{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:HQ64QZCAGOWQX7H45WIF2IRWZQ","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":"f27ffd26cb4eff9d18754b33850ee66247176f4b45f4d91470279ac232e1f0cf","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2015-12-11T22:19:49Z","title_canon_sha256":"c9e49cb2262d31588f169a12c456e38f8c5420fc97184b42221d244c343c6e48"},"schema_version":"1.0","source":{"id":"1512.03840","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1512.03840","created_at":"2026-05-18T01:24:23Z"},{"alias_kind":"arxiv_version","alias_value":"1512.03840v1","created_at":"2026-05-18T01:24:23Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1512.03840","created_at":"2026-05-18T01:24:23Z"},{"alias_kind":"pith_short_12","alias_value":"HQ64QZCAGOWQ","created_at":"2026-05-18T12:29:25Z"},{"alias_kind":"pith_short_16","alias_value":"HQ64QZCAGOWQX7H4","created_at":"2026-05-18T12:29:25Z"},{"alias_kind":"pith_short_8","alias_value":"HQ64QZCA","created_at":"2026-05-18T12:29:25Z"}],"graph_snapshots":[{"event_id":"sha256:f29694e49775ad6c6f7a6c7fa0abb4b94aeebed5b3d3b6de0c94dcdace550ff9","target":"graph","created_at":"2026-05-18T01:24:23Z","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":"Fix an integer $d \\geq 0$, a field $\\mathbb{F}$, and a vector space $V$ over $\\mathbb{F}$ with dimension $d+1$. By a decomposition of $V$ we mean a sequence $\\{V_i\\}_{i=0}^d$ of $1$-dimensional $\\mathbb{F}$-subspaces of $V$ such that $V = \\sum_{i=0}^d V_i$ (direct sum). Consider $\\mathbb{F}$-linear transformations $A$, $B$ from $V$ to $V$. Then $A,B$ is called an LR pair whenever there exists a decomposition $\\{V_i\\}_{i=0}^d$ of $V$ such that $A V_i = V_{i-1}$ and $B V_i = V_{i+1}$ for $0 \\leq i \\leq d$, where $V_{-1}=0$ and $V_{d+1}=0$. By an LR triple we mean a $3$-tuple $A,B,C$ of $\\mathbb{","authors_text":"Kazumasa Nomura","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2015-12-11T22:19:49Z","title":"An LR pair that can be extended to an LR triple"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1512.03840","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:de465148e8045f1257343278333badefaf3b8ecedddfcea12998443afbe9731f","target":"record","created_at":"2026-05-18T01:24:23Z","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":"f27ffd26cb4eff9d18754b33850ee66247176f4b45f4d91470279ac232e1f0cf","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2015-12-11T22:19:49Z","title_canon_sha256":"c9e49cb2262d31588f169a12c456e38f8c5420fc97184b42221d244c343c6e48"},"schema_version":"1.0","source":{"id":"1512.03840","kind":"arxiv","version":1}},"canonical_sha256":"3c3dc8644033ad0bfcfced905d2236cc1d4f084c480b311d22a5cd8079ab2fad","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"3c3dc8644033ad0bfcfced905d2236cc1d4f084c480b311d22a5cd8079ab2fad","first_computed_at":"2026-05-18T01:24:23.869353Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:24:23.869353Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"jK65pMejURu/d/7+rOdKerDJeMto3bn+SPerCSNAx+vLlyci9OLHhnnbFYL2k60KM7sN7TneGcHPtgeqEzlKBQ==","signature_status":"signed_v1","signed_at":"2026-05-18T01:24:23.869941Z","signed_message":"canonical_sha256_bytes"},"source_id":"1512.03840","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:de465148e8045f1257343278333badefaf3b8ecedddfcea12998443afbe9731f","sha256:f29694e49775ad6c6f7a6c7fa0abb4b94aeebed5b3d3b6de0c94dcdace550ff9"],"state_sha256":"214dc1e2ec32895d14eb8680c2cb52038d14a5d2859d85caa30a5f2ae06d6b87"}