{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:UCU3YWLFMEFGUGAIV66OPYZBOZ","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":"9a68ccc2d121df62658ae07479fcbacc9ac4efb4a6e0a5917ea0c91b02519f4a","cross_cats_sorted":["math-ph","math.GR","math.MP","math.PR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-01-02T20:34:06Z","title_canon_sha256":"a1bd960f57b2b59ae6ed5e26a6a1e48718692bc9165c9a7d507535100fbed4a4"},"schema_version":"1.0","source":{"id":"1501.00476","kind":"arxiv","version":5}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1501.00476","created_at":"2026-05-18T01:08:26Z"},{"alias_kind":"arxiv_version","alias_value":"1501.00476v5","created_at":"2026-05-18T01:08:26Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1501.00476","created_at":"2026-05-18T01:08:26Z"},{"alias_kind":"pith_short_12","alias_value":"UCU3YWLFMEFG","created_at":"2026-05-18T12:29:44Z"},{"alias_kind":"pith_short_16","alias_value":"UCU3YWLFMEFGUGAI","created_at":"2026-05-18T12:29:44Z"},{"alias_kind":"pith_short_8","alias_value":"UCU3YWLF","created_at":"2026-05-18T12:29:44Z"}],"graph_snapshots":[{"event_id":"sha256:1e24636f9075fd87ccf3fd838c5f4e66292eeb10c3cced84e2b2567c7f1695df","target":"graph","created_at":"2026-05-18T01:08:26Z","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":"The connective constant $\\mu(G)$ of an infinite transitive graph $G$ is the exponential growth rate of the number of self-avoiding walks from a given origin. In earlier work of Grimmett and Li, a locality theorem was proved for connective constants, namely, that the connective constants of two graphs are close in value whenever the graphs agree on a large ball around the origin. A condition of the theorem was that the graphs support so-called 'unimodular graph height functions'. When the graphs are Cayley graphs of infinite, finitely generated groups, there is a special type of unimodular grap","authors_text":"Geoffrey R. Grimmett, Zhongyang Li","cross_cats":["math-ph","math.GR","math.MP","math.PR"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-01-02T20:34:06Z","title":"Connective constants and height functions for Cayley graphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1501.00476","kind":"arxiv","version":5},"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":"sha256:0bb20122d3677beddfd75039474a236eb611a56a26009120e0c6fbc4042c4e8a","target":"record","created_at":"2026-05-18T01:08:26Z","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":"9a68ccc2d121df62658ae07479fcbacc9ac4efb4a6e0a5917ea0c91b02519f4a","cross_cats_sorted":["math-ph","math.GR","math.MP","math.PR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-01-02T20:34:06Z","title_canon_sha256":"a1bd960f57b2b59ae6ed5e26a6a1e48718692bc9165c9a7d507535100fbed4a4"},"schema_version":"1.0","source":{"id":"1501.00476","kind":"arxiv","version":5}},"canonical_sha256":"a0a9bc5965610a6a1808afbce7e32176499a6a393dda968a0883ae6bd37e88c0","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"a0a9bc5965610a6a1808afbce7e32176499a6a393dda968a0883ae6bd37e88c0","first_computed_at":"2026-05-18T01:08:26.731501Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:08:26.731501Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Fr9PiGV4J0zrzBc6F2I8G+S8s902KJDRc11xb1bLJT4AxWJ0AJ/97XVE6G2tMWgN8FA800vIWrSbNg9wVaV0AQ==","signature_status":"signed_v1","signed_at":"2026-05-18T01:08:26.732186Z","signed_message":"canonical_sha256_bytes"},"source_id":"1501.00476","source_kind":"arxiv","source_version":5}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:0bb20122d3677beddfd75039474a236eb611a56a26009120e0c6fbc4042c4e8a","sha256:1e24636f9075fd87ccf3fd838c5f4e66292eeb10c3cced84e2b2567c7f1695df"],"state_sha256":"a63159788cbb8164776e7bcf21a98e0b6b46b96228c3437ea3973beecc62ef6a"}