{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:ORIS4JYJ5P32YAJQMFZTLHY4QT","short_pith_number":"pith:ORIS4JYJ","canonical_record":{"source":{"id":"1601.06391","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-01-24T14:51:12Z","cross_cats_sorted":["math.PR"],"title_canon_sha256":"8d9c3d87c9a7e30d2b66eb358649a46d34605d2e8af0b445cfb851a9debc4b10","abstract_canon_sha256":"d941ec2025a1be1014d82e3d1bcd47a140e8b644a15aa9f79e41ee184565b516"},"schema_version":"1.0"},"canonical_sha256":"74512e2709ebf7ac01306173359f1c84dc16161fee4c35c6e816de5a0148911f","source":{"kind":"arxiv","id":"1601.06391","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1601.06391","created_at":"2026-05-18T01:22:04Z"},{"alias_kind":"arxiv_version","alias_value":"1601.06391v1","created_at":"2026-05-18T01:22:04Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1601.06391","created_at":"2026-05-18T01:22:04Z"},{"alias_kind":"pith_short_12","alias_value":"ORIS4JYJ5P32","created_at":"2026-05-18T12:30:36Z"},{"alias_kind":"pith_short_16","alias_value":"ORIS4JYJ5P32YAJQ","created_at":"2026-05-18T12:30:36Z"},{"alias_kind":"pith_short_8","alias_value":"ORIS4JYJ","created_at":"2026-05-18T12:30:36Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:ORIS4JYJ5P32YAJQMFZTLHY4QT","target":"record","payload":{"canonical_record":{"source":{"id":"1601.06391","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-01-24T14:51:12Z","cross_cats_sorted":["math.PR"],"title_canon_sha256":"8d9c3d87c9a7e30d2b66eb358649a46d34605d2e8af0b445cfb851a9debc4b10","abstract_canon_sha256":"d941ec2025a1be1014d82e3d1bcd47a140e8b644a15aa9f79e41ee184565b516"},"schema_version":"1.0"},"canonical_sha256":"74512e2709ebf7ac01306173359f1c84dc16161fee4c35c6e816de5a0148911f","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:22:04.559415Z","signature_b64":"ZcL2+PMr2TpUXuESUcAiWnY75uQI6EORJcgIccebNJu5CHbOiKu2uETXEut1QZKSxxLDluoUv+G8bmBylYHcCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"74512e2709ebf7ac01306173359f1c84dc16161fee4c35c6e816de5a0148911f","last_reissued_at":"2026-05-18T01:22:04.558948Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:22:04.558948Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1601.06391","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":"2026-05-18T01:22:04Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"UophXVbR7+OSewvAaML0yqYp6waANicBEEmAClHt1MkJ/Uq1EVV3D1dKod221isTndK9oshNdUWmwJ9V8W8PCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-01T10:36:56.966114Z"},"content_sha256":"a2d56ae8bbada4237e161c0d21ecce5ad82f2347a5457dd24af8d52b505aebd5","schema_version":"1.0","event_id":"sha256:a2d56ae8bbada4237e161c0d21ecce5ad82f2347a5457dd24af8d52b505aebd5"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:ORIS4JYJ5P32YAJQMFZTLHY4QT","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Randomly juggling backwards","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.PR"],"primary_cat":"math.CO","authors_text":"Allen Knutson","submitted_at":"2016-01-24T14:51:12Z","abstract_excerpt":"We recall the directed graph of _juggling states_, closed walks within which give juggling patterns, as studied by Ron Graham in [w/Chung, w/Butler]. Various random walks in this graph have been studied before by several authors, and their equilibrium distributions computed. We motivate a random walk on the reverse graph (and an enrichment thereof) from a very classical linear algebra problem, leading to a particularly simple equilibrium: a Boltzmann distribution closely related to the Poincar\\'e series of the b-Grassmannian in infinite-dimensional space.\n  We determine the most likely asympto"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1601.06391","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":"2026-05-18T01:22:04Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ZTMZXeu0VA3Vw7WEfT7tcAaywpOAV0QruALnSHQacZ4YVLBwHnbcRVdvKJf72bPJ6Yv8Um575GXNn8mO852TCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-01T10:36:56.966476Z"},"content_sha256":"32f540c03961593cec1191c37a71b4b01aab4c2fb2ac58f368b829748f17fd01","schema_version":"1.0","event_id":"sha256:32f540c03961593cec1191c37a71b4b01aab4c2fb2ac58f368b829748f17fd01"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/ORIS4JYJ5P32YAJQMFZTLHY4QT/bundle.json","state_url":"https://pith.science/pith/ORIS4JYJ5P32YAJQMFZTLHY4QT/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/ORIS4JYJ5P32YAJQMFZTLHY4QT/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":"2026-06-01T10:36:56Z","links":{"resolver":"https://pith.science/pith/ORIS4JYJ5P32YAJQMFZTLHY4QT","bundle":"https://pith.science/pith/ORIS4JYJ5P32YAJQMFZTLHY4QT/bundle.json","state":"https://pith.science/pith/ORIS4JYJ5P32YAJQMFZTLHY4QT/state.json","well_known_bundle":"https://pith.science/.well-known/pith/ORIS4JYJ5P32YAJQMFZTLHY4QT/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:ORIS4JYJ5P32YAJQMFZTLHY4QT","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":"d941ec2025a1be1014d82e3d1bcd47a140e8b644a15aa9f79e41ee184565b516","cross_cats_sorted":["math.PR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-01-24T14:51:12Z","title_canon_sha256":"8d9c3d87c9a7e30d2b66eb358649a46d34605d2e8af0b445cfb851a9debc4b10"},"schema_version":"1.0","source":{"id":"1601.06391","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1601.06391","created_at":"2026-05-18T01:22:04Z"},{"alias_kind":"arxiv_version","alias_value":"1601.06391v1","created_at":"2026-05-18T01:22:04Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1601.06391","created_at":"2026-05-18T01:22:04Z"},{"alias_kind":"pith_short_12","alias_value":"ORIS4JYJ5P32","created_at":"2026-05-18T12:30:36Z"},{"alias_kind":"pith_short_16","alias_value":"ORIS4JYJ5P32YAJQ","created_at":"2026-05-18T12:30:36Z"},{"alias_kind":"pith_short_8","alias_value":"ORIS4JYJ","created_at":"2026-05-18T12:30:36Z"}],"graph_snapshots":[{"event_id":"sha256:32f540c03961593cec1191c37a71b4b01aab4c2fb2ac58f368b829748f17fd01","target":"graph","created_at":"2026-05-18T01:22:04Z","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 recall the directed graph of _juggling states_, closed walks within which give juggling patterns, as studied by Ron Graham in [w/Chung, w/Butler]. Various random walks in this graph have been studied before by several authors, and their equilibrium distributions computed. We motivate a random walk on the reverse graph (and an enrichment thereof) from a very classical linear algebra problem, leading to a particularly simple equilibrium: a Boltzmann distribution closely related to the Poincar\\'e series of the b-Grassmannian in infinite-dimensional space.\n  We determine the most likely asympto","authors_text":"Allen Knutson","cross_cats":["math.PR"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-01-24T14:51:12Z","title":"Randomly juggling backwards"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1601.06391","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:a2d56ae8bbada4237e161c0d21ecce5ad82f2347a5457dd24af8d52b505aebd5","target":"record","created_at":"2026-05-18T01:22:04Z","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":"d941ec2025a1be1014d82e3d1bcd47a140e8b644a15aa9f79e41ee184565b516","cross_cats_sorted":["math.PR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-01-24T14:51:12Z","title_canon_sha256":"8d9c3d87c9a7e30d2b66eb358649a46d34605d2e8af0b445cfb851a9debc4b10"},"schema_version":"1.0","source":{"id":"1601.06391","kind":"arxiv","version":1}},"canonical_sha256":"74512e2709ebf7ac01306173359f1c84dc16161fee4c35c6e816de5a0148911f","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"74512e2709ebf7ac01306173359f1c84dc16161fee4c35c6e816de5a0148911f","first_computed_at":"2026-05-18T01:22:04.558948Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:22:04.558948Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"ZcL2+PMr2TpUXuESUcAiWnY75uQI6EORJcgIccebNJu5CHbOiKu2uETXEut1QZKSxxLDluoUv+G8bmBylYHcCQ==","signature_status":"signed_v1","signed_at":"2026-05-18T01:22:04.559415Z","signed_message":"canonical_sha256_bytes"},"source_id":"1601.06391","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:a2d56ae8bbada4237e161c0d21ecce5ad82f2347a5457dd24af8d52b505aebd5","sha256:32f540c03961593cec1191c37a71b4b01aab4c2fb2ac58f368b829748f17fd01"],"state_sha256":"f8d7f33cfc99cc6a99a7c61464deb8de43e3415737532ab12c4233c2930e4a3a"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"18mKHsEL98iZNTq7otvMt2Fis4mV9Xs/gxNHZKQ9p1GmCKfyAhhO/N8WhIPwiqVwagEzn5RnaJfxrEhATEgJCw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-01T10:36:56.968412Z","bundle_sha256":"d84a0c414002a7104fd427106807bfb7e0d29d745688015b2d61cb0b246daec0"}}