{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2011:I7WFXSGWSD332QAIP66XCUVIQE","short_pith_number":"pith:I7WFXSGW","canonical_record":{"source":{"id":"1109.6725","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cond-mat.stat-mech","submitted_at":"2011-09-30T06:43:53Z","cross_cats_sorted":["cond-mat.dis-nn","math-ph","math.MP"],"title_canon_sha256":"d06419f2cad4877deaae176b780ae5aa7d010f676f0667ea84a0eec72fbd1f5b","abstract_canon_sha256":"f6fc1f7b697749b2540efd3b770ef1822336e36bcc496e2c7201d75a78031c68"},"schema_version":"1.0"},"canonical_sha256":"47ec5bc8d690f7bd40087fbd7152a8812b9e44da26aeb7ce10a9189fbd8fafbc","source":{"kind":"arxiv","id":"1109.6725","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1109.6725","created_at":"2026-05-18T04:00:13Z"},{"alias_kind":"arxiv_version","alias_value":"1109.6725v2","created_at":"2026-05-18T04:00:13Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1109.6725","created_at":"2026-05-18T04:00:13Z"},{"alias_kind":"pith_short_12","alias_value":"I7WFXSGWSD33","created_at":"2026-05-18T12:26:30Z"},{"alias_kind":"pith_short_16","alias_value":"I7WFXSGWSD332QAI","created_at":"2026-05-18T12:26:30Z"},{"alias_kind":"pith_short_8","alias_value":"I7WFXSGW","created_at":"2026-05-18T12:26:30Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2011:I7WFXSGWSD332QAIP66XCUVIQE","target":"record","payload":{"canonical_record":{"source":{"id":"1109.6725","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cond-mat.stat-mech","submitted_at":"2011-09-30T06:43:53Z","cross_cats_sorted":["cond-mat.dis-nn","math-ph","math.MP"],"title_canon_sha256":"d06419f2cad4877deaae176b780ae5aa7d010f676f0667ea84a0eec72fbd1f5b","abstract_canon_sha256":"f6fc1f7b697749b2540efd3b770ef1822336e36bcc496e2c7201d75a78031c68"},"schema_version":"1.0"},"canonical_sha256":"47ec5bc8d690f7bd40087fbd7152a8812b9e44da26aeb7ce10a9189fbd8fafbc","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:00:13.473762Z","signature_b64":"oa5gaVLZENo6N8s+B+1Qwmi3YyXnAOGDWpJzzWWM+AVkfKntFoDix7adqO0YgHW7yvk/qBGhX3iMiXx+RNaCCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"47ec5bc8d690f7bd40087fbd7152a8812b9e44da26aeb7ce10a9189fbd8fafbc","last_reissued_at":"2026-05-18T04:00:13.473079Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:00:13.473079Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1109.6725","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":"2026-05-18T04:00:13Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"1zo86F2uNvt1tctGKUsoWmJCuSafxB1mNJjQfY67pk1P/qOUOux3LpqBCXUTIMIxFs3ZaGiLgMbbye6tC23KBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-25T08:08:39.785658Z"},"content_sha256":"ea2288e3414dd6dd7a2d6dcbe4a0857b008ddb2385ee3a33083171c6f8d896ae","schema_version":"1.0","event_id":"sha256:ea2288e3414dd6dd7a2d6dcbe4a0857b008ddb2385ee3a33083171c6f8d896ae"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2011:I7WFXSGWSD332QAIP66XCUVIQE","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Cayley Trees and Bethe Lattices, a concise analysis for mathematicians and physicists","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cond-mat.dis-nn","math-ph","math.MP"],"primary_cat":"cond-mat.stat-mech","authors_text":"Massimo Ostilli","submitted_at":"2011-09-30T06:43:53Z","abstract_excerpt":"We review critically the concepts and the applications of Cayley Trees and Bethe Lattices in statistical mechanics in a tentative effort to remove widespread misuse of these simple, but yet important - and different - ideal graphs. We illustrate, in particular, two rigorous techniques to deal with Bethe Lattices, based respectively on self-similarity and on the Kolmogorov consistency theorem, linking the latter with the Cavity and Belief Propagation methods, more known to the physics community."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1109.6725","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":"2026-05-18T04:00:13Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"1RJ9XrQmE3oJOpEdMKzQrZxk3IeKFP2EB03sfy6RvTuGM10baY5NrloDGhDcqT6vruBx7r9GZDShUqzGqW4FCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-25T08:08:39.786003Z"},"content_sha256":"55eeb7c9baf35e96ef8c3552a049dec1d547e2f229259d1cff4358c5e1441582","schema_version":"1.0","event_id":"sha256:55eeb7c9baf35e96ef8c3552a049dec1d547e2f229259d1cff4358c5e1441582"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/I7WFXSGWSD332QAIP66XCUVIQE/bundle.json","state_url":"https://pith.science/pith/I7WFXSGWSD332QAIP66XCUVIQE/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/I7WFXSGWSD332QAIP66XCUVIQE/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":"2026-06-25T08:08:39Z","links":{"resolver":"https://pith.science/pith/I7WFXSGWSD332QAIP66XCUVIQE","bundle":"https://pith.science/pith/I7WFXSGWSD332QAIP66XCUVIQE/bundle.json","state":"https://pith.science/pith/I7WFXSGWSD332QAIP66XCUVIQE/state.json","well_known_bundle":"https://pith.science/.well-known/pith/I7WFXSGWSD332QAIP66XCUVIQE/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:I7WFXSGWSD332QAIP66XCUVIQE","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":"f6fc1f7b697749b2540efd3b770ef1822336e36bcc496e2c7201d75a78031c68","cross_cats_sorted":["cond-mat.dis-nn","math-ph","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cond-mat.stat-mech","submitted_at":"2011-09-30T06:43:53Z","title_canon_sha256":"d06419f2cad4877deaae176b780ae5aa7d010f676f0667ea84a0eec72fbd1f5b"},"schema_version":"1.0","source":{"id":"1109.6725","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1109.6725","created_at":"2026-05-18T04:00:13Z"},{"alias_kind":"arxiv_version","alias_value":"1109.6725v2","created_at":"2026-05-18T04:00:13Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1109.6725","created_at":"2026-05-18T04:00:13Z"},{"alias_kind":"pith_short_12","alias_value":"I7WFXSGWSD33","created_at":"2026-05-18T12:26:30Z"},{"alias_kind":"pith_short_16","alias_value":"I7WFXSGWSD332QAI","created_at":"2026-05-18T12:26:30Z"},{"alias_kind":"pith_short_8","alias_value":"I7WFXSGW","created_at":"2026-05-18T12:26:30Z"}],"graph_snapshots":[{"event_id":"sha256:55eeb7c9baf35e96ef8c3552a049dec1d547e2f229259d1cff4358c5e1441582","target":"graph","created_at":"2026-05-18T04:00:13Z","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 review critically the concepts and the applications of Cayley Trees and Bethe Lattices in statistical mechanics in a tentative effort to remove widespread misuse of these simple, but yet important - and different - ideal graphs. We illustrate, in particular, two rigorous techniques to deal with Bethe Lattices, based respectively on self-similarity and on the Kolmogorov consistency theorem, linking the latter with the Cavity and Belief Propagation methods, more known to the physics community.","authors_text":"Massimo Ostilli","cross_cats":["cond-mat.dis-nn","math-ph","math.MP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cond-mat.stat-mech","submitted_at":"2011-09-30T06:43:53Z","title":"Cayley Trees and Bethe Lattices, a concise analysis for mathematicians and physicists"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1109.6725","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:ea2288e3414dd6dd7a2d6dcbe4a0857b008ddb2385ee3a33083171c6f8d896ae","target":"record","created_at":"2026-05-18T04:00:13Z","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":"f6fc1f7b697749b2540efd3b770ef1822336e36bcc496e2c7201d75a78031c68","cross_cats_sorted":["cond-mat.dis-nn","math-ph","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cond-mat.stat-mech","submitted_at":"2011-09-30T06:43:53Z","title_canon_sha256":"d06419f2cad4877deaae176b780ae5aa7d010f676f0667ea84a0eec72fbd1f5b"},"schema_version":"1.0","source":{"id":"1109.6725","kind":"arxiv","version":2}},"canonical_sha256":"47ec5bc8d690f7bd40087fbd7152a8812b9e44da26aeb7ce10a9189fbd8fafbc","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"47ec5bc8d690f7bd40087fbd7152a8812b9e44da26aeb7ce10a9189fbd8fafbc","first_computed_at":"2026-05-18T04:00:13.473079Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:00:13.473079Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"oa5gaVLZENo6N8s+B+1Qwmi3YyXnAOGDWpJzzWWM+AVkfKntFoDix7adqO0YgHW7yvk/qBGhX3iMiXx+RNaCCA==","signature_status":"signed_v1","signed_at":"2026-05-18T04:00:13.473762Z","signed_message":"canonical_sha256_bytes"},"source_id":"1109.6725","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:ea2288e3414dd6dd7a2d6dcbe4a0857b008ddb2385ee3a33083171c6f8d896ae","sha256:55eeb7c9baf35e96ef8c3552a049dec1d547e2f229259d1cff4358c5e1441582"],"state_sha256":"61ff21792e1dd2efab8cb27eb35f785259e57a90ff132feb5803f77f7c59d15a"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"y/IZJGkU9Aoje2EIVm3DG1KZvU9cJ0KFliZHX/qboe9UVmzwuOoNt601dC920JIurbf9w+/eXFFxnf/sxke/Cg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-25T08:08:39.788035Z","bundle_sha256":"967d833182f652443f2661c28e928c45a435602371afbe6957e8ce88c67c3e60"}}