{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:E4C3LAHGPC4IAH22B7L4DTS7UN","short_pith_number":"pith:E4C3LAHG","canonical_record":{"source":{"id":"1209.3936","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2012-09-18T12:51:45Z","cross_cats_sorted":["math.AT"],"title_canon_sha256":"33de4291368e33c677481766a92166fce24e5539a363cfa24c13e218992d65bb","abstract_canon_sha256":"017c3903a72058bcee54ec7e1a0d449bc89f8f0fddbaebc59033940ad62ac6b0"},"schema_version":"1.0"},"canonical_sha256":"2705b580e678b8801f5a0fd7c1ce5fa349785a8d6af21dea32836cd36f8a0b57","source":{"kind":"arxiv","id":"1209.3936","version":4},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1209.3936","created_at":"2026-05-18T00:21:38Z"},{"alias_kind":"arxiv_version","alias_value":"1209.3936v4","created_at":"2026-05-18T00:21:38Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1209.3936","created_at":"2026-05-18T00:21:38Z"},{"alias_kind":"pith_short_12","alias_value":"E4C3LAHGPC4I","created_at":"2026-05-18T12:27:04Z"},{"alias_kind":"pith_short_16","alias_value":"E4C3LAHGPC4IAH22","created_at":"2026-05-18T12:27:04Z"},{"alias_kind":"pith_short_8","alias_value":"E4C3LAHG","created_at":"2026-05-18T12:27:04Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:E4C3LAHGPC4IAH22B7L4DTS7UN","target":"record","payload":{"canonical_record":{"source":{"id":"1209.3936","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2012-09-18T12:51:45Z","cross_cats_sorted":["math.AT"],"title_canon_sha256":"33de4291368e33c677481766a92166fce24e5539a363cfa24c13e218992d65bb","abstract_canon_sha256":"017c3903a72058bcee54ec7e1a0d449bc89f8f0fddbaebc59033940ad62ac6b0"},"schema_version":"1.0"},"canonical_sha256":"2705b580e678b8801f5a0fd7c1ce5fa349785a8d6af21dea32836cd36f8a0b57","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:21:38.217369Z","signature_b64":"o/jDmePOO5HqMSIkEZqrIo8dt9jjz/RVh8Za3YXWpRQDxbim//B9Id2mKuEY1Jvb54jmBVr43otfc/gzBquECA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2705b580e678b8801f5a0fd7c1ce5fa349785a8d6af21dea32836cd36f8a0b57","last_reissued_at":"2026-05-18T00:21:38.216573Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:21:38.216573Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1209.3936","source_version":4,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:21:38Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"RV3KlL25yMLqAToz5n7t0e0v1xEy3mSBVySunQrG3xAFv+wvCqxbxgVOkXUM9cPQYZ+6mGeUHUC7z6KRlW9aDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-24T22:57:27.025745Z"},"content_sha256":"807b556949480877d95497409dc34aee648f3ff79ea18bc674495900d9f5d14b","schema_version":"1.0","event_id":"sha256:807b556949480877d95497409dc34aee648f3ff79ea18bc674495900d9f5d14b"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:E4C3LAHGPC4IAH22B7L4DTS7UN","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Topological Entropy Conjecture","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AT"],"primary_cat":"math.DS","authors_text":"Lvlin Luo","submitted_at":"2012-09-18T12:51:45Z","abstract_excerpt":"In 1974, M. Shub stated Topological Entropy Conjecture, that is, the inequality $\\log\\rho\\leq ent(f)$ is valid or not, where $f$ is a continuous self-map on a compact manifold $M$, $ent(f)$ is the topological entropy of $f$ and $\\rho$ is the maximum absolute eigenvalue of $f_*$ which is the linear transformation induced by $f$ on the homology group $H_{*}(M;\\mathbb{Z})=\\bigoplus\\limits_{i=0}^n{H_{i}(M;\\mathbb{Z})}$. In 1986, A. B. Katok gave a counterexample such that the inequality $\\log\\rho\\leq ent(f)$ is invalid. In this paper, we define $f$-\\v{C}ech homology group $\\check{H}_{i}(X,f;\\mathb"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1209.3936","kind":"arxiv","version":4},"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":"2026-05-18T00:21:38Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"/oelhDl/Uen2792BzvK9GquwkFJxh/Udqjlt9lV9hbuAwhKoihsAtlEZk/H/iB5L+sSsl5VfvOXSCeNOIaFaBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-24T22:57:27.026093Z"},"content_sha256":"7eb85a5420fcfcad584d81c1bd858ac179ee024e47a0d0bfd9a04df64f01b986","schema_version":"1.0","event_id":"sha256:7eb85a5420fcfcad584d81c1bd858ac179ee024e47a0d0bfd9a04df64f01b986"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/E4C3LAHGPC4IAH22B7L4DTS7UN/bundle.json","state_url":"https://pith.science/pith/E4C3LAHGPC4IAH22B7L4DTS7UN/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/E4C3LAHGPC4IAH22B7L4DTS7UN/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":"2026-06-24T22:57:27Z","links":{"resolver":"https://pith.science/pith/E4C3LAHGPC4IAH22B7L4DTS7UN","bundle":"https://pith.science/pith/E4C3LAHGPC4IAH22B7L4DTS7UN/bundle.json","state":"https://pith.science/pith/E4C3LAHGPC4IAH22B7L4DTS7UN/state.json","well_known_bundle":"https://pith.science/.well-known/pith/E4C3LAHGPC4IAH22B7L4DTS7UN/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:E4C3LAHGPC4IAH22B7L4DTS7UN","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":"017c3903a72058bcee54ec7e1a0d449bc89f8f0fddbaebc59033940ad62ac6b0","cross_cats_sorted":["math.AT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2012-09-18T12:51:45Z","title_canon_sha256":"33de4291368e33c677481766a92166fce24e5539a363cfa24c13e218992d65bb"},"schema_version":"1.0","source":{"id":"1209.3936","kind":"arxiv","version":4}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1209.3936","created_at":"2026-05-18T00:21:38Z"},{"alias_kind":"arxiv_version","alias_value":"1209.3936v4","created_at":"2026-05-18T00:21:38Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1209.3936","created_at":"2026-05-18T00:21:38Z"},{"alias_kind":"pith_short_12","alias_value":"E4C3LAHGPC4I","created_at":"2026-05-18T12:27:04Z"},{"alias_kind":"pith_short_16","alias_value":"E4C3LAHGPC4IAH22","created_at":"2026-05-18T12:27:04Z"},{"alias_kind":"pith_short_8","alias_value":"E4C3LAHG","created_at":"2026-05-18T12:27:04Z"}],"graph_snapshots":[{"event_id":"sha256:7eb85a5420fcfcad584d81c1bd858ac179ee024e47a0d0bfd9a04df64f01b986","target":"graph","created_at":"2026-05-18T00:21:38Z","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":"In 1974, M. Shub stated Topological Entropy Conjecture, that is, the inequality $\\log\\rho\\leq ent(f)$ is valid or not, where $f$ is a continuous self-map on a compact manifold $M$, $ent(f)$ is the topological entropy of $f$ and $\\rho$ is the maximum absolute eigenvalue of $f_*$ which is the linear transformation induced by $f$ on the homology group $H_{*}(M;\\mathbb{Z})=\\bigoplus\\limits_{i=0}^n{H_{i}(M;\\mathbb{Z})}$. In 1986, A. B. Katok gave a counterexample such that the inequality $\\log\\rho\\leq ent(f)$ is invalid. In this paper, we define $f$-\\v{C}ech homology group $\\check{H}_{i}(X,f;\\mathb","authors_text":"Lvlin Luo","cross_cats":["math.AT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2012-09-18T12:51:45Z","title":"Topological Entropy Conjecture"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1209.3936","kind":"arxiv","version":4},"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":"sha256:807b556949480877d95497409dc34aee648f3ff79ea18bc674495900d9f5d14b","target":"record","created_at":"2026-05-18T00:21:38Z","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":"017c3903a72058bcee54ec7e1a0d449bc89f8f0fddbaebc59033940ad62ac6b0","cross_cats_sorted":["math.AT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2012-09-18T12:51:45Z","title_canon_sha256":"33de4291368e33c677481766a92166fce24e5539a363cfa24c13e218992d65bb"},"schema_version":"1.0","source":{"id":"1209.3936","kind":"arxiv","version":4}},"canonical_sha256":"2705b580e678b8801f5a0fd7c1ce5fa349785a8d6af21dea32836cd36f8a0b57","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"2705b580e678b8801f5a0fd7c1ce5fa349785a8d6af21dea32836cd36f8a0b57","first_computed_at":"2026-05-18T00:21:38.216573Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:21:38.216573Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"o/jDmePOO5HqMSIkEZqrIo8dt9jjz/RVh8Za3YXWpRQDxbim//B9Id2mKuEY1Jvb54jmBVr43otfc/gzBquECA==","signature_status":"signed_v1","signed_at":"2026-05-18T00:21:38.217369Z","signed_message":"canonical_sha256_bytes"},"source_id":"1209.3936","source_kind":"arxiv","source_version":4}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:807b556949480877d95497409dc34aee648f3ff79ea18bc674495900d9f5d14b","sha256:7eb85a5420fcfcad584d81c1bd858ac179ee024e47a0d0bfd9a04df64f01b986"],"state_sha256":"82a1a396f82168d30bdf8a76128862e46ae4a95ff5d6af92d0843c29a70a8dba"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"nMrjzSD3osXLAZpPbD14Gyc/6igKzba5CbLNLZm2M6CdY3iZYk00UVkBZhlhA2VdZsScb7m532kQsFVf7XrRDA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-24T22:57:27.028074Z","bundle_sha256":"b58593bbc5ff48c9750128a798e6a021c14aecb1d02bd02015afacaf142fdbf3"}}