{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2017:6P5VSY4KMXAD3LF6U6PZOCZXM5","short_pith_number":"pith:6P5VSY4K","canonical_record":{"source":{"id":"1710.09670","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2017-10-26T12:55:31Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"38f84ed4af1e63ff3541a26111b16f411db6c4e438efe48a7928f9ea2ffe2c1b","abstract_canon_sha256":"b1e5b28ca0767a4a5a2eff9ccf34999d2ebf8f1791781d349659ac7d4d3abd6e"},"schema_version":"1.0"},"canonical_sha256":"f3fb59638a65c03dacbea79f970b37675c96b221bda553689abfefe7b1b3eb4d","source":{"kind":"arxiv","id":"1710.09670","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1710.09670","created_at":"2026-05-18T00:31:56Z"},{"alias_kind":"arxiv_version","alias_value":"1710.09670v1","created_at":"2026-05-18T00:31:56Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1710.09670","created_at":"2026-05-18T00:31:56Z"},{"alias_kind":"pith_short_12","alias_value":"6P5VSY4KMXAD","created_at":"2026-05-18T12:31:03Z"},{"alias_kind":"pith_short_16","alias_value":"6P5VSY4KMXAD3LF6","created_at":"2026-05-18T12:31:03Z"},{"alias_kind":"pith_short_8","alias_value":"6P5VSY4K","created_at":"2026-05-18T12:31:03Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2017:6P5VSY4KMXAD3LF6U6PZOCZXM5","target":"record","payload":{"canonical_record":{"source":{"id":"1710.09670","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2017-10-26T12:55:31Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"38f84ed4af1e63ff3541a26111b16f411db6c4e438efe48a7928f9ea2ffe2c1b","abstract_canon_sha256":"b1e5b28ca0767a4a5a2eff9ccf34999d2ebf8f1791781d349659ac7d4d3abd6e"},"schema_version":"1.0"},"canonical_sha256":"f3fb59638a65c03dacbea79f970b37675c96b221bda553689abfefe7b1b3eb4d","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:31:56.674771Z","signature_b64":"TxmBPdoSEOovy+m0S4wYKBq0NWm6edpuBtZ0C7uqtSpwlrUS/RojqtBi7wBrcdkfqqis41iZB8qf8+/5SWFEBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f3fb59638a65c03dacbea79f970b37675c96b221bda553689abfefe7b1b3eb4d","last_reissued_at":"2026-05-18T00:31:56.674426Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:31:56.674426Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1710.09670","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":"2026-05-18T00:31:56Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"2gT53Xjcpl0oBOGg+eK+hcST1Dp8uP/ccfSMTe1HtPgLK8kBrI0WGFhNrf7S6totXNg0rQYvNi0KTh3GldLPBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-30T09:00:43.040898Z"},"content_sha256":"99ea3a40cf4da4fc9df6633a6cdadb453103cd29d9b7ed8011d9bf75bfcbe7f2","schema_version":"1.0","event_id":"sha256:99ea3a40cf4da4fc9df6633a6cdadb453103cd29d9b7ed8011d9bf75bfcbe7f2"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2017:6P5VSY4KMXAD3LF6U6PZOCZXM5","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Spitzer's identity for discrete random walks","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.PR","authors_text":"A.J.E.M. Janssen, Johan S.H. van Leeuwaarden","submitted_at":"2017-10-26T12:55:31Z","abstract_excerpt":"Spitzer's identity describes the position of a reflected random walk over time in terms of a bivariate transform. Among its many applications in probability theory are congestion levels in queues and random walkers in physics. We present a new derivation of Spitzer's identity under the assumption that the increments of the random walk have bounded jumps to the left. This mild assumption facilitates a proof of Spitzer's identity that only uses basic properties of analytic functions and contour integration. The main novelty, believed to be of broader interest, is a reversed approach that recogni"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1710.09670","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":"2026-05-18T00:31:56Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Ok9L3T1ghvIyZeM+zLZejU7WDT+/JYTElu7pdsQBdRY4LH43dEM5/4sBhPGAfzrE3UK0m1E8H1+wwDaxIjJ5Bw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-30T09:00:43.041543Z"},"content_sha256":"194c9dbeea88e5f4e0e2cd412131a2e959da255e90ad3d938926e3b7b3995295","schema_version":"1.0","event_id":"sha256:194c9dbeea88e5f4e0e2cd412131a2e959da255e90ad3d938926e3b7b3995295"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/6P5VSY4KMXAD3LF6U6PZOCZXM5/bundle.json","state_url":"https://pith.science/pith/6P5VSY4KMXAD3LF6U6PZOCZXM5/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/6P5VSY4KMXAD3LF6U6PZOCZXM5/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":"2026-05-30T09:00:43Z","links":{"resolver":"https://pith.science/pith/6P5VSY4KMXAD3LF6U6PZOCZXM5","bundle":"https://pith.science/pith/6P5VSY4KMXAD3LF6U6PZOCZXM5/bundle.json","state":"https://pith.science/pith/6P5VSY4KMXAD3LF6U6PZOCZXM5/state.json","well_known_bundle":"https://pith.science/.well-known/pith/6P5VSY4KMXAD3LF6U6PZOCZXM5/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2017:6P5VSY4KMXAD3LF6U6PZOCZXM5","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":"b1e5b28ca0767a4a5a2eff9ccf34999d2ebf8f1791781d349659ac7d4d3abd6e","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2017-10-26T12:55:31Z","title_canon_sha256":"38f84ed4af1e63ff3541a26111b16f411db6c4e438efe48a7928f9ea2ffe2c1b"},"schema_version":"1.0","source":{"id":"1710.09670","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1710.09670","created_at":"2026-05-18T00:31:56Z"},{"alias_kind":"arxiv_version","alias_value":"1710.09670v1","created_at":"2026-05-18T00:31:56Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1710.09670","created_at":"2026-05-18T00:31:56Z"},{"alias_kind":"pith_short_12","alias_value":"6P5VSY4KMXAD","created_at":"2026-05-18T12:31:03Z"},{"alias_kind":"pith_short_16","alias_value":"6P5VSY4KMXAD3LF6","created_at":"2026-05-18T12:31:03Z"},{"alias_kind":"pith_short_8","alias_value":"6P5VSY4K","created_at":"2026-05-18T12:31:03Z"}],"graph_snapshots":[{"event_id":"sha256:194c9dbeea88e5f4e0e2cd412131a2e959da255e90ad3d938926e3b7b3995295","target":"graph","created_at":"2026-05-18T00:31:56Z","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":"Spitzer's identity describes the position of a reflected random walk over time in terms of a bivariate transform. Among its many applications in probability theory are congestion levels in queues and random walkers in physics. We present a new derivation of Spitzer's identity under the assumption that the increments of the random walk have bounded jumps to the left. This mild assumption facilitates a proof of Spitzer's identity that only uses basic properties of analytic functions and contour integration. The main novelty, believed to be of broader interest, is a reversed approach that recogni","authors_text":"A.J.E.M. Janssen, Johan S.H. van Leeuwaarden","cross_cats":["math.CO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2017-10-26T12:55:31Z","title":"Spitzer's identity for discrete random walks"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1710.09670","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":"sha256:99ea3a40cf4da4fc9df6633a6cdadb453103cd29d9b7ed8011d9bf75bfcbe7f2","target":"record","created_at":"2026-05-18T00:31:56Z","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":"b1e5b28ca0767a4a5a2eff9ccf34999d2ebf8f1791781d349659ac7d4d3abd6e","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2017-10-26T12:55:31Z","title_canon_sha256":"38f84ed4af1e63ff3541a26111b16f411db6c4e438efe48a7928f9ea2ffe2c1b"},"schema_version":"1.0","source":{"id":"1710.09670","kind":"arxiv","version":1}},"canonical_sha256":"f3fb59638a65c03dacbea79f970b37675c96b221bda553689abfefe7b1b3eb4d","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"f3fb59638a65c03dacbea79f970b37675c96b221bda553689abfefe7b1b3eb4d","first_computed_at":"2026-05-18T00:31:56.674426Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:31:56.674426Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"TxmBPdoSEOovy+m0S4wYKBq0NWm6edpuBtZ0C7uqtSpwlrUS/RojqtBi7wBrcdkfqqis41iZB8qf8+/5SWFEBA==","signature_status":"signed_v1","signed_at":"2026-05-18T00:31:56.674771Z","signed_message":"canonical_sha256_bytes"},"source_id":"1710.09670","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:99ea3a40cf4da4fc9df6633a6cdadb453103cd29d9b7ed8011d9bf75bfcbe7f2","sha256:194c9dbeea88e5f4e0e2cd412131a2e959da255e90ad3d938926e3b7b3995295"],"state_sha256":"17406824e38642106fb80f6bf02a017a8830468bad747fdb09a1a5f6520a05de"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"b6RE/dGWQGWOA9UXvBYTilISbI1dRFL3GXdG/P7STMPImC5Mq798W20u8BgOq2APKrjVhFYXSmI6N3AjaBFNCg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-30T09:00:43.045242Z","bundle_sha256":"8275e3bbe6205c38fc99999f8a86f20e5fd8060a235f495efceb8cef30368f40"}}