{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2014:SMKVHBYFI22FBZZB6GFA7RTXU5","short_pith_number":"pith:SMKVHBYF","canonical_record":{"source":{"id":"1401.1034","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2014-01-06T10:53:21Z","cross_cats_sorted":[],"title_canon_sha256":"2af77ff05503deb9118a718bafa5ad32af1e550f964a57acc1ff20c25bd99bee","abstract_canon_sha256":"6db27eb35fc9c8615810f4cbf5af5661845d08c45676a01d69bb46209b9b0447"},"schema_version":"1.0"},"canonical_sha256":"931553870546b450e721f18a0fc677a7777c1694b902c9cf4cc4cb2981b5e4ad","source":{"kind":"arxiv","id":"1401.1034","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1401.1034","created_at":"2026-05-18T03:03:11Z"},{"alias_kind":"arxiv_version","alias_value":"1401.1034v1","created_at":"2026-05-18T03:03:11Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1401.1034","created_at":"2026-05-18T03:03:11Z"},{"alias_kind":"pith_short_12","alias_value":"SMKVHBYFI22F","created_at":"2026-05-18T12:28:49Z"},{"alias_kind":"pith_short_16","alias_value":"SMKVHBYFI22FBZZB","created_at":"2026-05-18T12:28:49Z"},{"alias_kind":"pith_short_8","alias_value":"SMKVHBYF","created_at":"2026-05-18T12:28:49Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2014:SMKVHBYFI22FBZZB6GFA7RTXU5","target":"record","payload":{"canonical_record":{"source":{"id":"1401.1034","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2014-01-06T10:53:21Z","cross_cats_sorted":[],"title_canon_sha256":"2af77ff05503deb9118a718bafa5ad32af1e550f964a57acc1ff20c25bd99bee","abstract_canon_sha256":"6db27eb35fc9c8615810f4cbf5af5661845d08c45676a01d69bb46209b9b0447"},"schema_version":"1.0"},"canonical_sha256":"931553870546b450e721f18a0fc677a7777c1694b902c9cf4cc4cb2981b5e4ad","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:03:11.109873Z","signature_b64":"JPltRTpRRB5cEz1zk7izONIhvUmJj3iF7gHB3yZe0KuQyQBmH96iG0kdO+IAjiGhVaocem4xm/DRTe4itk2ACw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"931553870546b450e721f18a0fc677a7777c1694b902c9cf4cc4cb2981b5e4ad","last_reissued_at":"2026-05-18T03:03:11.109163Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:03:11.109163Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1401.1034","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":"2026-05-18T03:03:11Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Hjb1hTO4VrMsUQ1fCPVwIDyJEo1p8jxpICGOLVzv/YB9MLCgFPuSLMxuvWX/E4u6Aw2Th/J08kvSHFne3v/RAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-26T11:37:43.009198Z"},"content_sha256":"b4ffeadf79c2d80e44aef50d309dccf9487129a0918e3268b97b99a77c537009","schema_version":"1.0","event_id":"sha256:b4ffeadf79c2d80e44aef50d309dccf9487129a0918e3268b97b99a77c537009"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2014:SMKVHBYFI22FBZZB6GFA7RTXU5","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Recurrence for vertex-reinforced random walks on Z with weak reinforcements","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Arvind Singh","submitted_at":"2014-01-06T10:53:21Z","abstract_excerpt":"We prove that any vertex-reinforced random walk on the integer lattice with non-decreasing reinforcement sequence $w$ satisfying $w(k) = o(k^{\\alpha})$ for some $\\alpha < 1/2$ is recurrent. This improves on previous results of Volkov (2006) and Schapira (2012)."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1401.1034","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":""},"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":"2026-05-18T03:03:11Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"in6Ax76OA+xofnnQpvEmkqcCz8xapC8eOGKKaxTO6se1pxi29do/ZSybOpofvBGF0MnWFdaQS3Z7HWJ4+XlIAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-26T11:37:43.009561Z"},"content_sha256":"68e00f2149f91cb1195203ce3e5b6e2a5129520301afb9cb78d49a5acc868da3","schema_version":"1.0","event_id":"sha256:68e00f2149f91cb1195203ce3e5b6e2a5129520301afb9cb78d49a5acc868da3"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/SMKVHBYFI22FBZZB6GFA7RTXU5/bundle.json","state_url":"https://pith.science/pith/SMKVHBYFI22FBZZB6GFA7RTXU5/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/SMKVHBYFI22FBZZB6GFA7RTXU5/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":"2026-05-26T11:37:43Z","links":{"resolver":"https://pith.science/pith/SMKVHBYFI22FBZZB6GFA7RTXU5","bundle":"https://pith.science/pith/SMKVHBYFI22FBZZB6GFA7RTXU5/bundle.json","state":"https://pith.science/pith/SMKVHBYFI22FBZZB6GFA7RTXU5/state.json","well_known_bundle":"https://pith.science/.well-known/pith/SMKVHBYFI22FBZZB6GFA7RTXU5/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:SMKVHBYFI22FBZZB6GFA7RTXU5","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":"6db27eb35fc9c8615810f4cbf5af5661845d08c45676a01d69bb46209b9b0447","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2014-01-06T10:53:21Z","title_canon_sha256":"2af77ff05503deb9118a718bafa5ad32af1e550f964a57acc1ff20c25bd99bee"},"schema_version":"1.0","source":{"id":"1401.1034","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1401.1034","created_at":"2026-05-18T03:03:11Z"},{"alias_kind":"arxiv_version","alias_value":"1401.1034v1","created_at":"2026-05-18T03:03:11Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1401.1034","created_at":"2026-05-18T03:03:11Z"},{"alias_kind":"pith_short_12","alias_value":"SMKVHBYFI22F","created_at":"2026-05-18T12:28:49Z"},{"alias_kind":"pith_short_16","alias_value":"SMKVHBYFI22FBZZB","created_at":"2026-05-18T12:28:49Z"},{"alias_kind":"pith_short_8","alias_value":"SMKVHBYF","created_at":"2026-05-18T12:28:49Z"}],"graph_snapshots":[{"event_id":"sha256:68e00f2149f91cb1195203ce3e5b6e2a5129520301afb9cb78d49a5acc868da3","target":"graph","created_at":"2026-05-18T03:03:11Z","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 prove that any vertex-reinforced random walk on the integer lattice with non-decreasing reinforcement sequence $w$ satisfying $w(k) = o(k^{\\alpha})$ for some $\\alpha < 1/2$ is recurrent. This improves on previous results of Volkov (2006) and Schapira (2012).","authors_text":"Arvind Singh","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2014-01-06T10:53:21Z","title":"Recurrence for vertex-reinforced random walks on Z with weak reinforcements"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1401.1034","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:b4ffeadf79c2d80e44aef50d309dccf9487129a0918e3268b97b99a77c537009","target":"record","created_at":"2026-05-18T03:03:11Z","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":"6db27eb35fc9c8615810f4cbf5af5661845d08c45676a01d69bb46209b9b0447","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2014-01-06T10:53:21Z","title_canon_sha256":"2af77ff05503deb9118a718bafa5ad32af1e550f964a57acc1ff20c25bd99bee"},"schema_version":"1.0","source":{"id":"1401.1034","kind":"arxiv","version":1}},"canonical_sha256":"931553870546b450e721f18a0fc677a7777c1694b902c9cf4cc4cb2981b5e4ad","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"931553870546b450e721f18a0fc677a7777c1694b902c9cf4cc4cb2981b5e4ad","first_computed_at":"2026-05-18T03:03:11.109163Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:03:11.109163Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"JPltRTpRRB5cEz1zk7izONIhvUmJj3iF7gHB3yZe0KuQyQBmH96iG0kdO+IAjiGhVaocem4xm/DRTe4itk2ACw==","signature_status":"signed_v1","signed_at":"2026-05-18T03:03:11.109873Z","signed_message":"canonical_sha256_bytes"},"source_id":"1401.1034","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:b4ffeadf79c2d80e44aef50d309dccf9487129a0918e3268b97b99a77c537009","sha256:68e00f2149f91cb1195203ce3e5b6e2a5129520301afb9cb78d49a5acc868da3"],"state_sha256":"2fdb93d49b84ea5ba4030690adc0a7343d40227d5285c61c7a4c00a9ca45d37d"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ocqLyOp4k/cluK9XcfraxYFRfSJrihgc0EgdjRFvBg2xc7a18Xuc+0ouHvQmdLYd43pqzH6rYLoTX/MEp9XtBg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-26T11:37:43.011873Z","bundle_sha256":"236a57494d1ee41c17b7050d852e493c66529b49c291f94a07a371ca588d0632"}}