{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:OTCTC7SWOGFAF2ZDHAPXK3UCOL","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":"ebaa3b69bf702eb53cbbe1e261617feb1e4c5327a7953e7b498721da5d00841e","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2026-06-09T22:52:43Z","title_canon_sha256":"4feb9315dbfc071c16b1350d5ab2396f1a555bc22628f6dbf2f1443d7d382aaa"},"schema_version":"1.0","source":{"id":"2606.11503","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2606.11503","created_at":"2026-06-11T01:09:52Z"},{"alias_kind":"arxiv_version","alias_value":"2606.11503v1","created_at":"2026-06-11T01:09:52Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.11503","created_at":"2026-06-11T01:09:52Z"},{"alias_kind":"pith_short_12","alias_value":"OTCTC7SWOGFA","created_at":"2026-06-11T01:09:52Z"},{"alias_kind":"pith_short_16","alias_value":"OTCTC7SWOGFAF2ZD","created_at":"2026-06-11T01:09:52Z"},{"alias_kind":"pith_short_8","alias_value":"OTCTC7SW","created_at":"2026-06-11T01:09:52Z"}],"graph_snapshots":[{"event_id":"sha256:a4f279d41d201c36d6d3ac205a6c0b4d804db6a177f5e0f01111bdff041214e9","target":"graph","created_at":"2026-06-11T01:09:52Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2606.11503/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"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","authors_text":"Augusto Teixeira, Caio Alves, Carlos Gustavo Moreira, Rangel Baldasso","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2026-06-09T22:52:43Z","title":"Percolation on hierarchical lattices"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.11503","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":"sha256:33e359a544ce4b51b56ca6a005fae40b6fc31392c73dbd88847918be6ca71fdf","target":"record","created_at":"2026-06-11T01:09:52Z","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":"ebaa3b69bf702eb53cbbe1e261617feb1e4c5327a7953e7b498721da5d00841e","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2026-06-09T22:52:43Z","title_canon_sha256":"4feb9315dbfc071c16b1350d5ab2396f1a555bc22628f6dbf2f1443d7d382aaa"},"schema_version":"1.0","source":{"id":"2606.11503","kind":"arxiv","version":1}},"canonical_sha256":"74c5317e56718a02eb23381f756e8272d10f2596cc44942ce8349096a2d7484c","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"74c5317e56718a02eb23381f756e8272d10f2596cc44942ce8349096a2d7484c","first_computed_at":"2026-06-11T01:09:52.714188Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-06-11T01:09:52.714188Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"mxciDFWS08z7Y9BQLdNoYW+1S2uJN3RovXp2lejS6TpdwYvFgWZjMH2Tgu3ePRL8tCXuH4FT7uPYcdbxfwx7CQ==","signature_status":"signed_v1","signed_at":"2026-06-11T01:09:52.714996Z","signed_message":"canonical_sha256_bytes"},"source_id":"2606.11503","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:33e359a544ce4b51b56ca6a005fae40b6fc31392c73dbd88847918be6ca71fdf","sha256:a4f279d41d201c36d6d3ac205a6c0b4d804db6a177f5e0f01111bdff041214e9"],"state_sha256":"41b1a4839f56d0c45540a196ae1edba0947db8c205607bd3b250aa0c302707da"}