ac239b458b25f2fb42b454726286f19effc502b94ab0b431c16931e2513","abstract_canon_sha256":"6403c7bf54038431496689592541d773d4015a45c5b9557feb331da598b64df4"},"schema_version":"1.0"},"canonical_sha256":"bd3a21510f195454435d6356105b530bb00ceee005ad25830d3dbd39749a2ab6","source":{"kind":"arxiv","id":"2509.09191","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2509.09191","created_at":"2026-07-05T12:09:31Z"},{"alias_kind":"arxiv_version","alias_value":"2509.09191v1","created_at":"2026-07-05T12:09:31Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2509.09191","created_at":"2026-07-05T12:09:31Z"},{"alias_kind":"pith_short_12","alias_value":"XU5CCUIPDFKF","created_at":"2026-07-05T12:09:31Z"},{"alias_kind":"pith_short_16","alias_value":"XU5CCUIPDFKFIQ25","created_at":"2026-07-05T12:09:31Z"},{"alias_kind":"pith_short_8","alias_value":"XU5CCUIP","created_at":"2026-07-05T12:09:31Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2025:XU5CCUIPDFKFIQ25MNLBAW2TBO","target":"record","payload":{"canonical_record":{"source":{"id":"2509.09191","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math-ph","submitted_at":"2025-09-11T07:02:48Z","cross_cats_sorted":["math.MP"],"title_canon_sha256":"d7ec2ac239b458b25f2fb42b454726286f19effc502b94ab0b431c16931e2513","abstract_canon_sha256":"6403c7bf54038431496689592541d773d4015a45c5b9557feb331da598b64df4"},"schema_version":"1.0"},"canonical_sha256":"bd3a21510f195454435d6356105b530bb00ceee005ad25830d3dbd39749a2ab6","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T12:09:31.284404Z","signature_b64":"N4YVl5qZQZgCyXXw5pNJvUBaPB1Q7lTMSX6bJAUuGR+O6HtznJzGC9YiJIsN+tw/idBPTTjjkER6NlgXS70GAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"bd3a21510f195454435d6356105b530bb00ceee005ad25830d3dbd39749a2ab6","last_reissued_at":"2026-07-05T12:09:31.283884Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T12:09:31.283884Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2509.09191","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":"2026-07-05T12:09:31Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"dQnrmOql3FfuLzoE/rS1xlRdgIXAeQ1sYi7ImlECIdUifjHc3lEAP8tL+HmHyaWuqZ6RgMNzNKvDOTDoTRexAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-10T21:57:47.567411Z"},"content_sha256":"4591c9df660239da9dfd98ce79f654f1b2174c64e329d7c72f3be0a6785858e5","schema_version":"1.0","event_id":"sha256:4591c9df660239da9dfd98ce79f654f1b2174c64e329d7c72f3be0a6785858e5"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2025:XU5CCUIPDFKFIQ25MNLBAW2TBO","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Permutation-Based Distances for Groups and Group-Valued Time Series","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.MP"],"primary_cat":"math-ph","authors_text":"Jos\\'e M. Amig\\'o, Roberto Dale","submitted_at":"2025-09-11T07:02:48Z","abstract_excerpt":"Permutations on a set, endowed with function composition, build a group called a symmetric group. In addition to their algebraic structure, symmetric groups have two metrics that are of particular interest to us here: the Cayley distance and the Kendall tau distance. In fact, the aim of this paper is to introduce the concept of distance in a general finite group based on them. The main tool that we use to this end is Cayley's theorem, which states that any finite group is isomorphic to a subgroup of a certain symmetric group. We also discuss the advantages and disadvantage of these permutation"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2509.09191","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/2509.09191/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":"2026-07-05T12:09:31Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"0BI9n/sU+80+ZOLVXVWoHmiyTGiZxG2dKQ+7VoUMX/ULNMJFRSSu7pqye5mkJ2A/BcADgva5DLh9bBKuXx+rCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-10T21:57:47.567932Z"},"content_sha256":"bdcdee903880739d93f7b5cc3ce39e564b613a9319f05d72a9eb327da204a75b","schema_version":"1.0","event_id":"sha256:bdcdee903880739d93f7b5cc3ce39e564b613a9319f05d72a9eb327da204a75b"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/XU5CCUIPDFKFIQ25MNLBAW2TBO/bundle.json","state_url":"https://pith.science/pith/XU5CCUIPDFKFIQ25MNLBAW2TBO/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/XU5CCUIPDFKFIQ25MNLBAW2TBO/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":"2026-08-10T21:57:47Z","links":{"resolver":"https://pith.science/pith/XU5CCUIPDFKFIQ25MNLBAW2TBO","bundle":"https://pith.science/pith/XU5CCUIPDFKFIQ25MNLBAW2TBO/bundle.json","state":"https://pith.science/pith/XU5CCUIPDFKFIQ25MNLBAW2TBO/state.json","well_known_bundle":"https://pith.science/.well-known/pith/XU5CCUIPDFKFIQ25MNLBAW2TBO/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2025:XU5CCUIPDFKFIQ25MNLBAW2TBO","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":"6403c7bf54038431496689592541d773d4015a45c5b9557feb331da598b64df4","cross_cats_sorted":["math.MP"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math-ph","submitted_at":"2025-09-11T07:02:48Z","title_canon_sha256":"d7ec2ac239b458b25f2fb42b454726286f19effc502b94ab0b431c16931e2513"},"schema_version":"1.0","source":{"id":"2509.09191","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2509.09191","created_at":"2026-07-05T12:09:31Z"},{"alias_kind":"arxiv_version","alias_value":"2509.09191v1","created_at":"2026-07-05T12:09:31Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2509.09191","created_at":"2026-07-05T12:09:31Z"},{"alias_kind":"pith_short_12","alias_value":"XU5CCUIPDFKF","created_at":"2026-07-05T12:09:31Z"},{"alias_kind":"pith_short_16","alias_value":"XU5CCUIPDFKFIQ25","created_at":"2026-07-05T12:09:31Z"},{"alias_kind":"pith_short_8","alias_value":"XU5CCUIP","created_at":"2026-07-05T12:09:31Z"}],"graph_snapshots":[{"event_id":"sha256:bdcdee903880739d93f7b5cc3ce39e564b613a9319f05d72a9eb327da204a75b","target":"graph","created_at":"2026-07-05T12:09:31Z","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/2509.09191/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Permutations on a set, endowed with function composition, build a group called a symmetric group. In addition to their algebraic structure, symmetric groups have two metrics that are of particular interest to us here: the Cayley distance and the Kendall tau distance. In fact, the aim of this paper is to introduce the concept of distance in a general finite group based on them. The main tool that we use to this end is Cayley's theorem, which states that any finite group is isomorphic to a subgroup of a certain symmetric group. We also discuss the advantages and disadvantage of these permutation","authors_text":"Jos\\'e M. Amig\\'o, Roberto Dale","cross_cats":["math.MP"],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math-ph","submitted_at":"2025-09-11T07:02:48Z","title":"Permutation-Based Distances for Groups and Group-Valued Time Series"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2509.09191","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":"sha256:4591c9df660239da9dfd98ce79f654f1b2174c64e329d7c72f3be0a6785858e5","target":"record","created_at":"2026-07-05T12:09:31Z","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":"6403c7bf54038431496689592541d773d4015a45c5b9557feb331da598b64df4","cross_cats_sorted":["math.MP"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math-ph","submitted_at":"2025-09-11T07:02:48Z","title_canon_sha256":"d7ec2ac239b458b25f2fb42b454726286f19effc502b94ab0b431c16931e2513"},"schema_version":"1.0","source":{"id":"2509.09191","kind":"arxiv","version":1}},"canonical_sha256":"bd3a21510f195454435d6356105b530bb00ceee005ad25830d3dbd39749a2ab6","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"bd3a21510f195454435d6356105b530bb00ceee005ad25830d3dbd39749a2ab6","first_computed_at":"2026-07-05T12:09:31.283884Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T12:09:31.283884Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"N4YVl5qZQZgCyXXw5pNJvUBaPB1Q7lTMSX6bJAUuGR+O6HtznJzGC9YiJIsN+tw/idBPTTjjkER6NlgXS70GAA==","signature_status":"signed_v1","signed_at":"2026-07-05T12:09:31.284404Z","signed_message":"canonical_sha256_bytes"},"source_id":"2509.09191","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:4591c9df660239da9dfd98ce79f654f1b2174c64e329d7c72f3be0a6785858e5","sha256:bdcdee903880739d93f7b5cc3ce39e564b613a9319f05d72a9eb327da204a75b"],"state_sha256":"96946146665cef737fff0239bf47a3d2b3246d125ab19846328be57ffb6dfeea"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Wnysh7H3dGMg2eZYOGRYobAGXOrITyAvQz9aMF14ojmFwdI7YXl20dzL8dDi0DvuhywPTvTM01lQXHsuN8ezCQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-10T21:57:47.571923Z","bundle_sha256":"6524e8de29d5dc395e2f19df04b1223993f01504239bdbd7855587064134ebc1"}}