{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2011:6KDXELOWAFQGJZPKXJCFH5BYJP","short_pith_number":"pith:6KDXELOW","canonical_record":{"source":{"id":"1108.4192","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2011-08-21T17:46:01Z","cross_cats_sorted":[],"title_canon_sha256":"d7e46e60aa782dd39d7a285f5be31a40a33d4b6f9b5f0d439c6201239dfc0940","abstract_canon_sha256":"c2999847ad41bc648fba50a671b1b12e5e137152140eceb0b38777cfe8ae3162"},"schema_version":"1.0"},"canonical_sha256":"f287722dd6016064e5eaba4453f4384bf2de3269f97f8adbfa09770355adfe6d","source":{"kind":"arxiv","id":"1108.4192","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1108.4192","created_at":"2026-05-18T04:14:55Z"},{"alias_kind":"arxiv_version","alias_value":"1108.4192v1","created_at":"2026-05-18T04:14:55Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1108.4192","created_at":"2026-05-18T04:14:55Z"},{"alias_kind":"pith_short_12","alias_value":"6KDXELOWAFQG","created_at":"2026-05-18T12:26:22Z"},{"alias_kind":"pith_short_16","alias_value":"6KDXELOWAFQGJZPK","created_at":"2026-05-18T12:26:22Z"},{"alias_kind":"pith_short_8","alias_value":"6KDXELOW","created_at":"2026-05-18T12:26:22Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2011:6KDXELOWAFQGJZPKXJCFH5BYJP","target":"record","payload":{"canonical_record":{"source":{"id":"1108.4192","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2011-08-21T17:46:01Z","cross_cats_sorted":[],"title_canon_sha256":"d7e46e60aa782dd39d7a285f5be31a40a33d4b6f9b5f0d439c6201239dfc0940","abstract_canon_sha256":"c2999847ad41bc648fba50a671b1b12e5e137152140eceb0b38777cfe8ae3162"},"schema_version":"1.0"},"canonical_sha256":"f287722dd6016064e5eaba4453f4384bf2de3269f97f8adbfa09770355adfe6d","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:14:55.973170Z","signature_b64":"2dsofjuQOG4ngcQlzy6fCB7eW3bQkuVW+O9T4x8ywSJ0k1mhAAr7LhHM6bSMcwEQu4xjMevWyKUGG0EBwr3KAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f287722dd6016064e5eaba4453f4384bf2de3269f97f8adbfa09770355adfe6d","last_reissued_at":"2026-05-18T04:14:55.972701Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:14:55.972701Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1108.4192","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T04:14:55Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"oAh6mqIgAM1TaDmEcl/+bSl4G3d6ZKXIQeEid2638+lErvR65Q1zkq0rljn3V089CH3gX0e+JdQBTBowkI0bDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-31T11:50:41.105985Z"},"content_sha256":"a42a48e83c49d7b1cbcaa099ed2af0e175d2844f4a34f57eeccea13ba9298f3d","schema_version":"1.0","event_id":"sha256:a42a48e83c49d7b1cbcaa099ed2af0e175d2844f4a34f57eeccea13ba9298f3d"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2011:6KDXELOWAFQGJZPKXJCFH5BYJP","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Harmonic models and spanning forests of residually finite groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Hanfeng Li, Lewis Bowen","submitted_at":"2011-08-21T17:46:01Z","abstract_excerpt":"We prove a number of identities relating the sofic entropy of a certain class of non-expansive algebraic dynamical systems, the sofic entropy of the Wired Spanning Forest and the tree entropy of Cayley graphs of residually finite groups. We also show that homoclinic points and periodic points in harmonic models are dense under general conditions."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1108.4192","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":""},"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":"2026-05-18T04:14:55Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"RjWp+3jf3NE1JLv7EKg18+e9sAdJfi6x4ah6kXHPSstpT2waUJ5LBCgYL57IlQuxXzTqVqGR/cmj/6UWowAuBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-31T11:50:41.106675Z"},"content_sha256":"024e205c0a1f9ae2547abbeeae5da497a2e7613b8a06325eeb59eb43faa9ea1b","schema_version":"1.0","event_id":"sha256:024e205c0a1f9ae2547abbeeae5da497a2e7613b8a06325eeb59eb43faa9ea1b"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/6KDXELOWAFQGJZPKXJCFH5BYJP/bundle.json","state_url":"https://pith.science/pith/6KDXELOWAFQGJZPKXJCFH5BYJP/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/6KDXELOWAFQGJZPKXJCFH5BYJP/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":"2026-05-31T11:50:41Z","links":{"resolver":"https://pith.science/pith/6KDXELOWAFQGJZPKXJCFH5BYJP","bundle":"https://pith.science/pith/6KDXELOWAFQGJZPKXJCFH5BYJP/bundle.json","state":"https://pith.science/pith/6KDXELOWAFQGJZPKXJCFH5BYJP/state.json","well_known_bundle":"https://pith.science/.well-known/pith/6KDXELOWAFQGJZPKXJCFH5BYJP/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:6KDXELOWAFQGJZPKXJCFH5BYJP","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":"c2999847ad41bc648fba50a671b1b12e5e137152140eceb0b38777cfe8ae3162","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2011-08-21T17:46:01Z","title_canon_sha256":"d7e46e60aa782dd39d7a285f5be31a40a33d4b6f9b5f0d439c6201239dfc0940"},"schema_version":"1.0","source":{"id":"1108.4192","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1108.4192","created_at":"2026-05-18T04:14:55Z"},{"alias_kind":"arxiv_version","alias_value":"1108.4192v1","created_at":"2026-05-18T04:14:55Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1108.4192","created_at":"2026-05-18T04:14:55Z"},{"alias_kind":"pith_short_12","alias_value":"6KDXELOWAFQG","created_at":"2026-05-18T12:26:22Z"},{"alias_kind":"pith_short_16","alias_value":"6KDXELOWAFQGJZPK","created_at":"2026-05-18T12:26:22Z"},{"alias_kind":"pith_short_8","alias_value":"6KDXELOW","created_at":"2026-05-18T12:26:22Z"}],"graph_snapshots":[{"event_id":"sha256:024e205c0a1f9ae2547abbeeae5da497a2e7613b8a06325eeb59eb43faa9ea1b","target":"graph","created_at":"2026-05-18T04:14:55Z","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 prove a number of identities relating the sofic entropy of a certain class of non-expansive algebraic dynamical systems, the sofic entropy of the Wired Spanning Forest and the tree entropy of Cayley graphs of residually finite groups. We also show that homoclinic points and periodic points in harmonic models are dense under general conditions.","authors_text":"Hanfeng Li, Lewis Bowen","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2011-08-21T17:46:01Z","title":"Harmonic models and spanning forests of residually finite groups"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1108.4192","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:a42a48e83c49d7b1cbcaa099ed2af0e175d2844f4a34f57eeccea13ba9298f3d","target":"record","created_at":"2026-05-18T04:14:55Z","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":"c2999847ad41bc648fba50a671b1b12e5e137152140eceb0b38777cfe8ae3162","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2011-08-21T17:46:01Z","title_canon_sha256":"d7e46e60aa782dd39d7a285f5be31a40a33d4b6f9b5f0d439c6201239dfc0940"},"schema_version":"1.0","source":{"id":"1108.4192","kind":"arxiv","version":1}},"canonical_sha256":"f287722dd6016064e5eaba4453f4384bf2de3269f97f8adbfa09770355adfe6d","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"f287722dd6016064e5eaba4453f4384bf2de3269f97f8adbfa09770355adfe6d","first_computed_at":"2026-05-18T04:14:55.972701Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:14:55.972701Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"2dsofjuQOG4ngcQlzy6fCB7eW3bQkuVW+O9T4x8ywSJ0k1mhAAr7LhHM6bSMcwEQu4xjMevWyKUGG0EBwr3KAQ==","signature_status":"signed_v1","signed_at":"2026-05-18T04:14:55.973170Z","signed_message":"canonical_sha256_bytes"},"source_id":"1108.4192","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:a42a48e83c49d7b1cbcaa099ed2af0e175d2844f4a34f57eeccea13ba9298f3d","sha256:024e205c0a1f9ae2547abbeeae5da497a2e7613b8a06325eeb59eb43faa9ea1b"],"state_sha256":"a66eacfe34c1625e5c39bd66fc111dbeb4d985437f23f56b518a6f5d4a56bbeb"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"kR/cSaTbWwpAvnvil1eZKFxXDOR93vcXFhCGd0vgkj8hACQ7Locy7qND5nXMEqzNtTWT0zNiiUgPjHm4m/VqBg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-31T11:50:41.110297Z","bundle_sha256":"0ac22a6fdbccde54a3b5113d3b09648f13d77cb59196d0e81a773c4b854cb951"}}