{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2018:VPIN5RTBBYMCED67POK72H7KWQ","short_pith_number":"pith:VPIN5RTB","canonical_record":{"source":{"id":"1801.00621","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2018-01-02T11:59:56Z","cross_cats_sorted":[],"title_canon_sha256":"ab3028c6bb1a59270fc2a10aef26fa5ea811269e28d2b1a43b3969770f857bfe","abstract_canon_sha256":"64ed26f9d3b5af61455568389e6dc76b74dc60701455170dda8353a524f220fb"},"schema_version":"1.0"},"canonical_sha256":"abd0dec6610e18220fdf7b95fd1feab43de0af3c6aafac112db3f85c31f3538f","source":{"kind":"arxiv","id":"1801.00621","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1801.00621","created_at":"2026-05-18T00:26:52Z"},{"alias_kind":"arxiv_version","alias_value":"1801.00621v1","created_at":"2026-05-18T00:26:52Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1801.00621","created_at":"2026-05-18T00:26:52Z"},{"alias_kind":"pith_short_12","alias_value":"VPIN5RTBBYMC","created_at":"2026-05-18T12:32:59Z"},{"alias_kind":"pith_short_16","alias_value":"VPIN5RTBBYMCED67","created_at":"2026-05-18T12:32:59Z"},{"alias_kind":"pith_short_8","alias_value":"VPIN5RTB","created_at":"2026-05-18T12:32:59Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2018:VPIN5RTBBYMCED67POK72H7KWQ","target":"record","payload":{"canonical_record":{"source":{"id":"1801.00621","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2018-01-02T11:59:56Z","cross_cats_sorted":[],"title_canon_sha256":"ab3028c6bb1a59270fc2a10aef26fa5ea811269e28d2b1a43b3969770f857bfe","abstract_canon_sha256":"64ed26f9d3b5af61455568389e6dc76b74dc60701455170dda8353a524f220fb"},"schema_version":"1.0"},"canonical_sha256":"abd0dec6610e18220fdf7b95fd1feab43de0af3c6aafac112db3f85c31f3538f","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:26:52.866734Z","signature_b64":"VNBDiGGm0f1+tIoOl9w3rD49YwjWnTkJp7r8KWxRvFs38UT1JubGba5XLNMcAmsiKATF9NLF5BqE93OQ2GIIAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"abd0dec6610e18220fdf7b95fd1feab43de0af3c6aafac112db3f85c31f3538f","last_reissued_at":"2026-05-18T00:26:52.866136Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:26:52.866136Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1801.00621","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":"2026-05-18T00:26:52Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"qBCpEmL2kQ48THSa85rzuAJrhDprnYP2A0ghCajBufn2WcWBFNg04lWcU7+KuKQusY7Z+E4+tiobCDL8n/SACg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-07T18:49:24.323350Z"},"content_sha256":"f48fabde190fd9e42a0e8ce07fe890fe511216984b88fe96fcc15b2c7c1663b6","schema_version":"1.0","event_id":"sha256:f48fabde190fd9e42a0e8ce07fe890fe511216984b88fe96fcc15b2c7c1663b6"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2018:VPIN5RTBBYMCED67POK72H7KWQ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"FKG (and other inequalities) via (generalized) FK representation (and iterated folding)","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Alberto Gandolfi","submitted_at":"2018-01-02T11:59:56Z","abstract_excerpt":"In this paper we prove several inequalities by means of diagrammatic expansions, a technique already used in [BG13]. This time we show that iterations of the folding of a probability leads to the proof of some in- equalities by means of a generalized random cluster representation of the iterated foldings. One of the inequalities is the well known FKG inequal- ity, which ends up being proven, quite unexpectedly, by means of the (generalized) FK representation. Although most of the results are not new, we hope that the techniques will find applications in other contexts."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1801.00621","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":"2026-05-18T00:26:52Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"uaZznQ0vcZP61zexUdZcOQ9cBq4RteVb9XgMbFFcXUGapion5g9GFS+LO/Y3dmEPU/6s4OBgq21eqZr9iDcRBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-07T18:49:24.324027Z"},"content_sha256":"af64f80cdf1ea7a3676428716e5ad9ebc1a49e3c709afaf1f307a02c2d96cd29","schema_version":"1.0","event_id":"sha256:af64f80cdf1ea7a3676428716e5ad9ebc1a49e3c709afaf1f307a02c2d96cd29"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/VPIN5RTBBYMCED67POK72H7KWQ/bundle.json","state_url":"https://pith.science/pith/VPIN5RTBBYMCED67POK72H7KWQ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/VPIN5RTBBYMCED67POK72H7KWQ/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":"2026-06-07T18:49:24Z","links":{"resolver":"https://pith.science/pith/VPIN5RTBBYMCED67POK72H7KWQ","bundle":"https://pith.science/pith/VPIN5RTBBYMCED67POK72H7KWQ/bundle.json","state":"https://pith.science/pith/VPIN5RTBBYMCED67POK72H7KWQ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/VPIN5RTBBYMCED67POK72H7KWQ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:VPIN5RTBBYMCED67POK72H7KWQ","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":"64ed26f9d3b5af61455568389e6dc76b74dc60701455170dda8353a524f220fb","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2018-01-02T11:59:56Z","title_canon_sha256":"ab3028c6bb1a59270fc2a10aef26fa5ea811269e28d2b1a43b3969770f857bfe"},"schema_version":"1.0","source":{"id":"1801.00621","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1801.00621","created_at":"2026-05-18T00:26:52Z"},{"alias_kind":"arxiv_version","alias_value":"1801.00621v1","created_at":"2026-05-18T00:26:52Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1801.00621","created_at":"2026-05-18T00:26:52Z"},{"alias_kind":"pith_short_12","alias_value":"VPIN5RTBBYMC","created_at":"2026-05-18T12:32:59Z"},{"alias_kind":"pith_short_16","alias_value":"VPIN5RTBBYMCED67","created_at":"2026-05-18T12:32:59Z"},{"alias_kind":"pith_short_8","alias_value":"VPIN5RTB","created_at":"2026-05-18T12:32:59Z"}],"graph_snapshots":[{"event_id":"sha256:af64f80cdf1ea7a3676428716e5ad9ebc1a49e3c709afaf1f307a02c2d96cd29","target":"graph","created_at":"2026-05-18T00:26:52Z","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":"In this paper we prove several inequalities by means of diagrammatic expansions, a technique already used in [BG13]. This time we show that iterations of the folding of a probability leads to the proof of some in- equalities by means of a generalized random cluster representation of the iterated foldings. One of the inequalities is the well known FKG inequal- ity, which ends up being proven, quite unexpectedly, by means of the (generalized) FK representation. Although most of the results are not new, we hope that the techniques will find applications in other contexts.","authors_text":"Alberto Gandolfi","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2018-01-02T11:59:56Z","title":"FKG (and other inequalities) via (generalized) FK representation (and iterated folding)"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1801.00621","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:f48fabde190fd9e42a0e8ce07fe890fe511216984b88fe96fcc15b2c7c1663b6","target":"record","created_at":"2026-05-18T00:26:52Z","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":"64ed26f9d3b5af61455568389e6dc76b74dc60701455170dda8353a524f220fb","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2018-01-02T11:59:56Z","title_canon_sha256":"ab3028c6bb1a59270fc2a10aef26fa5ea811269e28d2b1a43b3969770f857bfe"},"schema_version":"1.0","source":{"id":"1801.00621","kind":"arxiv","version":1}},"canonical_sha256":"abd0dec6610e18220fdf7b95fd1feab43de0af3c6aafac112db3f85c31f3538f","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"abd0dec6610e18220fdf7b95fd1feab43de0af3c6aafac112db3f85c31f3538f","first_computed_at":"2026-05-18T00:26:52.866136Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:26:52.866136Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"VNBDiGGm0f1+tIoOl9w3rD49YwjWnTkJp7r8KWxRvFs38UT1JubGba5XLNMcAmsiKATF9NLF5BqE93OQ2GIIAg==","signature_status":"signed_v1","signed_at":"2026-05-18T00:26:52.866734Z","signed_message":"canonical_sha256_bytes"},"source_id":"1801.00621","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:f48fabde190fd9e42a0e8ce07fe890fe511216984b88fe96fcc15b2c7c1663b6","sha256:af64f80cdf1ea7a3676428716e5ad9ebc1a49e3c709afaf1f307a02c2d96cd29"],"state_sha256":"52312e94c8ae6d57559185620b65a444ca0f89607c4f0b9418dcd1dbfb0dcbee"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"IHZLRNsr/jMtfJ6ptouHsCHv833kaESUS2VRkhi2pPEjlRA63CbahWrJEO6hbz5e7OAnrf7h3rTTyGVkl+n+Cw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-07T18:49:24.327706Z","bundle_sha256":"2a967778925c78b043ec0cdb890d62fcb48fd73922d311de29b78c9381446469"}}