{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:U6BYT65WZELJJ3QGKTSEB6WLO5","short_pith_number":"pith:U6BYT65W","canonical_record":{"source":{"id":"2606.00850","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CV","submitted_at":"2026-05-30T18:49:19Z","cross_cats_sorted":["math.FA","math.OA"],"title_canon_sha256":"a6f06813c7720908ad86de8c3be0a546fe6c5a540b92bf2ed140ecdf17c68cdd","abstract_canon_sha256":"e15cf58ed1cba52c99cc7a6967cbdbb52fc1192e85806fdb4eea98539614560a"},"schema_version":"1.0"},"canonical_sha256":"a78389fbb6c91694ee0654e440facb777ba46e12f8ca00722f2a90470c11f19b","source":{"kind":"arxiv","id":"2606.00850","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2606.00850","created_at":"2026-06-02T01:04:07Z"},{"alias_kind":"arxiv_version","alias_value":"2606.00850v1","created_at":"2026-06-02T01:04:07Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.00850","created_at":"2026-06-02T01:04:07Z"},{"alias_kind":"pith_short_12","alias_value":"U6BYT65WZELJ","created_at":"2026-06-02T01:04:07Z"},{"alias_kind":"pith_short_16","alias_value":"U6BYT65WZELJJ3QG","created_at":"2026-06-02T01:04:07Z"},{"alias_kind":"pith_short_8","alias_value":"U6BYT65W","created_at":"2026-06-02T01:04:07Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:U6BYT65WZELJJ3QGKTSEB6WLO5","target":"record","payload":{"canonical_record":{"source":{"id":"2606.00850","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CV","submitted_at":"2026-05-30T18:49:19Z","cross_cats_sorted":["math.FA","math.OA"],"title_canon_sha256":"a6f06813c7720908ad86de8c3be0a546fe6c5a540b92bf2ed140ecdf17c68cdd","abstract_canon_sha256":"e15cf58ed1cba52c99cc7a6967cbdbb52fc1192e85806fdb4eea98539614560a"},"schema_version":"1.0"},"canonical_sha256":"a78389fbb6c91694ee0654e440facb777ba46e12f8ca00722f2a90470c11f19b","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-02T01:04:07.613935Z","signature_b64":"oAYSpvhHRe+xQIw7kHH7Ga2hLbxfk4Y9ABVDx6adXBS8WRUuK5htRQkkx2e30OilOL/UE6Q7NIYWlpvIDBgsDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a78389fbb6c91694ee0654e440facb777ba46e12f8ca00722f2a90470c11f19b","last_reissued_at":"2026-06-02T01:04:07.613504Z","signature_status":"signed_v1","first_computed_at":"2026-06-02T01:04:07.613504Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2606.00850","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":"2026-06-02T01:04:07Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"OQTsPjvw719LPQ5IitV/Hz2G1JJnqzlxtfkZIx3pW2joIAVbC5DTrSMIukJNDoDeA1mJLzSGXrleBXS4w676AA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-08T17:18:38.894412Z"},"content_sha256":"5ef12ed29b76f60e02a3da23c7b5cbb8521562ed09edbc883000e7a6dc6b0331","schema_version":"1.0","event_id":"sha256:5ef12ed29b76f60e02a3da23c7b5cbb8521562ed09edbc883000e7a6dc6b0331"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:U6BYT65WZELJJ3QGKTSEB6WLO5","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"A Toeplitz corona theorem for the pentablock and applications","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.FA","math.OA"],"primary_cat":"math.CV","authors_text":"Nitin Tomar, Sourav Pal","submitted_at":"2026-05-30T18:49:19Z","abstract_excerpt":"We state and prove a Toeplitz corona theorem for the pentablock $\\mathbb{P}$, a domain in $\\mathbb{C}^3$ given by \\[ \\mathbb{P}=\\{(a_{21}, \\text{tr}(A), \\det(A)) \\in \\mathbb C^3 : A=[a_{ij}] \\in M_2(\\mathbb C), \\|A\\|<1\\}. \\] By two different applications of this theorem, we obtain a few new characterizations in the Toeplitz corona theorems for the bidisc and the symmetrized bidisc."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.00850","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2606.00850/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2026-06-02T01:04:07Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"rIgLL7NplrIj3HT3f1+u6mYmk+UsFJg0VrdQI7WoFt44kqv5hYVyvtW8zKP4AhLthJ2ygpQAd+j17vXDsGlJCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-08T17:18:38.894903Z"},"content_sha256":"01ebe590e9f412c19fd0c32b7af49de8eed8c7eae1127ab9f7968643bfea0e0a","schema_version":"1.0","event_id":"sha256:01ebe590e9f412c19fd0c32b7af49de8eed8c7eae1127ab9f7968643bfea0e0a"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/U6BYT65WZELJJ3QGKTSEB6WLO5/bundle.json","state_url":"https://pith.science/pith/U6BYT65WZELJJ3QGKTSEB6WLO5/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/U6BYT65WZELJJ3QGKTSEB6WLO5/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":"2026-08-08T17:18:38Z","links":{"resolver":"https://pith.science/pith/U6BYT65WZELJJ3QGKTSEB6WLO5","bundle":"https://pith.science/pith/U6BYT65WZELJJ3QGKTSEB6WLO5/bundle.json","state":"https://pith.science/pith/U6BYT65WZELJJ3QGKTSEB6WLO5/state.json","well_known_bundle":"https://pith.science/.well-known/pith/U6BYT65WZELJJ3QGKTSEB6WLO5/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:U6BYT65WZELJJ3QGKTSEB6WLO5","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":"e15cf58ed1cba52c99cc7a6967cbdbb52fc1192e85806fdb4eea98539614560a","cross_cats_sorted":["math.FA","math.OA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CV","submitted_at":"2026-05-30T18:49:19Z","title_canon_sha256":"a6f06813c7720908ad86de8c3be0a546fe6c5a540b92bf2ed140ecdf17c68cdd"},"schema_version":"1.0","source":{"id":"2606.00850","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2606.00850","created_at":"2026-06-02T01:04:07Z"},{"alias_kind":"arxiv_version","alias_value":"2606.00850v1","created_at":"2026-06-02T01:04:07Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.00850","created_at":"2026-06-02T01:04:07Z"},{"alias_kind":"pith_short_12","alias_value":"U6BYT65WZELJ","created_at":"2026-06-02T01:04:07Z"},{"alias_kind":"pith_short_16","alias_value":"U6BYT65WZELJJ3QG","created_at":"2026-06-02T01:04:07Z"},{"alias_kind":"pith_short_8","alias_value":"U6BYT65W","created_at":"2026-06-02T01:04:07Z"}],"graph_snapshots":[{"event_id":"sha256:01ebe590e9f412c19fd0c32b7af49de8eed8c7eae1127ab9f7968643bfea0e0a","target":"graph","created_at":"2026-06-02T01:04:07Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2606.00850/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We state and prove a Toeplitz corona theorem for the pentablock $\\mathbb{P}$, a domain in $\\mathbb{C}^3$ given by \\[ \\mathbb{P}=\\{(a_{21}, \\text{tr}(A), \\det(A)) \\in \\mathbb C^3 : A=[a_{ij}] \\in M_2(\\mathbb C), \\|A\\|<1\\}. \\] By two different applications of this theorem, we obtain a few new characterizations in the Toeplitz corona theorems for the bidisc and the symmetrized bidisc.","authors_text":"Nitin Tomar, Sourav Pal","cross_cats":["math.FA","math.OA"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CV","submitted_at":"2026-05-30T18:49:19Z","title":"A Toeplitz corona theorem for the pentablock and applications"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.00850","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":"sha256:5ef12ed29b76f60e02a3da23c7b5cbb8521562ed09edbc883000e7a6dc6b0331","target":"record","created_at":"2026-06-02T01:04:07Z","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":"e15cf58ed1cba52c99cc7a6967cbdbb52fc1192e85806fdb4eea98539614560a","cross_cats_sorted":["math.FA","math.OA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CV","submitted_at":"2026-05-30T18:49:19Z","title_canon_sha256":"a6f06813c7720908ad86de8c3be0a546fe6c5a540b92bf2ed140ecdf17c68cdd"},"schema_version":"1.0","source":{"id":"2606.00850","kind":"arxiv","version":1}},"canonical_sha256":"a78389fbb6c91694ee0654e440facb777ba46e12f8ca00722f2a90470c11f19b","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"a78389fbb6c91694ee0654e440facb777ba46e12f8ca00722f2a90470c11f19b","first_computed_at":"2026-06-02T01:04:07.613504Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-06-02T01:04:07.613504Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"oAYSpvhHRe+xQIw7kHH7Ga2hLbxfk4Y9ABVDx6adXBS8WRUuK5htRQkkx2e30OilOL/UE6Q7NIYWlpvIDBgsDQ==","signature_status":"signed_v1","signed_at":"2026-06-02T01:04:07.613935Z","signed_message":"canonical_sha256_bytes"},"source_id":"2606.00850","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:5ef12ed29b76f60e02a3da23c7b5cbb8521562ed09edbc883000e7a6dc6b0331","sha256:01ebe590e9f412c19fd0c32b7af49de8eed8c7eae1127ab9f7968643bfea0e0a"],"state_sha256":"435c928e9fcd77ef4ad7d87142349eba3a8041629dee7832acdc4be5f3ab25df"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"3TOABgPnBykhPoWhk3QBc437lS8guz33LCuFGOQjkeMnMmNKIfGd+PJRfqbz26ZSfM6Vjzl6Yatk2E4My2i2AA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-08T17:18:38.899246Z","bundle_sha256":"4821b8a8d342aaabfe75fb78e919615c86f768459eecabe5adaff39408040f94"}}