{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2024:6AW3OAMXREJO4263X7B7REIN6S","short_pith_number":"pith:6AW3OAMX","canonical_record":{"source":{"id":"2405.18169","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","primary_cat":"math.PR","submitted_at":"2024-05-28T13:30:06Z","cross_cats_sorted":["math.CO","math.GR"],"title_canon_sha256":"9eae7c72e0986001a2887943a68be18bdabc7c594b7ba1c76a5314ee218f539e","abstract_canon_sha256":"da066df7b73fcb17d0ad3366b043a3f51b08558c2e795714cb6fe906d855f19a"},"schema_version":"1.0"},"canonical_sha256":"f02db701978912ee6bdbbfc3f8910df48ce94d46250372ab9355966460d59149","source":{"kind":"arxiv","id":"2405.18169","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2405.18169","created_at":"2026-07-05T09:54:27Z"},{"alias_kind":"arxiv_version","alias_value":"2405.18169v2","created_at":"2026-07-05T09:54:27Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2405.18169","created_at":"2026-07-05T09:54:27Z"},{"alias_kind":"pith_short_12","alias_value":"6AW3OAMXREJO","created_at":"2026-07-05T09:54:27Z"},{"alias_kind":"pith_short_16","alias_value":"6AW3OAMXREJO4263","created_at":"2026-07-05T09:54:27Z"},{"alias_kind":"pith_short_8","alias_value":"6AW3OAMX","created_at":"2026-07-05T09:54:27Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2024:6AW3OAMXREJO4263X7B7REIN6S","target":"record","payload":{"canonical_record":{"source":{"id":"2405.18169","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","primary_cat":"math.PR","submitted_at":"2024-05-28T13:30:06Z","cross_cats_sorted":["math.CO","math.GR"],"title_canon_sha256":"9eae7c72e0986001a2887943a68be18bdabc7c594b7ba1c76a5314ee218f539e","abstract_canon_sha256":"da066df7b73fcb17d0ad3366b043a3f51b08558c2e795714cb6fe906d855f19a"},"schema_version":"1.0"},"canonical_sha256":"f02db701978912ee6bdbbfc3f8910df48ce94d46250372ab9355966460d59149","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T09:54:27.041039Z","signature_b64":"5jpY+UbNFcFfp890lQvkRv9xLj5mZYbyW4I7QY03ENxIjajxqNTTVrJh5F0kyDphqlZ/nxAnB1fObCGcQiQGCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f02db701978912ee6bdbbfc3f8910df48ce94d46250372ab9355966460d59149","last_reissued_at":"2026-07-05T09:54:27.040555Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T09:54:27.040555Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2405.18169","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":"2026-07-05T09:54:27Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"xcWYiFnsbc5GhQjQ6JCh38TynHXBoz4F5kn71C5xf0CaNry89R5mjZZl8Hqs7gYWr8Wyvt1Yc3auW5GFZW9LBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-05T00:28:26.762890Z"},"content_sha256":"96af0f84134fb2c89d021c4792e1cd6db038853c983f6dddd9d3bcfde3b5e7f2","schema_version":"1.0","event_id":"sha256:96af0f84134fb2c89d021c4792e1cd6db038853c983f6dddd9d3bcfde3b5e7f2"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2024:6AW3OAMXREJO4263X7B7REIN6S","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"An extension of the cogrowth formula to arbitrary subsets of the tree","license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","headline":"","cross_cats":["math.CO","math.GR"],"primary_cat":"math.PR","authors_text":"Doron Puder","submitted_at":"2024-05-28T13:30:06Z","abstract_excerpt":"What is the probability that a random walk in the free group ends in a proper power? Or in a primitive element? We present a formula that computes the exponential decay rate of the probability that a random walk on a regular tree ends in a given subset, in terms of the exponential decay rate of the analogous probability of the non-backtracking random walk. This generalizes the well-known cogrowth formula of Grigorchuk, Cohen and Northshield. We also extend the formula to arbitrary subsets of the biregular tree."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2405.18169","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2405.18169/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":"2026-07-05T09:54:27Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"PqJOgxmhdv3xl88HE8gom/1FqRyao0Wmrh95CAUeiGvk2L93KD0cDjRotcgwD0XFOG2+5gXFHVS+Ypa57KhdDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-05T00:28:26.763390Z"},"content_sha256":"b1507b0c4207de334513b61f6097358291ff9bb8979a5358e68b6f60abebd99c","schema_version":"1.0","event_id":"sha256:b1507b0c4207de334513b61f6097358291ff9bb8979a5358e68b6f60abebd99c"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/6AW3OAMXREJO4263X7B7REIN6S/bundle.json","state_url":"https://pith.science/pith/6AW3OAMXREJO4263X7B7REIN6S/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/6AW3OAMXREJO4263X7B7REIN6S/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":"2026-08-05T00:28:26Z","links":{"resolver":"https://pith.science/pith/6AW3OAMXREJO4263X7B7REIN6S","bundle":"https://pith.science/pith/6AW3OAMXREJO4263X7B7REIN6S/bundle.json","state":"https://pith.science/pith/6AW3OAMXREJO4263X7B7REIN6S/state.json","well_known_bundle":"https://pith.science/.well-known/pith/6AW3OAMXREJO4263X7B7REIN6S/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2024:6AW3OAMXREJO4263X7B7REIN6S","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":"da066df7b73fcb17d0ad3366b043a3f51b08558c2e795714cb6fe906d855f19a","cross_cats_sorted":["math.CO","math.GR"],"license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","primary_cat":"math.PR","submitted_at":"2024-05-28T13:30:06Z","title_canon_sha256":"9eae7c72e0986001a2887943a68be18bdabc7c594b7ba1c76a5314ee218f539e"},"schema_version":"1.0","source":{"id":"2405.18169","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2405.18169","created_at":"2026-07-05T09:54:27Z"},{"alias_kind":"arxiv_version","alias_value":"2405.18169v2","created_at":"2026-07-05T09:54:27Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2405.18169","created_at":"2026-07-05T09:54:27Z"},{"alias_kind":"pith_short_12","alias_value":"6AW3OAMXREJO","created_at":"2026-07-05T09:54:27Z"},{"alias_kind":"pith_short_16","alias_value":"6AW3OAMXREJO4263","created_at":"2026-07-05T09:54:27Z"},{"alias_kind":"pith_short_8","alias_value":"6AW3OAMX","created_at":"2026-07-05T09:54:27Z"}],"graph_snapshots":[{"event_id":"sha256:b1507b0c4207de334513b61f6097358291ff9bb8979a5358e68b6f60abebd99c","target":"graph","created_at":"2026-07-05T09:54:27Z","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/2405.18169/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"What is the probability that a random walk in the free group ends in a proper power? Or in a primitive element? We present a formula that computes the exponential decay rate of the probability that a random walk on a regular tree ends in a given subset, in terms of the exponential decay rate of the analogous probability of the non-backtracking random walk. This generalizes the well-known cogrowth formula of Grigorchuk, Cohen and Northshield. We also extend the formula to arbitrary subsets of the biregular tree.","authors_text":"Doron Puder","cross_cats":["math.CO","math.GR"],"headline":"","license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","primary_cat":"math.PR","submitted_at":"2024-05-28T13:30:06Z","title":"An extension of the cogrowth formula to arbitrary subsets of the tree"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2405.18169","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":"sha256:96af0f84134fb2c89d021c4792e1cd6db038853c983f6dddd9d3bcfde3b5e7f2","target":"record","created_at":"2026-07-05T09:54:27Z","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":"da066df7b73fcb17d0ad3366b043a3f51b08558c2e795714cb6fe906d855f19a","cross_cats_sorted":["math.CO","math.GR"],"license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","primary_cat":"math.PR","submitted_at":"2024-05-28T13:30:06Z","title_canon_sha256":"9eae7c72e0986001a2887943a68be18bdabc7c594b7ba1c76a5314ee218f539e"},"schema_version":"1.0","source":{"id":"2405.18169","kind":"arxiv","version":2}},"canonical_sha256":"f02db701978912ee6bdbbfc3f8910df48ce94d46250372ab9355966460d59149","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"f02db701978912ee6bdbbfc3f8910df48ce94d46250372ab9355966460d59149","first_computed_at":"2026-07-05T09:54:27.040555Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T09:54:27.040555Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"5jpY+UbNFcFfp890lQvkRv9xLj5mZYbyW4I7QY03ENxIjajxqNTTVrJh5F0kyDphqlZ/nxAnB1fObCGcQiQGCg==","signature_status":"signed_v1","signed_at":"2026-07-05T09:54:27.041039Z","signed_message":"canonical_sha256_bytes"},"source_id":"2405.18169","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:96af0f84134fb2c89d021c4792e1cd6db038853c983f6dddd9d3bcfde3b5e7f2","sha256:b1507b0c4207de334513b61f6097358291ff9bb8979a5358e68b6f60abebd99c"],"state_sha256":"77af7c3f7f862d45802c561b732d473fdedad563ad961d571a27db5c49924c1e"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"2JqhC51U9n6GqnclMyow83PF3lA/lph/R1Cf4RoZ233Zc/6I1CR6e76Q2dHALE6Y/T9gaOUaWK8tVfZx7qfNBg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-05T00:28:26.768971Z","bundle_sha256":"746211618056853259602a6e997063f98cd82a3d3e6bd377b5525d0098b3077e"}}