{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2018:V2S7MIMAG4ZDKNTDO4FIJFG37G","short_pith_number":"pith:V2S7MIMA","canonical_record":{"source":{"id":"1812.08282","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-12-19T22:56:42Z","cross_cats_sorted":[],"title_canon_sha256":"d893716d487a0c807923a4a74fc41d8a4f684ef597290b5bd6a6d27b9eca3869","abstract_canon_sha256":"650912daff864da6e0dc077a8190d4525c32a8ab8d2702f05cb7abdcbb18b1d7"},"schema_version":"1.0"},"canonical_sha256":"aea5f621803732353663770a8494dbf9b37fc250d8e1b003dab7b159e1cc8f8f","source":{"kind":"arxiv","id":"1812.08282","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1812.08282","created_at":"2026-05-17T23:57:51Z"},{"alias_kind":"arxiv_version","alias_value":"1812.08282v1","created_at":"2026-05-17T23:57:51Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1812.08282","created_at":"2026-05-17T23:57:51Z"},{"alias_kind":"pith_short_12","alias_value":"V2S7MIMAG4ZD","created_at":"2026-05-18T12:32:56Z"},{"alias_kind":"pith_short_16","alias_value":"V2S7MIMAG4ZDKNTD","created_at":"2026-05-18T12:32:56Z"},{"alias_kind":"pith_short_8","alias_value":"V2S7MIMA","created_at":"2026-05-18T12:32:56Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2018:V2S7MIMAG4ZDKNTDO4FIJFG37G","target":"record","payload":{"canonical_record":{"source":{"id":"1812.08282","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-12-19T22:56:42Z","cross_cats_sorted":[],"title_canon_sha256":"d893716d487a0c807923a4a74fc41d8a4f684ef597290b5bd6a6d27b9eca3869","abstract_canon_sha256":"650912daff864da6e0dc077a8190d4525c32a8ab8d2702f05cb7abdcbb18b1d7"},"schema_version":"1.0"},"canonical_sha256":"aea5f621803732353663770a8494dbf9b37fc250d8e1b003dab7b159e1cc8f8f","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:57:51.739621Z","signature_b64":"htUWqEofDeh5Aw6qi8iTvpTf/oQtLLqvLcmGamZZmL8TRh9nxnxsUoSXFwCR9k1qS22ji+/6K+xMNuSrUcYKCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"aea5f621803732353663770a8494dbf9b37fc250d8e1b003dab7b159e1cc8f8f","last_reissued_at":"2026-05-17T23:57:51.738928Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:57:51.738928Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1812.08282","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":"2026-05-17T23:57:51Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"x3glCmR3/MssR11OfubfTpxop0WNZZg2InI7Hud0mQ6w1AFVPZG/IIl0ocYQg3cHJVQLMYPzXq0OisCC2IzhAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-24T16:54:39.673522Z"},"content_sha256":"625240ba7ff0678854147cc31915ba12e68f0b42811e7d57d82687eb1d66cd6e","schema_version":"1.0","event_id":"sha256:625240ba7ff0678854147cc31915ba12e68f0b42811e7d57d82687eb1d66cd6e"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2018:V2S7MIMAG4ZDKNTDO4FIJFG37G","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"The maximum, spectrum and supremum for critical set sizes in (0,1)-matrices","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Liam K. Wright, Nicholas J. Cavenagh","submitted_at":"2018-12-19T22:56:42Z","abstract_excerpt":"If $D$ is a partially filled-in $(0,1)$-matrix with a unique completion to a $(0,1)$-matrix $M$ (with prescribed row and column sums), we say that $D$ is a {\\em defining set} for $M$. A {\\em critical set} is a minimal defining set (the deletion of any entry results in more than one completion). We give a new classification of critical sets in $(0,1)$-matrices and apply this theory to $\\Lambda_{2m}^m$, the set of $(0,1)$-matrices of dimensions $2m\\times 2m$ with uniform row and column sum $m$.\n  The smallest possible size for a defining set of a matrix in $\\Lambda_{2m}^m$ is $m^2$\n  \\cite{Cav},"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1812.08282","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":"2026-05-17T23:57:51Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"bheYxzq5jIsem7Ap/vYZhkAPS/KfSSAUuSjV8byhdrPSQLoeokJFTbKMqEyZBPU/XtABRg1p5ltxByLmvax4Bg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-24T16:54:39.673856Z"},"content_sha256":"0ea3c19df4d232dbfa3e1ca0fd7fb068c8d6041a75a3e68e2fd005db942b998a","schema_version":"1.0","event_id":"sha256:0ea3c19df4d232dbfa3e1ca0fd7fb068c8d6041a75a3e68e2fd005db942b998a"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/V2S7MIMAG4ZDKNTDO4FIJFG37G/bundle.json","state_url":"https://pith.science/pith/V2S7MIMAG4ZDKNTDO4FIJFG37G/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/V2S7MIMAG4ZDKNTDO4FIJFG37G/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":"2026-06-24T16:54:39Z","links":{"resolver":"https://pith.science/pith/V2S7MIMAG4ZDKNTDO4FIJFG37G","bundle":"https://pith.science/pith/V2S7MIMAG4ZDKNTDO4FIJFG37G/bundle.json","state":"https://pith.science/pith/V2S7MIMAG4ZDKNTDO4FIJFG37G/state.json","well_known_bundle":"https://pith.science/.well-known/pith/V2S7MIMAG4ZDKNTDO4FIJFG37G/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:V2S7MIMAG4ZDKNTDO4FIJFG37G","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":"650912daff864da6e0dc077a8190d4525c32a8ab8d2702f05cb7abdcbb18b1d7","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-12-19T22:56:42Z","title_canon_sha256":"d893716d487a0c807923a4a74fc41d8a4f684ef597290b5bd6a6d27b9eca3869"},"schema_version":"1.0","source":{"id":"1812.08282","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1812.08282","created_at":"2026-05-17T23:57:51Z"},{"alias_kind":"arxiv_version","alias_value":"1812.08282v1","created_at":"2026-05-17T23:57:51Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1812.08282","created_at":"2026-05-17T23:57:51Z"},{"alias_kind":"pith_short_12","alias_value":"V2S7MIMAG4ZD","created_at":"2026-05-18T12:32:56Z"},{"alias_kind":"pith_short_16","alias_value":"V2S7MIMAG4ZDKNTD","created_at":"2026-05-18T12:32:56Z"},{"alias_kind":"pith_short_8","alias_value":"V2S7MIMA","created_at":"2026-05-18T12:32:56Z"}],"graph_snapshots":[{"event_id":"sha256:0ea3c19df4d232dbfa3e1ca0fd7fb068c8d6041a75a3e68e2fd005db942b998a","target":"graph","created_at":"2026-05-17T23:57:51Z","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":"If $D$ is a partially filled-in $(0,1)$-matrix with a unique completion to a $(0,1)$-matrix $M$ (with prescribed row and column sums), we say that $D$ is a {\\em defining set} for $M$. A {\\em critical set} is a minimal defining set (the deletion of any entry results in more than one completion). We give a new classification of critical sets in $(0,1)$-matrices and apply this theory to $\\Lambda_{2m}^m$, the set of $(0,1)$-matrices of dimensions $2m\\times 2m$ with uniform row and column sum $m$.\n  The smallest possible size for a defining set of a matrix in $\\Lambda_{2m}^m$ is $m^2$\n  \\cite{Cav},","authors_text":"Liam K. Wright, Nicholas J. Cavenagh","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-12-19T22:56:42Z","title":"The maximum, spectrum and supremum for critical set sizes in (0,1)-matrices"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1812.08282","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":"sha256:625240ba7ff0678854147cc31915ba12e68f0b42811e7d57d82687eb1d66cd6e","target":"record","created_at":"2026-05-17T23:57:51Z","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":"650912daff864da6e0dc077a8190d4525c32a8ab8d2702f05cb7abdcbb18b1d7","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-12-19T22:56:42Z","title_canon_sha256":"d893716d487a0c807923a4a74fc41d8a4f684ef597290b5bd6a6d27b9eca3869"},"schema_version":"1.0","source":{"id":"1812.08282","kind":"arxiv","version":1}},"canonical_sha256":"aea5f621803732353663770a8494dbf9b37fc250d8e1b003dab7b159e1cc8f8f","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"aea5f621803732353663770a8494dbf9b37fc250d8e1b003dab7b159e1cc8f8f","first_computed_at":"2026-05-17T23:57:51.738928Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:57:51.738928Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"htUWqEofDeh5Aw6qi8iTvpTf/oQtLLqvLcmGamZZmL8TRh9nxnxsUoSXFwCR9k1qS22ji+/6K+xMNuSrUcYKCA==","signature_status":"signed_v1","signed_at":"2026-05-17T23:57:51.739621Z","signed_message":"canonical_sha256_bytes"},"source_id":"1812.08282","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:625240ba7ff0678854147cc31915ba12e68f0b42811e7d57d82687eb1d66cd6e","sha256:0ea3c19df4d232dbfa3e1ca0fd7fb068c8d6041a75a3e68e2fd005db942b998a"],"state_sha256":"339632d42a17abd6ba25f77f7bcbdec3e8d96f5d381e852030b7b5fe089b3206"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"+CU7PruvzEdMqKLO+dkYdxeoDoYntz80ekyivzkHS5NFYUGA+tbVw3v8k9nG6lAxyYzezWRxFv+9ign0xuyXCg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-24T16:54:39.675747Z","bundle_sha256":"ab5e8617164d4139084c4437a22fc60c9020e20f5e6cfd55d7b44e1801d3e41a"}}