{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2013:A3OB52LHQTVEGET2GSFABBHWOJ","short_pith_number":"pith:A3OB52LH","canonical_record":{"source":{"id":"1303.6341","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2013-03-25T22:44:17Z","cross_cats_sorted":["math-ph","math.CO","math.MP"],"title_canon_sha256":"86ffe1327929e5205f2d9854b6592d232f9f4e3e874e303b9b9c4bbb920f7ac2","abstract_canon_sha256":"290531aa2ab94a3dfac2b76c91172f319acbb2b5097962b9f272ceb570ed87af"},"schema_version":"1.0"},"canonical_sha256":"06dc1ee96784ea43127a348a0084f672656ce6010d8eb98c6c67a52f9743a086","source":{"kind":"arxiv","id":"1303.6341","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1303.6341","created_at":"2026-05-18T01:50:52Z"},{"alias_kind":"arxiv_version","alias_value":"1303.6341v2","created_at":"2026-05-18T01:50:52Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1303.6341","created_at":"2026-05-18T01:50:52Z"},{"alias_kind":"pith_short_12","alias_value":"A3OB52LHQTVE","created_at":"2026-05-18T12:27:38Z"},{"alias_kind":"pith_short_16","alias_value":"A3OB52LHQTVEGET2","created_at":"2026-05-18T12:27:38Z"},{"alias_kind":"pith_short_8","alias_value":"A3OB52LH","created_at":"2026-05-18T12:27:38Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2013:A3OB52LHQTVEGET2GSFABBHWOJ","target":"record","payload":{"canonical_record":{"source":{"id":"1303.6341","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2013-03-25T22:44:17Z","cross_cats_sorted":["math-ph","math.CO","math.MP"],"title_canon_sha256":"86ffe1327929e5205f2d9854b6592d232f9f4e3e874e303b9b9c4bbb920f7ac2","abstract_canon_sha256":"290531aa2ab94a3dfac2b76c91172f319acbb2b5097962b9f272ceb570ed87af"},"schema_version":"1.0"},"canonical_sha256":"06dc1ee96784ea43127a348a0084f672656ce6010d8eb98c6c67a52f9743a086","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:50:52.875264Z","signature_b64":"q2GSbvFpQD+Du+NNedUV+waWa1eBmGj/FtoyuwMu3xutlubRunKFsdSnROTTlBJhDrK+goqXamQrNje3bXuSDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"06dc1ee96784ea43127a348a0084f672656ce6010d8eb98c6c67a52f9743a086","last_reissued_at":"2026-05-18T01:50:52.874631Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:50:52.874631Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1303.6341","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":"2026-05-18T01:50:52Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"L2c0vwg/9ngNlm/DxNmPkmee7WqvoP9ks2UUOQxkeASgG5PuQEBKR5CcjtJgfyJo/h/7V80K9+UdxMC363pPBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-23T11:32:18.678986Z"},"content_sha256":"90271f912d48ab53e1ae69d0af9d4ca24961e19d5da0daf7d98585f9f743a13a","schema_version":"1.0","event_id":"sha256:90271f912d48ab53e1ae69d0af9d4ca24961e19d5da0daf7d98585f9f743a13a"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2013:A3OB52LHQTVEGET2GSFABBHWOJ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Connectivity patterns in loop percolation I: the rationality phenomenon and constant term identities","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.CO","math.MP"],"primary_cat":"math.PR","authors_text":"Dan Romik","submitted_at":"2013-03-25T22:44:17Z","abstract_excerpt":"Loop percolation, also known as the dense O(1) loop model, is a variant of critical bond percolation in the square lattice Z^2 whose graph structure consists of a disjoint union of cycles. We study its connectivity pattern, which is a random noncrossing matching associated with a loop percolation configuration. These connectivity patterns exhibit a striking rationality property whereby probabilities of naturally-occurring events are dyadic rational numbers or rational functions of a size parameter n, but the reasons for this are not completely understood. We prove the rationality phenomenon in"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1303.6341","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":""},"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":"2026-05-18T01:50:52Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"wLwF8Rx6MpEC3z1GCuWSQhzPL8w3YAZVAXf5WWyhADSHD7QYa/j4U/UBQoWw3b08K1EsG/9mE9zBiwFz4RewAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-23T11:32:18.679348Z"},"content_sha256":"92a88d97de98190d1e7d568d1e2fab621589db889bc6134dc5528a5b2c78e795","schema_version":"1.0","event_id":"sha256:92a88d97de98190d1e7d568d1e2fab621589db889bc6134dc5528a5b2c78e795"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/A3OB52LHQTVEGET2GSFABBHWOJ/bundle.json","state_url":"https://pith.science/pith/A3OB52LHQTVEGET2GSFABBHWOJ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/A3OB52LHQTVEGET2GSFABBHWOJ/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":"2026-06-23T11:32:18Z","links":{"resolver":"https://pith.science/pith/A3OB52LHQTVEGET2GSFABBHWOJ","bundle":"https://pith.science/pith/A3OB52LHQTVEGET2GSFABBHWOJ/bundle.json","state":"https://pith.science/pith/A3OB52LHQTVEGET2GSFABBHWOJ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/A3OB52LHQTVEGET2GSFABBHWOJ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:A3OB52LHQTVEGET2GSFABBHWOJ","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":"290531aa2ab94a3dfac2b76c91172f319acbb2b5097962b9f272ceb570ed87af","cross_cats_sorted":["math-ph","math.CO","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2013-03-25T22:44:17Z","title_canon_sha256":"86ffe1327929e5205f2d9854b6592d232f9f4e3e874e303b9b9c4bbb920f7ac2"},"schema_version":"1.0","source":{"id":"1303.6341","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1303.6341","created_at":"2026-05-18T01:50:52Z"},{"alias_kind":"arxiv_version","alias_value":"1303.6341v2","created_at":"2026-05-18T01:50:52Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1303.6341","created_at":"2026-05-18T01:50:52Z"},{"alias_kind":"pith_short_12","alias_value":"A3OB52LHQTVE","created_at":"2026-05-18T12:27:38Z"},{"alias_kind":"pith_short_16","alias_value":"A3OB52LHQTVEGET2","created_at":"2026-05-18T12:27:38Z"},{"alias_kind":"pith_short_8","alias_value":"A3OB52LH","created_at":"2026-05-18T12:27:38Z"}],"graph_snapshots":[{"event_id":"sha256:92a88d97de98190d1e7d568d1e2fab621589db889bc6134dc5528a5b2c78e795","target":"graph","created_at":"2026-05-18T01:50:52Z","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":"Loop percolation, also known as the dense O(1) loop model, is a variant of critical bond percolation in the square lattice Z^2 whose graph structure consists of a disjoint union of cycles. We study its connectivity pattern, which is a random noncrossing matching associated with a loop percolation configuration. These connectivity patterns exhibit a striking rationality property whereby probabilities of naturally-occurring events are dyadic rational numbers or rational functions of a size parameter n, but the reasons for this are not completely understood. We prove the rationality phenomenon in","authors_text":"Dan Romik","cross_cats":["math-ph","math.CO","math.MP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2013-03-25T22:44:17Z","title":"Connectivity patterns in loop percolation I: the rationality phenomenon and constant term identities"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1303.6341","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":"sha256:90271f912d48ab53e1ae69d0af9d4ca24961e19d5da0daf7d98585f9f743a13a","target":"record","created_at":"2026-05-18T01:50:52Z","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":"290531aa2ab94a3dfac2b76c91172f319acbb2b5097962b9f272ceb570ed87af","cross_cats_sorted":["math-ph","math.CO","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2013-03-25T22:44:17Z","title_canon_sha256":"86ffe1327929e5205f2d9854b6592d232f9f4e3e874e303b9b9c4bbb920f7ac2"},"schema_version":"1.0","source":{"id":"1303.6341","kind":"arxiv","version":2}},"canonical_sha256":"06dc1ee96784ea43127a348a0084f672656ce6010d8eb98c6c67a52f9743a086","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"06dc1ee96784ea43127a348a0084f672656ce6010d8eb98c6c67a52f9743a086","first_computed_at":"2026-05-18T01:50:52.874631Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:50:52.874631Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"q2GSbvFpQD+Du+NNedUV+waWa1eBmGj/FtoyuwMu3xutlubRunKFsdSnROTTlBJhDrK+goqXamQrNje3bXuSDw==","signature_status":"signed_v1","signed_at":"2026-05-18T01:50:52.875264Z","signed_message":"canonical_sha256_bytes"},"source_id":"1303.6341","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:90271f912d48ab53e1ae69d0af9d4ca24961e19d5da0daf7d98585f9f743a13a","sha256:92a88d97de98190d1e7d568d1e2fab621589db889bc6134dc5528a5b2c78e795"],"state_sha256":"10ec46dcb49e68acc28645abdf8aec989d794c7bbf8db4e6465e813b54e1d6ed"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Pdg9IEPV1LrvFryIQO3d1pOiOa6GSGpI7scS4UFfrPwjY1H29Av5aeBoGw2J+X7yNDqloDtGLnm1QNxn8+nmCw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-23T11:32:18.681207Z","bundle_sha256":"f6310d5630acac2ee8543d96b1b9860a6e73e10fe7f79bbe91ca2bcfb49b4124"}}