{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:B4DQJX2PRDA4NSP35Y3VUAXCKW","short_pith_number":"pith:B4DQJX2P","canonical_record":{"source":{"id":"1605.00318","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2016-05-01T23:02:04Z","cross_cats_sorted":[],"title_canon_sha256":"2b3b78221751299b8813b368f5240354c04f257f0645fc8dbe95600ebfa9890d","abstract_canon_sha256":"9074c56fdca19463816c43a93511625906969e74a1f287e360f79c7875b8b131"},"schema_version":"1.0"},"canonical_sha256":"0f0704df4f88c1c6c9fbee375a02e255ab56b8e0b4a1766cfce54ad701f32fed","source":{"kind":"arxiv","id":"1605.00318","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1605.00318","created_at":"2026-05-18T00:26:13Z"},{"alias_kind":"arxiv_version","alias_value":"1605.00318v3","created_at":"2026-05-18T00:26:13Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1605.00318","created_at":"2026-05-18T00:26:13Z"},{"alias_kind":"pith_short_12","alias_value":"B4DQJX2PRDA4","created_at":"2026-05-18T12:30:07Z"},{"alias_kind":"pith_short_16","alias_value":"B4DQJX2PRDA4NSP3","created_at":"2026-05-18T12:30:07Z"},{"alias_kind":"pith_short_8","alias_value":"B4DQJX2P","created_at":"2026-05-18T12:30:07Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:B4DQJX2PRDA4NSP35Y3VUAXCKW","target":"record","payload":{"canonical_record":{"source":{"id":"1605.00318","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2016-05-01T23:02:04Z","cross_cats_sorted":[],"title_canon_sha256":"2b3b78221751299b8813b368f5240354c04f257f0645fc8dbe95600ebfa9890d","abstract_canon_sha256":"9074c56fdca19463816c43a93511625906969e74a1f287e360f79c7875b8b131"},"schema_version":"1.0"},"canonical_sha256":"0f0704df4f88c1c6c9fbee375a02e255ab56b8e0b4a1766cfce54ad701f32fed","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:26:13.080729Z","signature_b64":"3Q+lEP4ub9DKeTg82ywL6nDXfL7qudXCvCuiT1dVACZY1l2SCRaRHTvMyzSghsoISQruAtlh4L7anWXbb6ecDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0f0704df4f88c1c6c9fbee375a02e255ab56b8e0b4a1766cfce54ad701f32fed","last_reissued_at":"2026-05-18T00:26:13.080053Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:26:13.080053Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1605.00318","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:26:13Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"niiu2Ga0GZleQDTa1F0kaP3BUVyd2Ogbrm9EPOA+/9Cm4gfhhwuMAUlafQYxrRd7ceKbUueQcSZ1SD0LitCpCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-20T20:18:53.163220Z"},"content_sha256":"b82eae3a464168ca51a7dd4cab4694eaa0c58bd806b858a110a17357cf7c3725","schema_version":"1.0","event_id":"sha256:b82eae3a464168ca51a7dd4cab4694eaa0c58bd806b858a110a17357cf7c3725"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:B4DQJX2PRDA4NSP35Y3VUAXCKW","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"The Weyl product on quasi-Banach modulation spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Joachim Toft, Patrik Wahlberg, Yuanyuan Chen","submitted_at":"2016-05-01T23:02:04Z","abstract_excerpt":"We study the bilinear Weyl product acting on quasi-Banach modulation spaces. We find sufficient conditions for continuity of the Weyl product and we derive necessary conditions. The results extend known results for Banach modulation spaces."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1605.00318","kind":"arxiv","version":3},"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":"2026-05-18T00:26:13Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"w34mCS+ex5ttoqc0zMwT7D7HYvIIr5+QW0YnRL7yb3W65er3aUFRTgXhus2M8ZeFVt95OIzNeWTCT3yBo7MiCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-20T20:18:53.163581Z"},"content_sha256":"838d868fa3c741a42f1f450578050cf0d8b5cd1dea3b1b4dc46ca1bda2069390","schema_version":"1.0","event_id":"sha256:838d868fa3c741a42f1f450578050cf0d8b5cd1dea3b1b4dc46ca1bda2069390"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/B4DQJX2PRDA4NSP35Y3VUAXCKW/bundle.json","state_url":"https://pith.science/pith/B4DQJX2PRDA4NSP35Y3VUAXCKW/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/B4DQJX2PRDA4NSP35Y3VUAXCKW/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":"2026-06-20T20:18:53Z","links":{"resolver":"https://pith.science/pith/B4DQJX2PRDA4NSP35Y3VUAXCKW","bundle":"https://pith.science/pith/B4DQJX2PRDA4NSP35Y3VUAXCKW/bundle.json","state":"https://pith.science/pith/B4DQJX2PRDA4NSP35Y3VUAXCKW/state.json","well_known_bundle":"https://pith.science/.well-known/pith/B4DQJX2PRDA4NSP35Y3VUAXCKW/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:B4DQJX2PRDA4NSP35Y3VUAXCKW","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":"9074c56fdca19463816c43a93511625906969e74a1f287e360f79c7875b8b131","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2016-05-01T23:02:04Z","title_canon_sha256":"2b3b78221751299b8813b368f5240354c04f257f0645fc8dbe95600ebfa9890d"},"schema_version":"1.0","source":{"id":"1605.00318","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1605.00318","created_at":"2026-05-18T00:26:13Z"},{"alias_kind":"arxiv_version","alias_value":"1605.00318v3","created_at":"2026-05-18T00:26:13Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1605.00318","created_at":"2026-05-18T00:26:13Z"},{"alias_kind":"pith_short_12","alias_value":"B4DQJX2PRDA4","created_at":"2026-05-18T12:30:07Z"},{"alias_kind":"pith_short_16","alias_value":"B4DQJX2PRDA4NSP3","created_at":"2026-05-18T12:30:07Z"},{"alias_kind":"pith_short_8","alias_value":"B4DQJX2P","created_at":"2026-05-18T12:30:07Z"}],"graph_snapshots":[{"event_id":"sha256:838d868fa3c741a42f1f450578050cf0d8b5cd1dea3b1b4dc46ca1bda2069390","target":"graph","created_at":"2026-05-18T00:26:13Z","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":"We study the bilinear Weyl product acting on quasi-Banach modulation spaces. We find sufficient conditions for continuity of the Weyl product and we derive necessary conditions. The results extend known results for Banach modulation spaces.","authors_text":"Joachim Toft, Patrik Wahlberg, Yuanyuan Chen","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2016-05-01T23:02:04Z","title":"The Weyl product on quasi-Banach modulation spaces"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1605.00318","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:b82eae3a464168ca51a7dd4cab4694eaa0c58bd806b858a110a17357cf7c3725","target":"record","created_at":"2026-05-18T00:26:13Z","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":"9074c56fdca19463816c43a93511625906969e74a1f287e360f79c7875b8b131","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2016-05-01T23:02:04Z","title_canon_sha256":"2b3b78221751299b8813b368f5240354c04f257f0645fc8dbe95600ebfa9890d"},"schema_version":"1.0","source":{"id":"1605.00318","kind":"arxiv","version":3}},"canonical_sha256":"0f0704df4f88c1c6c9fbee375a02e255ab56b8e0b4a1766cfce54ad701f32fed","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"0f0704df4f88c1c6c9fbee375a02e255ab56b8e0b4a1766cfce54ad701f32fed","first_computed_at":"2026-05-18T00:26:13.080053Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:26:13.080053Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"3Q+lEP4ub9DKeTg82ywL6nDXfL7qudXCvCuiT1dVACZY1l2SCRaRHTvMyzSghsoISQruAtlh4L7anWXbb6ecDw==","signature_status":"signed_v1","signed_at":"2026-05-18T00:26:13.080729Z","signed_message":"canonical_sha256_bytes"},"source_id":"1605.00318","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:b82eae3a464168ca51a7dd4cab4694eaa0c58bd806b858a110a17357cf7c3725","sha256:838d868fa3c741a42f1f450578050cf0d8b5cd1dea3b1b4dc46ca1bda2069390"],"state_sha256":"e9e1a96639eeafb0a9f14117a0fcb47ca74a3252cfebc907ebb3f10dc91ace78"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"EIsqByzEB5oeO6rAPghDnz5bxpq1wROrQc586DAD0xF7Kca6FTpzl1YqLnv5LdizLr98y+N1kHDCFXautsjmAA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-20T20:18:53.165539Z","bundle_sha256":"ecf683f3c2823a89769e60862533df14d570012b608ae1db6dbcbc7d807e2c4e"}}