{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2026:OTCTC7SWOGFAF2ZDHAPXK3UCOL","short_pith_number":"pith:OTCTC7SW","schema_version":"1.0","canonical_sha256":"74c5317e56718a02eb23381f756e8272d10f2596cc44942ce8349096a2d7484c","source":{"kind":"arxiv","id":"2606.11503","version":1},"attestation_state":"computed","paper":{"title":"Percolation on hierarchical lattices","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Augusto Teixeira, Caio Alves, Carlos Gustavo Moreira, Rangel Baldasso","submitted_at":"2026-06-09T22:52:43Z","abstract_excerpt":"We consider independent Bernoulli percolation on top of sequences of hierarchical graphs.\n  Given a graph $G_{1}$ with two distinguished vertices $a_{1}$ and $b_{1}$, the hierarchical graph with seed $G_{1}$ is the sequence $\\big( G_{k} \\big)_{k \\geq 1}$ resulting from the inductive procedure, where the graph $G_{k+1}$ is obtained from $G_{k}$ by replacing each of its edges with a copy of $G_{1}$, attached by the vertices $a_{1}$ and $b_{1}$. We prove that, under sharp hypotheses, percolation on these graphs presents a unique phase transition. Second, we establish the existence of several crit"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"2606.11503","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2026-06-09T22:52:43Z","cross_cats_sorted":[],"title_canon_sha256":"4feb9315dbfc071c16b1350d5ab2396f1a555bc22628f6dbf2f1443d7d382aaa","abstract_canon_sha256":"ebaa3b69bf702eb53cbbe1e261617feb1e4c5327a7953e7b498721da5d00841e"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-11T01:09:52.714996Z","signature_b64":"mxciDFWS08z7Y9BQLdNoYW+1S2uJN3RovXp2lejS6TpdwYvFgWZjMH2Tgu3ePRL8tCXuH4FT7uPYcdbxfwx7CQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"74c5317e56718a02eb23381f756e8272d10f2596cc44942ce8349096a2d7484c","last_reissued_at":"2026-06-11T01:09:52.714188Z","signature_status":"signed_v1","first_computed_at":"2026-06-11T01:09:52.714188Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Percolation on hierarchical lattices","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Augusto Teixeira, Caio Alves, Carlos Gustavo Moreira, Rangel Baldasso","submitted_at":"2026-06-09T22:52:43Z","abstract_excerpt":"We consider independent Bernoulli percolation on top of sequences of hierarchical graphs.\n  Given a graph $G_{1}$ with two distinguished vertices $a_{1}$ and $b_{1}$, the hierarchical graph with seed $G_{1}$ is the sequence $\\big( G_{k} \\big)_{k \\geq 1}$ resulting from the inductive procedure, where the graph $G_{k+1}$ is obtained from $G_{k}$ by replacing each of its edges with a copy of $G_{1}$, attached by the vertices $a_{1}$ and $b_{1}$. We prove that, under sharp hypotheses, percolation on these graphs presents a unique phase transition. Second, we establish the existence of several crit"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.11503","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2606.11503/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"2606.11503","created_at":"2026-06-11T01:09:52.714309+00:00"},{"alias_kind":"arxiv_version","alias_value":"2606.11503v1","created_at":"2026-06-11T01:09:52.714309+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.11503","created_at":"2026-06-11T01:09:52.714309+00:00"},{"alias_kind":"pith_short_12","alias_value":"OTCTC7SWOGFA","created_at":"2026-06-11T01:09:52.714309+00:00"},{"alias_kind":"pith_short_16","alias_value":"OTCTC7SWOGFAF2ZD","created_at":"2026-06-11T01:09:52.714309+00:00"},{"alias_kind":"pith_short_8","alias_value":"OTCTC7SW","created_at":"2026-06-11T01:09:52.714309+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/OTCTC7SWOGFAF2ZDHAPXK3UCOL","json":"https://pith.science/pith/OTCTC7SWOGFAF2ZDHAPXK3UCOL.json","graph_json":"https://pith.science/api/pith-number/OTCTC7SWOGFAF2ZDHAPXK3UCOL/graph.json","events_json":"https://pith.science/api/pith-number/OTCTC7SWOGFAF2ZDHAPXK3UCOL/events.json","paper":"https://pith.science/paper/OTCTC7SW"},"agent_actions":{"view_html":"https://pith.science/pith/OTCTC7SWOGFAF2ZDHAPXK3UCOL","download_json":"https://pith.science/pith/OTCTC7SWOGFAF2ZDHAPXK3UCOL.json","view_paper":"https://pith.science/paper/OTCTC7SW","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2606.11503&json=true","fetch_graph":"https://pith.science/api/pith-number/OTCTC7SWOGFAF2ZDHAPXK3UCOL/graph.json","fetch_events":"https://pith.science/api/pith-number/OTCTC7SWOGFAF2ZDHAPXK3UCOL/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/OTCTC7SWOGFAF2ZDHAPXK3UCOL/action/timestamp_anchor","attest_storage":"https://pith.science/pith/OTCTC7SWOGFAF2ZDHAPXK3UCOL/action/storage_attestation","attest_author":"https://pith.science/pith/OTCTC7SWOGFAF2ZDHAPXK3UCOL/action/author_attestation","sign_citation":"https://pith.science/pith/OTCTC7SWOGFAF2ZDHAPXK3UCOL/action/citation_signature","submit_replication":"https://pith.science/pith/OTCTC7SWOGFAF2ZDHAPXK3UCOL/action/replication_record"}},"created_at":"2026-06-11T01:09:52.714309+00:00","updated_at":"2026-06-11T01:09:52.714309+00:00"}