{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2010:S5ZCGRU2GZCYFS4XCBC56YC37U","short_pith_number":"pith:S5ZCGRU2","canonical_record":{"source":{"id":"1001.1529","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2010-01-10T16:26:52Z","cross_cats_sorted":["math-ph","math.MP"],"title_canon_sha256":"95c10c6f5dbdc50b8bfe08dc2dd0672f3b190aadfffae812bfdb448c14d699e6","abstract_canon_sha256":"d8ac6224b7e7bce8bc6cca361d69815767e72a577f3fd62568715eebf699d119"},"schema_version":"1.0"},"canonical_sha256":"977223469a364582cb971045df605bfd36e756f5ec811ac380b28c682b89c5fb","source":{"kind":"arxiv","id":"1001.1529","version":4},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1001.1529","created_at":"2026-05-18T02:10:03Z"},{"alias_kind":"arxiv_version","alias_value":"1001.1529v4","created_at":"2026-05-18T02:10:03Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1001.1529","created_at":"2026-05-18T02:10:03Z"},{"alias_kind":"pith_short_12","alias_value":"S5ZCGRU2GZCY","created_at":"2026-05-18T12:26:13Z"},{"alias_kind":"pith_short_16","alias_value":"S5ZCGRU2GZCYFS4X","created_at":"2026-05-18T12:26:13Z"},{"alias_kind":"pith_short_8","alias_value":"S5ZCGRU2","created_at":"2026-05-18T12:26:13Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2010:S5ZCGRU2GZCYFS4XCBC56YC37U","target":"record","payload":{"canonical_record":{"source":{"id":"1001.1529","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2010-01-10T16:26:52Z","cross_cats_sorted":["math-ph","math.MP"],"title_canon_sha256":"95c10c6f5dbdc50b8bfe08dc2dd0672f3b190aadfffae812bfdb448c14d699e6","abstract_canon_sha256":"d8ac6224b7e7bce8bc6cca361d69815767e72a577f3fd62568715eebf699d119"},"schema_version":"1.0"},"canonical_sha256":"977223469a364582cb971045df605bfd36e756f5ec811ac380b28c682b89c5fb","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:10:03.326228Z","signature_b64":"CE/V2u2q4Fi1nBaAkcqqmgDyWDeePyD4/ca6HRmr4e7gRLJkDCiDEqFRvs4iat+cu922q1vsG6ODvD7zsCtdCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"977223469a364582cb971045df605bfd36e756f5ec811ac380b28c682b89c5fb","last_reissued_at":"2026-05-18T02:10:03.325468Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:10:03.325468Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1001.1529","source_version":4,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:10:03Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Pmd75blQ4HaGWLUyWkejw2qXifpPohUUnnyq381vDZVVjHfo6DISlMfYgJEWQkORz8Gdq8jGtGPBZU+dvGSbAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-20T06:24:12.464793Z"},"content_sha256":"9853509d196bed1780130d35989b527473c66adc1a9630790a3f087f98836384","schema_version":"1.0","event_id":"sha256:9853509d196bed1780130d35989b527473c66adc1a9630790a3f087f98836384"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2010:S5ZCGRU2GZCYFS4XCBC56YC37U","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Phase separation in random cluster models III: circuit regularity","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.MP"],"primary_cat":"math.PR","authors_text":"Alan Hammond","submitted_at":"2010-01-10T16:26:52Z","abstract_excerpt":"We study the droplet that results from conditioning the subcritical Fortuin-Kasteleyn planar random cluster model on the presence of an open circuit Gamma_0 encircling the origin and enclosing an area of at least (or exactly) n^2. In this paper, we prove that the resulting circuit is highly regular: we define a notion of a regeneration site in such a way that, for any such element v of Gamma_0, the circuit Gamma_0 cuts through the radial line segment through v only at v. We show that, provided that the conditioned circuit is centred at the origin in a natural sense, the set of regeneration sit"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1001.1529","kind":"arxiv","version":4},"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":"2026-05-18T02:10:03Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"7NNZzipW9brzuuZ4yPD0h4rVEGBAQGjZDC08nYfm4I9D2kzatchjhDfoTtd8UX0caOxOMqdPx+daiuCjdaFIBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-20T06:24:12.465128Z"},"content_sha256":"a4567ec2148691213af22dcb26a468a2f29e60e63b5252a1a8fdeef9308cbe7c","schema_version":"1.0","event_id":"sha256:a4567ec2148691213af22dcb26a468a2f29e60e63b5252a1a8fdeef9308cbe7c"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/S5ZCGRU2GZCYFS4XCBC56YC37U/bundle.json","state_url":"https://pith.science/pith/S5ZCGRU2GZCYFS4XCBC56YC37U/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/S5ZCGRU2GZCYFS4XCBC56YC37U/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":"2026-06-20T06:24:12Z","links":{"resolver":"https://pith.science/pith/S5ZCGRU2GZCYFS4XCBC56YC37U","bundle":"https://pith.science/pith/S5ZCGRU2GZCYFS4XCBC56YC37U/bundle.json","state":"https://pith.science/pith/S5ZCGRU2GZCYFS4XCBC56YC37U/state.json","well_known_bundle":"https://pith.science/.well-known/pith/S5ZCGRU2GZCYFS4XCBC56YC37U/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2010:S5ZCGRU2GZCYFS4XCBC56YC37U","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":"d8ac6224b7e7bce8bc6cca361d69815767e72a577f3fd62568715eebf699d119","cross_cats_sorted":["math-ph","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2010-01-10T16:26:52Z","title_canon_sha256":"95c10c6f5dbdc50b8bfe08dc2dd0672f3b190aadfffae812bfdb448c14d699e6"},"schema_version":"1.0","source":{"id":"1001.1529","kind":"arxiv","version":4}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1001.1529","created_at":"2026-05-18T02:10:03Z"},{"alias_kind":"arxiv_version","alias_value":"1001.1529v4","created_at":"2026-05-18T02:10:03Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1001.1529","created_at":"2026-05-18T02:10:03Z"},{"alias_kind":"pith_short_12","alias_value":"S5ZCGRU2GZCY","created_at":"2026-05-18T12:26:13Z"},{"alias_kind":"pith_short_16","alias_value":"S5ZCGRU2GZCYFS4X","created_at":"2026-05-18T12:26:13Z"},{"alias_kind":"pith_short_8","alias_value":"S5ZCGRU2","created_at":"2026-05-18T12:26:13Z"}],"graph_snapshots":[{"event_id":"sha256:a4567ec2148691213af22dcb26a468a2f29e60e63b5252a1a8fdeef9308cbe7c","target":"graph","created_at":"2026-05-18T02:10:03Z","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 study the droplet that results from conditioning the subcritical Fortuin-Kasteleyn planar random cluster model on the presence of an open circuit Gamma_0 encircling the origin and enclosing an area of at least (or exactly) n^2. In this paper, we prove that the resulting circuit is highly regular: we define a notion of a regeneration site in such a way that, for any such element v of Gamma_0, the circuit Gamma_0 cuts through the radial line segment through v only at v. We show that, provided that the conditioned circuit is centred at the origin in a natural sense, the set of regeneration sit","authors_text":"Alan Hammond","cross_cats":["math-ph","math.MP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2010-01-10T16:26:52Z","title":"Phase separation in random cluster models III: circuit regularity"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1001.1529","kind":"arxiv","version":4},"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":"sha256:9853509d196bed1780130d35989b527473c66adc1a9630790a3f087f98836384","target":"record","created_at":"2026-05-18T02:10:03Z","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":"d8ac6224b7e7bce8bc6cca361d69815767e72a577f3fd62568715eebf699d119","cross_cats_sorted":["math-ph","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2010-01-10T16:26:52Z","title_canon_sha256":"95c10c6f5dbdc50b8bfe08dc2dd0672f3b190aadfffae812bfdb448c14d699e6"},"schema_version":"1.0","source":{"id":"1001.1529","kind":"arxiv","version":4}},"canonical_sha256":"977223469a364582cb971045df605bfd36e756f5ec811ac380b28c682b89c5fb","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"977223469a364582cb971045df605bfd36e756f5ec811ac380b28c682b89c5fb","first_computed_at":"2026-05-18T02:10:03.325468Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:10:03.325468Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"CE/V2u2q4Fi1nBaAkcqqmgDyWDeePyD4/ca6HRmr4e7gRLJkDCiDEqFRvs4iat+cu922q1vsG6ODvD7zsCtdCQ==","signature_status":"signed_v1","signed_at":"2026-05-18T02:10:03.326228Z","signed_message":"canonical_sha256_bytes"},"source_id":"1001.1529","source_kind":"arxiv","source_version":4}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:9853509d196bed1780130d35989b527473c66adc1a9630790a3f087f98836384","sha256:a4567ec2148691213af22dcb26a468a2f29e60e63b5252a1a8fdeef9308cbe7c"],"state_sha256":"5ef1f187a2817afae452b4ec1127741d0f8c4cd07199429690ab3dea15e83b4e"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"jb+uTB84wSO7fhTFqqQUGrmvX1MqYKPeNPiEDgwufSEPQ23vYZ0QqSXTScX5pNxCIyFviwEO+9gfHIKZH2VcBw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-20T06:24:12.466963Z","bundle_sha256":"ce857734e0f29cf4f3a01cb379c08bc9c0967232947a89cdbeafb5816272d648"}}