{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2010:FCJAUN2ETITXFEQGQRESY2IIXZ","short_pith_number":"pith:FCJAUN2E","canonical_record":{"source":{"id":"1009.4037","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OA","submitted_at":"2010-09-21T10:10:52Z","cross_cats_sorted":[],"title_canon_sha256":"5763cdf194260d8aa5337d6db78bd915ff0fbc0dd8f1b81246c2765e68af311e","abstract_canon_sha256":"22912e400fdd16793b8ad78eb36e1cfc893aa8ff3e2ae21bcf4e71cc5a0798f3"},"schema_version":"1.0"},"canonical_sha256":"28920a37449a2772920684492c6908be40d57e0c4fcf24dafeaa8735b3149135","source":{"kind":"arxiv","id":"1009.4037","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1009.4037","created_at":"2026-05-18T04:35:14Z"},{"alias_kind":"arxiv_version","alias_value":"1009.4037v2","created_at":"2026-05-18T04:35:14Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1009.4037","created_at":"2026-05-18T04:35:14Z"},{"alias_kind":"pith_short_12","alias_value":"FCJAUN2ETITX","created_at":"2026-05-18T12:26:06Z"},{"alias_kind":"pith_short_16","alias_value":"FCJAUN2ETITXFEQG","created_at":"2026-05-18T12:26:06Z"},{"alias_kind":"pith_short_8","alias_value":"FCJAUN2E","created_at":"2026-05-18T12:26:06Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2010:FCJAUN2ETITXFEQGQRESY2IIXZ","target":"record","payload":{"canonical_record":{"source":{"id":"1009.4037","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OA","submitted_at":"2010-09-21T10:10:52Z","cross_cats_sorted":[],"title_canon_sha256":"5763cdf194260d8aa5337d6db78bd915ff0fbc0dd8f1b81246c2765e68af311e","abstract_canon_sha256":"22912e400fdd16793b8ad78eb36e1cfc893aa8ff3e2ae21bcf4e71cc5a0798f3"},"schema_version":"1.0"},"canonical_sha256":"28920a37449a2772920684492c6908be40d57e0c4fcf24dafeaa8735b3149135","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:35:14.547478Z","signature_b64":"h8RQD20UswkLbT0OQYs9uC2E/xw989PHpN3TxyomXxsXiRuQ6lHig0zBjAdZofaj5no0j566p/RfGz8SOVX6Dg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"28920a37449a2772920684492c6908be40d57e0c4fcf24dafeaa8735b3149135","last_reissued_at":"2026-05-18T04:35:14.546979Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:35:14.546979Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1009.4037","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T04:35:14Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"B5kiDrt7iADaLVfFGf7ZfeD72lqxZt+Beq1XBI3Jj79WRD4ieViC7fi2V4729anDaG0vWGdqrzzRSuVS8DRwBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-28T15:36:01.892859Z"},"content_sha256":"2ded172460db0860ec595ad515fae2f0f356025eacdc913fcc30b333fe5933fa","schema_version":"1.0","event_id":"sha256:2ded172460db0860ec595ad515fae2f0f356025eacdc913fcc30b333fe5933fa"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2010:FCJAUN2ETITXFEQGQRESY2IIXZ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Defect of a Kronecker product of unitary matrices","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.OA","authors_text":"Wojciech Tadej","submitted_at":"2010-09-21T10:10:52Z","abstract_excerpt":"The generalized defect D(U) of a unitary NxN matrix U with no zero entries is the dimension of the real space of directions, moving into which from U we do not disturb the moduli |U_ij| as well as the Gram matrix U'*U in the first order. Then the defect d(U) is equal to D(U) - (2N-1), that is the generalized defect diminished by the dimension of the manifold {Dr*U*Dc : Dr,Dc unitary diagonal}. Calculation of d(U) involves calculating the dimension of the space in R^(N^2) spanned by a certain set of vectors associated with U. We split this space into a direct sum, assuming that U is a Kronecker"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1009.4037","kind":"arxiv","version":2},"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":"2026-05-18T04:35:14Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"P6WNccOnEyXfDvu8DyosQPY69IorNCAk2nrOnIeToRrUk020rb9QqobbAxP18RRjwOS6HLJJTY29MgQWSoGJAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-28T15:36:01.893204Z"},"content_sha256":"1c88f754d1f4a39b1bc45b1f75300da3936ba0c7b4bc060947f59b57cb7ca4f9","schema_version":"1.0","event_id":"sha256:1c88f754d1f4a39b1bc45b1f75300da3936ba0c7b4bc060947f59b57cb7ca4f9"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/FCJAUN2ETITXFEQGQRESY2IIXZ/bundle.json","state_url":"https://pith.science/pith/FCJAUN2ETITXFEQGQRESY2IIXZ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/FCJAUN2ETITXFEQGQRESY2IIXZ/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":"2026-06-28T15:36:01Z","links":{"resolver":"https://pith.science/pith/FCJAUN2ETITXFEQGQRESY2IIXZ","bundle":"https://pith.science/pith/FCJAUN2ETITXFEQGQRESY2IIXZ/bundle.json","state":"https://pith.science/pith/FCJAUN2ETITXFEQGQRESY2IIXZ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/FCJAUN2ETITXFEQGQRESY2IIXZ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2010:FCJAUN2ETITXFEQGQRESY2IIXZ","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":"22912e400fdd16793b8ad78eb36e1cfc893aa8ff3e2ae21bcf4e71cc5a0798f3","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OA","submitted_at":"2010-09-21T10:10:52Z","title_canon_sha256":"5763cdf194260d8aa5337d6db78bd915ff0fbc0dd8f1b81246c2765e68af311e"},"schema_version":"1.0","source":{"id":"1009.4037","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1009.4037","created_at":"2026-05-18T04:35:14Z"},{"alias_kind":"arxiv_version","alias_value":"1009.4037v2","created_at":"2026-05-18T04:35:14Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1009.4037","created_at":"2026-05-18T04:35:14Z"},{"alias_kind":"pith_short_12","alias_value":"FCJAUN2ETITX","created_at":"2026-05-18T12:26:06Z"},{"alias_kind":"pith_short_16","alias_value":"FCJAUN2ETITXFEQG","created_at":"2026-05-18T12:26:06Z"},{"alias_kind":"pith_short_8","alias_value":"FCJAUN2E","created_at":"2026-05-18T12:26:06Z"}],"graph_snapshots":[{"event_id":"sha256:1c88f754d1f4a39b1bc45b1f75300da3936ba0c7b4bc060947f59b57cb7ca4f9","target":"graph","created_at":"2026-05-18T04:35:14Z","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":"The generalized defect D(U) of a unitary NxN matrix U with no zero entries is the dimension of the real space of directions, moving into which from U we do not disturb the moduli |U_ij| as well as the Gram matrix U'*U in the first order. Then the defect d(U) is equal to D(U) - (2N-1), that is the generalized defect diminished by the dimension of the manifold {Dr*U*Dc : Dr,Dc unitary diagonal}. Calculation of d(U) involves calculating the dimension of the space in R^(N^2) spanned by a certain set of vectors associated with U. We split this space into a direct sum, assuming that U is a Kronecker","authors_text":"Wojciech Tadej","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OA","submitted_at":"2010-09-21T10:10:52Z","title":"Defect of a Kronecker product of unitary matrices"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1009.4037","kind":"arxiv","version":2},"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":"sha256:2ded172460db0860ec595ad515fae2f0f356025eacdc913fcc30b333fe5933fa","target":"record","created_at":"2026-05-18T04:35:14Z","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":"22912e400fdd16793b8ad78eb36e1cfc893aa8ff3e2ae21bcf4e71cc5a0798f3","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OA","submitted_at":"2010-09-21T10:10:52Z","title_canon_sha256":"5763cdf194260d8aa5337d6db78bd915ff0fbc0dd8f1b81246c2765e68af311e"},"schema_version":"1.0","source":{"id":"1009.4037","kind":"arxiv","version":2}},"canonical_sha256":"28920a37449a2772920684492c6908be40d57e0c4fcf24dafeaa8735b3149135","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"28920a37449a2772920684492c6908be40d57e0c4fcf24dafeaa8735b3149135","first_computed_at":"2026-05-18T04:35:14.546979Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:35:14.546979Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"h8RQD20UswkLbT0OQYs9uC2E/xw989PHpN3TxyomXxsXiRuQ6lHig0zBjAdZofaj5no0j566p/RfGz8SOVX6Dg==","signature_status":"signed_v1","signed_at":"2026-05-18T04:35:14.547478Z","signed_message":"canonical_sha256_bytes"},"source_id":"1009.4037","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:2ded172460db0860ec595ad515fae2f0f356025eacdc913fcc30b333fe5933fa","sha256:1c88f754d1f4a39b1bc45b1f75300da3936ba0c7b4bc060947f59b57cb7ca4f9"],"state_sha256":"75fc77fed74c1cc5a94774ff4ab0d96aeae8d9d7862d399d021cad3f67372b73"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"/DPzsMsCUmmuUidMMUcLUjGtt2AcIzpuziX7dL7Mnt9Np5Z9X7YV3in33H38YioFazzphkTujxA33vstPXOGDw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-28T15:36:01.895238Z","bundle_sha256":"6254da8bd7bf7888a1f545441c535afb4336ec5dfc710aea8a71be6981fbccec"}}