{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2004:7Z2U2HRQWC6B4OJTB6PP4F6BAR","short_pith_number":"pith:7Z2U2HRQ","canonical_record":{"source":{"id":"math/0405601","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.PR","submitted_at":"2004-05-31T18:50:10Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"adacd9ab618e9873473b3efcc96ed12f8562ae574e00452fc716b888905d8ea5","abstract_canon_sha256":"83c8d80b6cc619fd019b72330892c794e34e16664c19dcdbe2f936fe8e17a7d3"},"schema_version":"1.0"},"canonical_sha256":"fe754d1e30b0bc1e39330f9efe17c1047b581a268fdb5bc40dcca5ce96c6cd0d","source":{"kind":"arxiv","id":"math/0405601","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/0405601","created_at":"2026-07-04T14:38:40Z"},{"alias_kind":"arxiv_version","alias_value":"math/0405601v1","created_at":"2026-07-04T14:38:40Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0405601","created_at":"2026-07-04T14:38:40Z"},{"alias_kind":"pith_short_12","alias_value":"7Z2U2HRQWC6B","created_at":"2026-07-04T14:38:40Z"},{"alias_kind":"pith_short_16","alias_value":"7Z2U2HRQWC6B4OJT","created_at":"2026-07-04T14:38:40Z"},{"alias_kind":"pith_short_8","alias_value":"7Z2U2HRQ","created_at":"2026-07-04T14:38:40Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2004:7Z2U2HRQWC6B4OJTB6PP4F6BAR","target":"record","payload":{"canonical_record":{"source":{"id":"math/0405601","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.PR","submitted_at":"2004-05-31T18:50:10Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"adacd9ab618e9873473b3efcc96ed12f8562ae574e00452fc716b888905d8ea5","abstract_canon_sha256":"83c8d80b6cc619fd019b72330892c794e34e16664c19dcdbe2f936fe8e17a7d3"},"schema_version":"1.0"},"canonical_sha256":"fe754d1e30b0bc1e39330f9efe17c1047b581a268fdb5bc40dcca5ce96c6cd0d","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-04T14:38:40.323029Z","signature_b64":"chCtmexIRFsvxw9IFnG2gnQsd2TVEsQ/uwyopWBskpULiMk8tBt4wLg3RlCEhjqtNa9pNZQFBYzaBz/X/gyaAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"fe754d1e30b0bc1e39330f9efe17c1047b581a268fdb5bc40dcca5ce96c6cd0d","last_reissued_at":"2026-07-04T14:38:40.322625Z","signature_status":"signed_v1","first_computed_at":"2026-07-04T14:38:40.322625Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"math/0405601","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":"2026-07-04T14:38:40Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"r2vepCa1myY3/JrA5biy7l/eFuzvWK6ZBFwwnZT92XWsmpjSmJG6ayz2cBtCRVOglx+7xzReut9Gx7wzIAt1CQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-10T23:28:05.405367Z"},"content_sha256":"066a21d6b74e2977c15a043cafda8c9ac0252975b891f645f126450428158b55","schema_version":"1.0","event_id":"sha256:066a21d6b74e2977c15a043cafda8c9ac0252975b891f645f126450428158b55"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2004:7Z2U2HRQWC6B4OJTB6PP4F6BAR","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Random walks with $k$-wise independent increments","license":"","headline":"","cross_cats":["math.CO"],"primary_cat":"math.PR","authors_text":"Dan Romik, Gady Kozma, Itai Benjamini","submitted_at":"2004-05-31T18:50:10Z","abstract_excerpt":"We construct examples of a random walk with pairwise-independent steps which is almost-surely bounded, and for any $m$ and $k$ a random walk with $k$-wise independent steps which has no stationary distribution modulo $m$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0405601","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/math/0405601/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":"2026-07-04T14:38:40Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"qi8nvmOl9Zj+wXI7nHjvBTyfzN9DmW+hm+Xw192/bU/lbmJvu5pVqXresPmiTWJ4xlsPCNsJnMOzo3oDlSGeDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-10T23:28:05.406053Z"},"content_sha256":"7c7ba6d13f8949c6a4caf491b4e8dde17b87396e279ec3d6d5d8de0acecf8245","schema_version":"1.0","event_id":"sha256:7c7ba6d13f8949c6a4caf491b4e8dde17b87396e279ec3d6d5d8de0acecf8245"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/7Z2U2HRQWC6B4OJTB6PP4F6BAR/bundle.json","state_url":"https://pith.science/pith/7Z2U2HRQWC6B4OJTB6PP4F6BAR/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/7Z2U2HRQWC6B4OJTB6PP4F6BAR/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":"2026-08-10T23:28:05Z","links":{"resolver":"https://pith.science/pith/7Z2U2HRQWC6B4OJTB6PP4F6BAR","bundle":"https://pith.science/pith/7Z2U2HRQWC6B4OJTB6PP4F6BAR/bundle.json","state":"https://pith.science/pith/7Z2U2HRQWC6B4OJTB6PP4F6BAR/state.json","well_known_bundle":"https://pith.science/.well-known/pith/7Z2U2HRQWC6B4OJTB6PP4F6BAR/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2004:7Z2U2HRQWC6B4OJTB6PP4F6BAR","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":"83c8d80b6cc619fd019b72330892c794e34e16664c19dcdbe2f936fe8e17a7d3","cross_cats_sorted":["math.CO"],"license":"","primary_cat":"math.PR","submitted_at":"2004-05-31T18:50:10Z","title_canon_sha256":"adacd9ab618e9873473b3efcc96ed12f8562ae574e00452fc716b888905d8ea5"},"schema_version":"1.0","source":{"id":"math/0405601","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/0405601","created_at":"2026-07-04T14:38:40Z"},{"alias_kind":"arxiv_version","alias_value":"math/0405601v1","created_at":"2026-07-04T14:38:40Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0405601","created_at":"2026-07-04T14:38:40Z"},{"alias_kind":"pith_short_12","alias_value":"7Z2U2HRQWC6B","created_at":"2026-07-04T14:38:40Z"},{"alias_kind":"pith_short_16","alias_value":"7Z2U2HRQWC6B4OJT","created_at":"2026-07-04T14:38:40Z"},{"alias_kind":"pith_short_8","alias_value":"7Z2U2HRQ","created_at":"2026-07-04T14:38:40Z"}],"graph_snapshots":[{"event_id":"sha256:7c7ba6d13f8949c6a4caf491b4e8dde17b87396e279ec3d6d5d8de0acecf8245","target":"graph","created_at":"2026-07-04T14:38:40Z","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/math/0405601/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We construct examples of a random walk with pairwise-independent steps which is almost-surely bounded, and for any $m$ and $k$ a random walk with $k$-wise independent steps which has no stationary distribution modulo $m$.","authors_text":"Dan Romik, Gady Kozma, Itai Benjamini","cross_cats":["math.CO"],"headline":"","license":"","primary_cat":"math.PR","submitted_at":"2004-05-31T18:50:10Z","title":"Random walks with $k$-wise independent increments"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0405601","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":"sha256:066a21d6b74e2977c15a043cafda8c9ac0252975b891f645f126450428158b55","target":"record","created_at":"2026-07-04T14:38:40Z","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":"83c8d80b6cc619fd019b72330892c794e34e16664c19dcdbe2f936fe8e17a7d3","cross_cats_sorted":["math.CO"],"license":"","primary_cat":"math.PR","submitted_at":"2004-05-31T18:50:10Z","title_canon_sha256":"adacd9ab618e9873473b3efcc96ed12f8562ae574e00452fc716b888905d8ea5"},"schema_version":"1.0","source":{"id":"math/0405601","kind":"arxiv","version":1}},"canonical_sha256":"fe754d1e30b0bc1e39330f9efe17c1047b581a268fdb5bc40dcca5ce96c6cd0d","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"fe754d1e30b0bc1e39330f9efe17c1047b581a268fdb5bc40dcca5ce96c6cd0d","first_computed_at":"2026-07-04T14:38:40.322625Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-04T14:38:40.322625Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"chCtmexIRFsvxw9IFnG2gnQsd2TVEsQ/uwyopWBskpULiMk8tBt4wLg3RlCEhjqtNa9pNZQFBYzaBz/X/gyaAw==","signature_status":"signed_v1","signed_at":"2026-07-04T14:38:40.323029Z","signed_message":"canonical_sha256_bytes"},"source_id":"math/0405601","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:066a21d6b74e2977c15a043cafda8c9ac0252975b891f645f126450428158b55","sha256:7c7ba6d13f8949c6a4caf491b4e8dde17b87396e279ec3d6d5d8de0acecf8245"],"state_sha256":"d5c49670bba6d4a14b5f3e3a0493208859416b22b06bf163c9f6618f0c6c4201"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"+ZQIBPQUX2BxRa7yDoNAzIw8RAB11eTemR1xsQRIEgKUzNapH8QEm3GGMmiDWBwax194gyTc78JfwqmlB4XsDw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-10T23:28:05.417484Z","bundle_sha256":"f21015a7e66bd723049edb48abb95bd59bf086b7698a7d528010e8731e3d277a"}}