{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2011:VQS3ALYQGIJUIUIGV4EEMSGPPJ","short_pith_number":"pith:VQS3ALYQ","canonical_record":{"source":{"id":"1104.2195","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2011-04-12T12:52:53Z","cross_cats_sorted":[],"title_canon_sha256":"d0b81eb961edac71ef204afa1b33bdf904682afa4db5029f8a2461517b2213d1","abstract_canon_sha256":"3e0c66920bc93d3b6fe9fc31b350aba699c9efc590663e0cb13a1e45816a0618"},"schema_version":"1.0"},"canonical_sha256":"ac25b02f103213445106af084648cf7a63b0db82a63571d9f7db54d4df372fe8","source":{"kind":"arxiv","id":"1104.2195","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1104.2195","created_at":"2026-05-18T04:21:51Z"},{"alias_kind":"arxiv_version","alias_value":"1104.2195v2","created_at":"2026-05-18T04:21:51Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1104.2195","created_at":"2026-05-18T04:21:51Z"},{"alias_kind":"pith_short_12","alias_value":"VQS3ALYQGIJU","created_at":"2026-05-18T12:26:44Z"},{"alias_kind":"pith_short_16","alias_value":"VQS3ALYQGIJUIUIG","created_at":"2026-05-18T12:26:44Z"},{"alias_kind":"pith_short_8","alias_value":"VQS3ALYQ","created_at":"2026-05-18T12:26:44Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2011:VQS3ALYQGIJUIUIGV4EEMSGPPJ","target":"record","payload":{"canonical_record":{"source":{"id":"1104.2195","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2011-04-12T12:52:53Z","cross_cats_sorted":[],"title_canon_sha256":"d0b81eb961edac71ef204afa1b33bdf904682afa4db5029f8a2461517b2213d1","abstract_canon_sha256":"3e0c66920bc93d3b6fe9fc31b350aba699c9efc590663e0cb13a1e45816a0618"},"schema_version":"1.0"},"canonical_sha256":"ac25b02f103213445106af084648cf7a63b0db82a63571d9f7db54d4df372fe8","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:21:51.468135Z","signature_b64":"SlW8Px6sgmUyHbDdv73mVgZMZjOwPpLHZPXlKxtE2FH6CjZ2CUdOXgFo4ZwGy+bPLHeoWT+7mLk2oPIQKUiTBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ac25b02f103213445106af084648cf7a63b0db82a63571d9f7db54d4df372fe8","last_reissued_at":"2026-05-18T04:21:51.467516Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:21:51.467516Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1104.2195","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T04:21:51Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"NamEMA8MhDIeny+sQ7pk85HvFNcUfNmNXwkVT9J0cYvWKSqWsQgTlb0Ls4Tl1dIOxkzjRcmxViZBQL5i89fSBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-09T07:42:59.883076Z"},"content_sha256":"455a73852dd2e7aa5241d234b595653ffdd33f341a830abd52618c17d3511a13","schema_version":"1.0","event_id":"sha256:455a73852dd2e7aa5241d234b595653ffdd33f341a830abd52618c17d3511a13"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2011:VQS3ALYQGIJUIUIGV4EEMSGPPJ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Topological Pressure for sub-additive potentials of amenable group actions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Bingbing Liang, Kesong Yan","submitted_at":"2011-04-12T12:52:53Z","abstract_excerpt":"We define the topological pressure for any sub-additive potentials of the countable discrete amenable group action and any given open cover. A local variational principle for the topological pressure is established."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1104.2195","kind":"arxiv","version":2},"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":"2026-05-18T04:21:51Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"KdPIiWBG6Qi+FFgcmDJcCIBFnMm5TuPiDuc5rTdcmLxw3fbgP1yFLHHGEoDWcct+YNveedFcT5uH1WjM+z5XBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-09T07:42:59.883703Z"},"content_sha256":"362a508b7416c2d51771369bf4fc6ded5f82abf6cd1f37f176d0e2bea50484b3","schema_version":"1.0","event_id":"sha256:362a508b7416c2d51771369bf4fc6ded5f82abf6cd1f37f176d0e2bea50484b3"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/VQS3ALYQGIJUIUIGV4EEMSGPPJ/bundle.json","state_url":"https://pith.science/pith/VQS3ALYQGIJUIUIGV4EEMSGPPJ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/VQS3ALYQGIJUIUIGV4EEMSGPPJ/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":"2026-06-09T07:42:59Z","links":{"resolver":"https://pith.science/pith/VQS3ALYQGIJUIUIGV4EEMSGPPJ","bundle":"https://pith.science/pith/VQS3ALYQGIJUIUIGV4EEMSGPPJ/bundle.json","state":"https://pith.science/pith/VQS3ALYQGIJUIUIGV4EEMSGPPJ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/VQS3ALYQGIJUIUIGV4EEMSGPPJ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:VQS3ALYQGIJUIUIGV4EEMSGPPJ","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":"3e0c66920bc93d3b6fe9fc31b350aba699c9efc590663e0cb13a1e45816a0618","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2011-04-12T12:52:53Z","title_canon_sha256":"d0b81eb961edac71ef204afa1b33bdf904682afa4db5029f8a2461517b2213d1"},"schema_version":"1.0","source":{"id":"1104.2195","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1104.2195","created_at":"2026-05-18T04:21:51Z"},{"alias_kind":"arxiv_version","alias_value":"1104.2195v2","created_at":"2026-05-18T04:21:51Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1104.2195","created_at":"2026-05-18T04:21:51Z"},{"alias_kind":"pith_short_12","alias_value":"VQS3ALYQGIJU","created_at":"2026-05-18T12:26:44Z"},{"alias_kind":"pith_short_16","alias_value":"VQS3ALYQGIJUIUIG","created_at":"2026-05-18T12:26:44Z"},{"alias_kind":"pith_short_8","alias_value":"VQS3ALYQ","created_at":"2026-05-18T12:26:44Z"}],"graph_snapshots":[{"event_id":"sha256:362a508b7416c2d51771369bf4fc6ded5f82abf6cd1f37f176d0e2bea50484b3","target":"graph","created_at":"2026-05-18T04:21:51Z","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 define the topological pressure for any sub-additive potentials of the countable discrete amenable group action and any given open cover. A local variational principle for the topological pressure is established.","authors_text":"Bingbing Liang, Kesong Yan","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2011-04-12T12:52:53Z","title":"Topological Pressure for sub-additive potentials of amenable group actions"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1104.2195","kind":"arxiv","version":2},"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":"sha256:455a73852dd2e7aa5241d234b595653ffdd33f341a830abd52618c17d3511a13","target":"record","created_at":"2026-05-18T04:21:51Z","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":"3e0c66920bc93d3b6fe9fc31b350aba699c9efc590663e0cb13a1e45816a0618","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2011-04-12T12:52:53Z","title_canon_sha256":"d0b81eb961edac71ef204afa1b33bdf904682afa4db5029f8a2461517b2213d1"},"schema_version":"1.0","source":{"id":"1104.2195","kind":"arxiv","version":2}},"canonical_sha256":"ac25b02f103213445106af084648cf7a63b0db82a63571d9f7db54d4df372fe8","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"ac25b02f103213445106af084648cf7a63b0db82a63571d9f7db54d4df372fe8","first_computed_at":"2026-05-18T04:21:51.467516Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:21:51.467516Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"SlW8Px6sgmUyHbDdv73mVgZMZjOwPpLHZPXlKxtE2FH6CjZ2CUdOXgFo4ZwGy+bPLHeoWT+7mLk2oPIQKUiTBw==","signature_status":"signed_v1","signed_at":"2026-05-18T04:21:51.468135Z","signed_message":"canonical_sha256_bytes"},"source_id":"1104.2195","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:455a73852dd2e7aa5241d234b595653ffdd33f341a830abd52618c17d3511a13","sha256:362a508b7416c2d51771369bf4fc6ded5f82abf6cd1f37f176d0e2bea50484b3"],"state_sha256":"fe284d3eff01e7b80582d88e736ec37d44940d17d17c27afe5ab53ed9c029291"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"VJFq8PFiuoCAXhyDhRWYs2pZ0EAdaRSoVRIiiGWpsVtQBCKKuCGVp5PWiYrIO0xjZj8MjTI47DctZB7wYY27Bw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-09T07:42:59.886947Z","bundle_sha256":"a4a9b7f868a04c73f63207e8dcaae047226270f68bb71686d60e9b788d193453"}}