{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2007:ONG73XZU4MWCLGXU62I3H5UY3N","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":"b02fa580a051358983c14f2f8459c5dd1a943b881b7589cb34ab7e2c3c672960","cross_cats_sorted":["math-ph","math.CO","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2007-11-23T03:12:09Z","title_canon_sha256":"77a4f5565f72c45f1ff05fee5242321c41d796c3c4e4b3e5f588adf13333d7ce"},"schema_version":"1.0","source":{"id":"0711.3664","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0711.3664","created_at":"2026-05-18T04:16:52Z"},{"alias_kind":"arxiv_version","alias_value":"0711.3664v3","created_at":"2026-05-18T04:16:52Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0711.3664","created_at":"2026-05-18T04:16:52Z"},{"alias_kind":"pith_short_12","alias_value":"ONG73XZU4MWC","created_at":"2026-05-18T12:25:55Z"},{"alias_kind":"pith_short_16","alias_value":"ONG73XZU4MWCLGXU","created_at":"2026-05-18T12:25:55Z"},{"alias_kind":"pith_short_8","alias_value":"ONG73XZU","created_at":"2026-05-18T12:25:55Z"}],"graph_snapshots":[{"event_id":"sha256:3912f60cdc8268ba7aef2e6d17777305f349eb2dc47d17162a1623e5b07a70cf","target":"graph","created_at":"2026-05-18T04:16:52Z","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":"Consider $k$-colorings of the complete tree of depth $\\ell$ and branching factor $\\Delta$. If we fix the coloring of the leaves, as $\\ell$ tends to $\\infty$, for what range of $k$ is the root uniformly distributed over all $k$ colors? This corresponds to the threshold for uniqueness of the infinite-volume Gibbs measure. It is straightforward to show the existence of colorings of the leaves which ``freeze'' the entire tree when $k\\le\\Delta+1$. For $k\\geq\\Delta+2$, Jonasson proved the root is ``unbiased'' for any fixed coloring of the leaves and thus the Gibbs measure is unique. What happens for","authors_text":"Dror Weitz, Eric Vigoda, Juan Vera, Nayantara Bhatnagar","cross_cats":["math-ph","math.CO","math.MP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2007-11-23T03:12:09Z","title":"Reconstruction for Colorings on Trees"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0711.3664","kind":"arxiv","version":3},"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:fe7b7037e6b8ea9ed220f16fb49d00c732bebf67ddf177b5c838ccf1e716dce7","target":"record","created_at":"2026-05-18T04:16:52Z","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":"b02fa580a051358983c14f2f8459c5dd1a943b881b7589cb34ab7e2c3c672960","cross_cats_sorted":["math-ph","math.CO","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2007-11-23T03:12:09Z","title_canon_sha256":"77a4f5565f72c45f1ff05fee5242321c41d796c3c4e4b3e5f588adf13333d7ce"},"schema_version":"1.0","source":{"id":"0711.3664","kind":"arxiv","version":3}},"canonical_sha256":"734dfddf34e32c259af4f691b3f698db4ac15e589991dccf862ef7ee8fdfde23","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"734dfddf34e32c259af4f691b3f698db4ac15e589991dccf862ef7ee8fdfde23","first_computed_at":"2026-05-18T04:16:52.519613Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:16:52.519613Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"dcIO+1XFR/cwwmADyWKlC/R7m+GSH4xHaq/sfRGEzpe6DVdecrZGkuslVtG8BHLNAd4wdSTmx/XROT6V8YrkBw==","signature_status":"signed_v1","signed_at":"2026-05-18T04:16:52.520090Z","signed_message":"canonical_sha256_bytes"},"source_id":"0711.3664","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:fe7b7037e6b8ea9ed220f16fb49d00c732bebf67ddf177b5c838ccf1e716dce7","sha256:3912f60cdc8268ba7aef2e6d17777305f349eb2dc47d17162a1623e5b07a70cf"],"state_sha256":"6fc4380424ff340d826404a9dc67a6aa62c481ddddcbb91436a332b838126d18"}