{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2005:7GZHZC4EHOSBO4KBYGJFPJULPI","short_pith_number":"pith:7GZHZC4E","canonical_record":{"source":{"id":"math-ph/0510024","kind":"arxiv","version":2},"metadata":{"license":"","primary_cat":"math-ph","submitted_at":"2005-10-06T14:12:22Z","cross_cats_sorted":["math.MP"],"title_canon_sha256":"ddb3d173a66eaba4e977d29e484b935243f5fafafe1af70a085cea9e337ea49d","abstract_canon_sha256":"6c3e70bfcfd9618854b41eb3a5f2d676f95ad9767c6f16a8e497ffd287fb3a20"},"schema_version":"1.0"},"canonical_sha256":"f9b27c8b843ba4177141c19257a68b7a0bec64ff711bd320373ab2d5d0541494","source":{"kind":"arxiv","id":"math-ph/0510024","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math-ph/0510024","created_at":"2026-05-18T02:35:38Z"},{"alias_kind":"arxiv_version","alias_value":"math-ph/0510024v2","created_at":"2026-05-18T02:35:38Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math-ph/0510024","created_at":"2026-05-18T02:35:38Z"},{"alias_kind":"pith_short_12","alias_value":"7GZHZC4EHOSB","created_at":"2026-05-18T12:25:53Z"},{"alias_kind":"pith_short_16","alias_value":"7GZHZC4EHOSBO4KB","created_at":"2026-05-18T12:25:53Z"},{"alias_kind":"pith_short_8","alias_value":"7GZHZC4E","created_at":"2026-05-18T12:25:53Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2005:7GZHZC4EHOSBO4KBYGJFPJULPI","target":"record","payload":{"canonical_record":{"source":{"id":"math-ph/0510024","kind":"arxiv","version":2},"metadata":{"license":"","primary_cat":"math-ph","submitted_at":"2005-10-06T14:12:22Z","cross_cats_sorted":["math.MP"],"title_canon_sha256":"ddb3d173a66eaba4e977d29e484b935243f5fafafe1af70a085cea9e337ea49d","abstract_canon_sha256":"6c3e70bfcfd9618854b41eb3a5f2d676f95ad9767c6f16a8e497ffd287fb3a20"},"schema_version":"1.0"},"canonical_sha256":"f9b27c8b843ba4177141c19257a68b7a0bec64ff711bd320373ab2d5d0541494","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:35:38.685417Z","signature_b64":"FSZEdsuPyX6d+JuAYR1szsM5QCI0CHdWgSrP5FiS7IXWC6T9C7Gskv5USbOfSwnxH9JqfYtqlBKgXLw3TZrsCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f9b27c8b843ba4177141c19257a68b7a0bec64ff711bd320373ab2d5d0541494","last_reissued_at":"2026-05-18T02:35:38.685001Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:35:38.685001Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"math-ph/0510024","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":"2026-05-18T02:35:38Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"n8/Q1Hx91rSS1tXOjvYkqttM6dOhtm7T6lRzSLrqyR+oxA/YZu7LBMmeeFwyfNO6IKAEOY+NyP+gQfeJ0LmuDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-01T21:21:01.920056Z"},"content_sha256":"bbf0d29287a25caa9de57fe318001ab755fa3d62c55922bff5a179bb172ee90a","schema_version":"1.0","event_id":"sha256:bbf0d29287a25caa9de57fe318001ab755fa3d62c55922bff5a179bb172ee90a"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2005:7GZHZC4EHOSBO4KBYGJFPJULPI","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On Inhomogeneous $p$-Adic Potts Model on a Cayley Tree","license":"","headline":"","cross_cats":["math.MP"],"primary_cat":"math-ph","authors_text":"Farrukh Mukhamedov, Utkir Rozikov","submitted_at":"2005-10-06T14:12:22Z","abstract_excerpt":"We consider a nearest-neighbor inhomogeneous $p$-adic Potts (with $q\\geq 2$ spin values) model on the Cayley tree of order $k\\geq 1$. The inhomogeneity means that the interaction $J_{xy}$ couplings depend on nearest-neighbors points $x, y $ of the Cayley tree. We study ($p-$ adic) Gibbs measures of the model. We show that (i) if $q\\notin p\\mathbb{N}$ then there is unique Gibbs measure for any $k\\geq 1$ and $\\forall J_{xy}$ with $|J_{xy}|<p^{-1/(p-1)}$. (ii) For $q\\in p\\mathbb{N}, p\\geq 3$ one can choose $J_{xy}$ and $k\\geq 1$ such that there exist at least two Gibbs measures which are translat"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math-ph/0510024","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":"2026-05-18T02:35:38Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"W8tPc+uaKjQz7iXyhGbXs1nApQT4VpDKrN/vmjAd0w38B5Re8UQKzFrIj8+HN+OWfPbqDDBA1czv9qMs9SJQDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-01T21:21:01.920414Z"},"content_sha256":"780ec9b5e0a3f7ba05edadd72a6b8cf1fe5ee9b5ac1eb9cf3b069abacf30d9d9","schema_version":"1.0","event_id":"sha256:780ec9b5e0a3f7ba05edadd72a6b8cf1fe5ee9b5ac1eb9cf3b069abacf30d9d9"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/7GZHZC4EHOSBO4KBYGJFPJULPI/bundle.json","state_url":"https://pith.science/pith/7GZHZC4EHOSBO4KBYGJFPJULPI/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/7GZHZC4EHOSBO4KBYGJFPJULPI/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":"2026-06-01T21:21:01Z","links":{"resolver":"https://pith.science/pith/7GZHZC4EHOSBO4KBYGJFPJULPI","bundle":"https://pith.science/pith/7GZHZC4EHOSBO4KBYGJFPJULPI/bundle.json","state":"https://pith.science/pith/7GZHZC4EHOSBO4KBYGJFPJULPI/state.json","well_known_bundle":"https://pith.science/.well-known/pith/7GZHZC4EHOSBO4KBYGJFPJULPI/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2005:7GZHZC4EHOSBO4KBYGJFPJULPI","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":"6c3e70bfcfd9618854b41eb3a5f2d676f95ad9767c6f16a8e497ffd287fb3a20","cross_cats_sorted":["math.MP"],"license":"","primary_cat":"math-ph","submitted_at":"2005-10-06T14:12:22Z","title_canon_sha256":"ddb3d173a66eaba4e977d29e484b935243f5fafafe1af70a085cea9e337ea49d"},"schema_version":"1.0","source":{"id":"math-ph/0510024","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math-ph/0510024","created_at":"2026-05-18T02:35:38Z"},{"alias_kind":"arxiv_version","alias_value":"math-ph/0510024v2","created_at":"2026-05-18T02:35:38Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math-ph/0510024","created_at":"2026-05-18T02:35:38Z"},{"alias_kind":"pith_short_12","alias_value":"7GZHZC4EHOSB","created_at":"2026-05-18T12:25:53Z"},{"alias_kind":"pith_short_16","alias_value":"7GZHZC4EHOSBO4KB","created_at":"2026-05-18T12:25:53Z"},{"alias_kind":"pith_short_8","alias_value":"7GZHZC4E","created_at":"2026-05-18T12:25:53Z"}],"graph_snapshots":[{"event_id":"sha256:780ec9b5e0a3f7ba05edadd72a6b8cf1fe5ee9b5ac1eb9cf3b069abacf30d9d9","target":"graph","created_at":"2026-05-18T02:35:38Z","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 consider a nearest-neighbor inhomogeneous $p$-adic Potts (with $q\\geq 2$ spin values) model on the Cayley tree of order $k\\geq 1$. The inhomogeneity means that the interaction $J_{xy}$ couplings depend on nearest-neighbors points $x, y $ of the Cayley tree. We study ($p-$ adic) Gibbs measures of the model. We show that (i) if $q\\notin p\\mathbb{N}$ then there is unique Gibbs measure for any $k\\geq 1$ and $\\forall J_{xy}$ with $|J_{xy}|<p^{-1/(p-1)}$. (ii) For $q\\in p\\mathbb{N}, p\\geq 3$ one can choose $J_{xy}$ and $k\\geq 1$ such that there exist at least two Gibbs measures which are translat","authors_text":"Farrukh Mukhamedov, Utkir Rozikov","cross_cats":["math.MP"],"headline":"","license":"","primary_cat":"math-ph","submitted_at":"2005-10-06T14:12:22Z","title":"On Inhomogeneous $p$-Adic Potts Model on a Cayley Tree"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math-ph/0510024","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:bbf0d29287a25caa9de57fe318001ab755fa3d62c55922bff5a179bb172ee90a","target":"record","created_at":"2026-05-18T02:35:38Z","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":"6c3e70bfcfd9618854b41eb3a5f2d676f95ad9767c6f16a8e497ffd287fb3a20","cross_cats_sorted":["math.MP"],"license":"","primary_cat":"math-ph","submitted_at":"2005-10-06T14:12:22Z","title_canon_sha256":"ddb3d173a66eaba4e977d29e484b935243f5fafafe1af70a085cea9e337ea49d"},"schema_version":"1.0","source":{"id":"math-ph/0510024","kind":"arxiv","version":2}},"canonical_sha256":"f9b27c8b843ba4177141c19257a68b7a0bec64ff711bd320373ab2d5d0541494","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"f9b27c8b843ba4177141c19257a68b7a0bec64ff711bd320373ab2d5d0541494","first_computed_at":"2026-05-18T02:35:38.685001Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:35:38.685001Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"FSZEdsuPyX6d+JuAYR1szsM5QCI0CHdWgSrP5FiS7IXWC6T9C7Gskv5USbOfSwnxH9JqfYtqlBKgXLw3TZrsCQ==","signature_status":"signed_v1","signed_at":"2026-05-18T02:35:38.685417Z","signed_message":"canonical_sha256_bytes"},"source_id":"math-ph/0510024","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:bbf0d29287a25caa9de57fe318001ab755fa3d62c55922bff5a179bb172ee90a","sha256:780ec9b5e0a3f7ba05edadd72a6b8cf1fe5ee9b5ac1eb9cf3b069abacf30d9d9"],"state_sha256":"27a44a3db35d097888ec547782608418df52827f85bca6865b235e2444528f94"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"+RVyN9RCj8yHbxk1LrwHmYdsgut3U7nzksxPqzeRpcFP3zBMsCMrqatrlTNQMTO6FTpRM7vjo//qB+NkkKfAAA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-01T21:21:01.922307Z","bundle_sha256":"f4eb629f75af0149c02aa4b3ae95a304a7c0306c1454e3b71ddd0c94f6ebd1fe"}}