{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2013:LRKRZC5RRVLLVXSJNTIQWLQB7T","short_pith_number":"pith:LRKRZC5R","canonical_record":{"source":{"id":"1304.0296","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2013-04-01T04:34:42Z","cross_cats_sorted":[],"title_canon_sha256":"07c2d700a97417bb9a8ad7e3db6f822e56b7d857954bea536df650c42372cfac","abstract_canon_sha256":"7e2151eefe9843299820f881446ac518cecc10e40ee74199df53cd79b29e12d9"},"schema_version":"1.0"},"canonical_sha256":"5c551c8bb18d56bade496cd10b2e01fcf4e477e36c7782613a819143084848e3","source":{"kind":"arxiv","id":"1304.0296","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1304.0296","created_at":"2026-05-18T03:29:17Z"},{"alias_kind":"arxiv_version","alias_value":"1304.0296v1","created_at":"2026-05-18T03:29:17Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1304.0296","created_at":"2026-05-18T03:29:17Z"},{"alias_kind":"pith_short_12","alias_value":"LRKRZC5RRVLL","created_at":"2026-05-18T12:27:51Z"},{"alias_kind":"pith_short_16","alias_value":"LRKRZC5RRVLLVXSJ","created_at":"2026-05-18T12:27:51Z"},{"alias_kind":"pith_short_8","alias_value":"LRKRZC5R","created_at":"2026-05-18T12:27:51Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2013:LRKRZC5RRVLLVXSJNTIQWLQB7T","target":"record","payload":{"canonical_record":{"source":{"id":"1304.0296","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2013-04-01T04:34:42Z","cross_cats_sorted":[],"title_canon_sha256":"07c2d700a97417bb9a8ad7e3db6f822e56b7d857954bea536df650c42372cfac","abstract_canon_sha256":"7e2151eefe9843299820f881446ac518cecc10e40ee74199df53cd79b29e12d9"},"schema_version":"1.0"},"canonical_sha256":"5c551c8bb18d56bade496cd10b2e01fcf4e477e36c7782613a819143084848e3","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:29:17.356721Z","signature_b64":"LEaQbUPEmxNZtZO21c0H1drQwE7cCUbOn4bBbBNY+lTt/3i5IkwL4WmWwBdM64f8uwbX3GfMQY+FgV0hmD/XCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5c551c8bb18d56bade496cd10b2e01fcf4e477e36c7782613a819143084848e3","last_reissued_at":"2026-05-18T03:29:17.355972Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:29:17.355972Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1304.0296","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":"2026-05-18T03:29:17Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"uBI000F8FxVs+bV2uSIkmoAmndj866HOCteQzbOL89B8zHfNbHS2wjPR+NzJ0mkQx2Fj0W/iz5L1FsaP9dKnAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-20T18:20:50.909781Z"},"content_sha256":"a275eb3e60b7b50fbbe56e6fca547c2aab07f595d12315ccfdc6aea549f4e20a","schema_version":"1.0","event_id":"sha256:a275eb3e60b7b50fbbe56e6fca547c2aab07f595d12315ccfdc6aea549f4e20a"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2013:LRKRZC5RRVLLVXSJNTIQWLQB7T","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Zero-dilation Index of a Finite Matrix","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Hwa-Long Gau, Kuo-Zhong Wang, Pei Yuan Wu","submitted_at":"2013-04-01T04:34:42Z","abstract_excerpt":"For an $n$-by-$n$ complex matrix $A$, we define its zero-dilation index $d(A)$ as the largest size of a zero matrix which can be dilated to $A$. This is the same as the maximum $k$ ($\\ge 1$) for which 0 is in the rank-$k$ numerical range of $A$. Using a result of Li and Sze, we show that if $d(A) > \\lfloor 2n/3\\rfloor$, then, under unitary similarity, $A$ has the zero matrix of size $3d(A)-2n$ as a direct summand. It complements the known fact that if $d(A)>\\lfloor n/2\\rfloor$, then 0 is an eigenvalue of $A$. We then use it to give a complete characterization of $n$-by-$n$ matrices $A$ with $d"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1304.0296","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":"2026-05-18T03:29:17Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"3DsGJqLCBXqjb1QFKxLR2KQNtDV36/mdUkQUP8ah2+HMU2Ic4MA/X8bwdlBcC2ewuGmEKCqHUDvHJ5/9zvy3Cw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-20T18:20:50.910124Z"},"content_sha256":"333697d20cf5a373390128cb66760919e1f6d95357e0544eac9b94bdc7a33b4e","schema_version":"1.0","event_id":"sha256:333697d20cf5a373390128cb66760919e1f6d95357e0544eac9b94bdc7a33b4e"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/LRKRZC5RRVLLVXSJNTIQWLQB7T/bundle.json","state_url":"https://pith.science/pith/LRKRZC5RRVLLVXSJNTIQWLQB7T/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/LRKRZC5RRVLLVXSJNTIQWLQB7T/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":"2026-06-20T18:20:50Z","links":{"resolver":"https://pith.science/pith/LRKRZC5RRVLLVXSJNTIQWLQB7T","bundle":"https://pith.science/pith/LRKRZC5RRVLLVXSJNTIQWLQB7T/bundle.json","state":"https://pith.science/pith/LRKRZC5RRVLLVXSJNTIQWLQB7T/state.json","well_known_bundle":"https://pith.science/.well-known/pith/LRKRZC5RRVLLVXSJNTIQWLQB7T/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:LRKRZC5RRVLLVXSJNTIQWLQB7T","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":"7e2151eefe9843299820f881446ac518cecc10e40ee74199df53cd79b29e12d9","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2013-04-01T04:34:42Z","title_canon_sha256":"07c2d700a97417bb9a8ad7e3db6f822e56b7d857954bea536df650c42372cfac"},"schema_version":"1.0","source":{"id":"1304.0296","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1304.0296","created_at":"2026-05-18T03:29:17Z"},{"alias_kind":"arxiv_version","alias_value":"1304.0296v1","created_at":"2026-05-18T03:29:17Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1304.0296","created_at":"2026-05-18T03:29:17Z"},{"alias_kind":"pith_short_12","alias_value":"LRKRZC5RRVLL","created_at":"2026-05-18T12:27:51Z"},{"alias_kind":"pith_short_16","alias_value":"LRKRZC5RRVLLVXSJ","created_at":"2026-05-18T12:27:51Z"},{"alias_kind":"pith_short_8","alias_value":"LRKRZC5R","created_at":"2026-05-18T12:27:51Z"}],"graph_snapshots":[{"event_id":"sha256:333697d20cf5a373390128cb66760919e1f6d95357e0544eac9b94bdc7a33b4e","target":"graph","created_at":"2026-05-18T03:29:17Z","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":"For an $n$-by-$n$ complex matrix $A$, we define its zero-dilation index $d(A)$ as the largest size of a zero matrix which can be dilated to $A$. This is the same as the maximum $k$ ($\\ge 1$) for which 0 is in the rank-$k$ numerical range of $A$. Using a result of Li and Sze, we show that if $d(A) > \\lfloor 2n/3\\rfloor$, then, under unitary similarity, $A$ has the zero matrix of size $3d(A)-2n$ as a direct summand. It complements the known fact that if $d(A)>\\lfloor n/2\\rfloor$, then 0 is an eigenvalue of $A$. We then use it to give a complete characterization of $n$-by-$n$ matrices $A$ with $d","authors_text":"Hwa-Long Gau, Kuo-Zhong Wang, Pei Yuan Wu","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2013-04-01T04:34:42Z","title":"Zero-dilation Index of a Finite Matrix"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1304.0296","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:a275eb3e60b7b50fbbe56e6fca547c2aab07f595d12315ccfdc6aea549f4e20a","target":"record","created_at":"2026-05-18T03:29:17Z","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":"7e2151eefe9843299820f881446ac518cecc10e40ee74199df53cd79b29e12d9","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2013-04-01T04:34:42Z","title_canon_sha256":"07c2d700a97417bb9a8ad7e3db6f822e56b7d857954bea536df650c42372cfac"},"schema_version":"1.0","source":{"id":"1304.0296","kind":"arxiv","version":1}},"canonical_sha256":"5c551c8bb18d56bade496cd10b2e01fcf4e477e36c7782613a819143084848e3","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"5c551c8bb18d56bade496cd10b2e01fcf4e477e36c7782613a819143084848e3","first_computed_at":"2026-05-18T03:29:17.355972Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:29:17.355972Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"LEaQbUPEmxNZtZO21c0H1drQwE7cCUbOn4bBbBNY+lTt/3i5IkwL4WmWwBdM64f8uwbX3GfMQY+FgV0hmD/XCg==","signature_status":"signed_v1","signed_at":"2026-05-18T03:29:17.356721Z","signed_message":"canonical_sha256_bytes"},"source_id":"1304.0296","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:a275eb3e60b7b50fbbe56e6fca547c2aab07f595d12315ccfdc6aea549f4e20a","sha256:333697d20cf5a373390128cb66760919e1f6d95357e0544eac9b94bdc7a33b4e"],"state_sha256":"913d7525ccbb4e524a87a40244e506b38172db7e9f48384b54d1ca64a8b23799"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"e+oXE/heX7vrACTuShfpuIAjssNne5h30ytfY5A1L4/hZOtozhRlLIJGa4f5HFXymGtOvb4DLUWdoRVGfxc1Cg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-20T18:20:50.912062Z","bundle_sha256":"fb64380cda5adac6ab3ef0fd0354d43302405fc2d93fbef0754c5c1c66acfbe8"}}