{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2018:2JR3MLZVQDD5JQCKHI7DYZ7U7Y","short_pith_number":"pith:2JR3MLZV","canonical_record":{"source":{"id":"1806.03601","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2018-06-10T07:31:39Z","cross_cats_sorted":[],"title_canon_sha256":"5c359b594f48dbd7dfa8eb4e7a7e199c512799254504e14bf1a6f8c5817629ba","abstract_canon_sha256":"3b9d40b1db3f315106d4e44b0d1b93ffc3d5683a3ead8ae25618951815e22693"},"schema_version":"1.0"},"canonical_sha256":"d263b62f3580c7d4c04a3a3e3c67f4fe221ffb3fcda8871e651204423af0a507","source":{"kind":"arxiv","id":"1806.03601","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1806.03601","created_at":"2026-05-18T00:13:42Z"},{"alias_kind":"arxiv_version","alias_value":"1806.03601v1","created_at":"2026-05-18T00:13:42Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1806.03601","created_at":"2026-05-18T00:13:42Z"},{"alias_kind":"pith_short_12","alias_value":"2JR3MLZVQDD5","created_at":"2026-05-18T12:32:02Z"},{"alias_kind":"pith_short_16","alias_value":"2JR3MLZVQDD5JQCK","created_at":"2026-05-18T12:32:02Z"},{"alias_kind":"pith_short_8","alias_value":"2JR3MLZV","created_at":"2026-05-18T12:32:02Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2018:2JR3MLZVQDD5JQCKHI7DYZ7U7Y","target":"record","payload":{"canonical_record":{"source":{"id":"1806.03601","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2018-06-10T07:31:39Z","cross_cats_sorted":[],"title_canon_sha256":"5c359b594f48dbd7dfa8eb4e7a7e199c512799254504e14bf1a6f8c5817629ba","abstract_canon_sha256":"3b9d40b1db3f315106d4e44b0d1b93ffc3d5683a3ead8ae25618951815e22693"},"schema_version":"1.0"},"canonical_sha256":"d263b62f3580c7d4c04a3a3e3c67f4fe221ffb3fcda8871e651204423af0a507","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:13:42.793624Z","signature_b64":"dfc2tCyiSU/9jcHCYA/6hEz62ddjgs5J7gZ37W2D5MM/mj78rtNecGjVbivqlx+9xX+a9SkfnGZUxLfV5Kw2CQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d263b62f3580c7d4c04a3a3e3c67f4fe221ffb3fcda8871e651204423af0a507","last_reissued_at":"2026-05-18T00:13:42.792897Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:13:42.792897Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1806.03601","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":"2026-05-18T00:13:42Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"L54p2fSIZBa1LUCHtqTwiwsF1Hyinh75SJ6ldAED0HEFrNvDATFb5zjEu6pyUTx5UfWhDqyWIDad6KaWE8LyCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-06T20:18:05.220866Z"},"content_sha256":"97b4b0a4875f5b20e2a470ff3d1527a685bc9b2686c596128985d8cd9012e8b5","schema_version":"1.0","event_id":"sha256:97b4b0a4875f5b20e2a470ff3d1527a685bc9b2686c596128985d8cd9012e8b5"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2018:2JR3MLZVQDD5JQCKHI7DYZ7U7Y","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Strongly Independent Matrices and Applications on the Rigidity of $A$-Invariant Measures on $n$-Torus","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Enhui Shi, Huichi Huang, Hui Xu","submitted_at":"2018-06-10T07:31:39Z","abstract_excerpt":"We introduce the notion of strongly independent matrices and show the existence of strongly independent matrices in $GL(n,\\mathbb{Z})$ over $\\mathbb{Z}^n\\setminus\\{0\\}$ when $2n+1$ is a prime number. As an application of strong independence, we give a measure rigidity result for endomorphisms on $n$-torus $\\mathbb{T}^n$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1806.03601","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":"2026-05-18T00:13:42Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"vxluhYNBZONffXhQQU6r56QQ7hzTq0HSUr+no/rq3C74YcvY9Y2+a/GN9ulUquaiwlTTY+pi+0LG0VX96Br+BA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-06T20:18:05.221211Z"},"content_sha256":"e69a115da03d282658f22ccd01c24136d0c657f1c1104e08f7a7247bd0b27bff","schema_version":"1.0","event_id":"sha256:e69a115da03d282658f22ccd01c24136d0c657f1c1104e08f7a7247bd0b27bff"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/2JR3MLZVQDD5JQCKHI7DYZ7U7Y/bundle.json","state_url":"https://pith.science/pith/2JR3MLZVQDD5JQCKHI7DYZ7U7Y/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/2JR3MLZVQDD5JQCKHI7DYZ7U7Y/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":"2026-06-06T20:18:05Z","links":{"resolver":"https://pith.science/pith/2JR3MLZVQDD5JQCKHI7DYZ7U7Y","bundle":"https://pith.science/pith/2JR3MLZVQDD5JQCKHI7DYZ7U7Y/bundle.json","state":"https://pith.science/pith/2JR3MLZVQDD5JQCKHI7DYZ7U7Y/state.json","well_known_bundle":"https://pith.science/.well-known/pith/2JR3MLZVQDD5JQCKHI7DYZ7U7Y/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:2JR3MLZVQDD5JQCKHI7DYZ7U7Y","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":"3b9d40b1db3f315106d4e44b0d1b93ffc3d5683a3ead8ae25618951815e22693","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2018-06-10T07:31:39Z","title_canon_sha256":"5c359b594f48dbd7dfa8eb4e7a7e199c512799254504e14bf1a6f8c5817629ba"},"schema_version":"1.0","source":{"id":"1806.03601","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1806.03601","created_at":"2026-05-18T00:13:42Z"},{"alias_kind":"arxiv_version","alias_value":"1806.03601v1","created_at":"2026-05-18T00:13:42Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1806.03601","created_at":"2026-05-18T00:13:42Z"},{"alias_kind":"pith_short_12","alias_value":"2JR3MLZVQDD5","created_at":"2026-05-18T12:32:02Z"},{"alias_kind":"pith_short_16","alias_value":"2JR3MLZVQDD5JQCK","created_at":"2026-05-18T12:32:02Z"},{"alias_kind":"pith_short_8","alias_value":"2JR3MLZV","created_at":"2026-05-18T12:32:02Z"}],"graph_snapshots":[{"event_id":"sha256:e69a115da03d282658f22ccd01c24136d0c657f1c1104e08f7a7247bd0b27bff","target":"graph","created_at":"2026-05-18T00:13:42Z","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 introduce the notion of strongly independent matrices and show the existence of strongly independent matrices in $GL(n,\\mathbb{Z})$ over $\\mathbb{Z}^n\\setminus\\{0\\}$ when $2n+1$ is a prime number. As an application of strong independence, we give a measure rigidity result for endomorphisms on $n$-torus $\\mathbb{T}^n$.","authors_text":"Enhui Shi, Huichi Huang, Hui Xu","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2018-06-10T07:31:39Z","title":"Strongly Independent Matrices and Applications on the Rigidity of $A$-Invariant Measures on $n$-Torus"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1806.03601","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":"sha256:97b4b0a4875f5b20e2a470ff3d1527a685bc9b2686c596128985d8cd9012e8b5","target":"record","created_at":"2026-05-18T00:13:42Z","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":"3b9d40b1db3f315106d4e44b0d1b93ffc3d5683a3ead8ae25618951815e22693","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2018-06-10T07:31:39Z","title_canon_sha256":"5c359b594f48dbd7dfa8eb4e7a7e199c512799254504e14bf1a6f8c5817629ba"},"schema_version":"1.0","source":{"id":"1806.03601","kind":"arxiv","version":1}},"canonical_sha256":"d263b62f3580c7d4c04a3a3e3c67f4fe221ffb3fcda8871e651204423af0a507","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"d263b62f3580c7d4c04a3a3e3c67f4fe221ffb3fcda8871e651204423af0a507","first_computed_at":"2026-05-18T00:13:42.792897Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:13:42.792897Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"dfc2tCyiSU/9jcHCYA/6hEz62ddjgs5J7gZ37W2D5MM/mj78rtNecGjVbivqlx+9xX+a9SkfnGZUxLfV5Kw2CQ==","signature_status":"signed_v1","signed_at":"2026-05-18T00:13:42.793624Z","signed_message":"canonical_sha256_bytes"},"source_id":"1806.03601","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:97b4b0a4875f5b20e2a470ff3d1527a685bc9b2686c596128985d8cd9012e8b5","sha256:e69a115da03d282658f22ccd01c24136d0c657f1c1104e08f7a7247bd0b27bff"],"state_sha256":"7b5b3783eaeb54a6e2ecf81b89bcdc76ee4d0fee8b5db40b3ed2514c430a88a2"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Rf5iJAb0q5J1ZCtATEMpl5s9myHNEW8wLpcY/fvw4hFHYmTX/zdwgPOZl/xLm4MY3XiZdDAWkONmY2SBLuGPDg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-06T20:18:05.223576Z","bundle_sha256":"91ff843856be790484b415669ea3e85fcd470791e295383818137c943527dca5"}}