{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:7YYNROFDAHDERWISLPG3ZHL74G","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":"9ff480f851daa767cd0029b4371f5e5d4b87b3bfbaa8c0f9a7a81aa0fe53c0fc","cross_cats_sorted":["math.CV","math.OA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2013-02-21T15:43:57Z","title_canon_sha256":"752d495d21faf64b27a0f9ef0800ec4027240767409e2f1d2b9c114addcbd2fd"},"schema_version":"1.0","source":{"id":"1302.5312","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1302.5312","created_at":"2026-05-18T03:12:34Z"},{"alias_kind":"arxiv_version","alias_value":"1302.5312v3","created_at":"2026-05-18T03:12:34Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1302.5312","created_at":"2026-05-18T03:12:34Z"},{"alias_kind":"pith_short_12","alias_value":"7YYNROFDAHDE","created_at":"2026-05-18T12:27:38Z"},{"alias_kind":"pith_short_16","alias_value":"7YYNROFDAHDERWIS","created_at":"2026-05-18T12:27:38Z"},{"alias_kind":"pith_short_8","alias_value":"7YYNROFD","created_at":"2026-05-18T12:27:38Z"}],"graph_snapshots":[{"event_id":"sha256:ddfba22995b9c7b66a5484601260e5161bdc6a31d6c4a3e63aedda30a0989b04","target":"graph","created_at":"2026-05-18T03:12:34Z","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 this note we establish a vector-valued version of Beurling's Theorem (the Lax-Halmos Theorem) for the polydisc. As an application of the main result, we provide necessary and sufficient conditions for the completion problem in $H^\\infty(\\mathbb{D}^n)$.","authors_text":"Amol Sasane, Brett D. Wick, Jaydeb Sarkar","cross_cats":["math.CV","math.OA"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2013-02-21T15:43:57Z","title":"Doubly commuting submodules of the Hardy module over polydiscs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1302.5312","kind":"arxiv","version":3},"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:bc501606ed7d92da6c00eda8325df9ffaf5fb462694f8496b3afef32f6254a8e","target":"record","created_at":"2026-05-18T03:12:34Z","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":"9ff480f851daa767cd0029b4371f5e5d4b87b3bfbaa8c0f9a7a81aa0fe53c0fc","cross_cats_sorted":["math.CV","math.OA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2013-02-21T15:43:57Z","title_canon_sha256":"752d495d21faf64b27a0f9ef0800ec4027240767409e2f1d2b9c114addcbd2fd"},"schema_version":"1.0","source":{"id":"1302.5312","kind":"arxiv","version":3}},"canonical_sha256":"fe30d8b8a301c648d9125bcdbc9d7fe1a2addb898d599ed5792eb042852429ca","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"fe30d8b8a301c648d9125bcdbc9d7fe1a2addb898d599ed5792eb042852429ca","first_computed_at":"2026-05-18T03:12:34.964663Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:12:34.964663Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"l/tutxUuYC2YqMcoZmMYtHDjoXIZOzptTwVLFXdAVF9Ep/MiO+Nk98K0CJYnGvSUU8uJy3bs1jgbRR9rpA3sAQ==","signature_status":"signed_v1","signed_at":"2026-05-18T03:12:34.965501Z","signed_message":"canonical_sha256_bytes"},"source_id":"1302.5312","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:bc501606ed7d92da6c00eda8325df9ffaf5fb462694f8496b3afef32f6254a8e","sha256:ddfba22995b9c7b66a5484601260e5161bdc6a31d6c4a3e63aedda30a0989b04"],"state_sha256":"3edfe9f4ac7357d1cecff24043b8dcc71d0c5b6aad78ba69d1ff36207a717b28"}