{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:5K6UF7FIYUJYUO7XC76GPZ65QZ","short_pith_number":"pith:5K6UF7FI","canonical_record":{"source":{"id":"2606.06503","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GM","submitted_at":"2026-05-22T20:16:21Z","cross_cats_sorted":[],"title_canon_sha256":"18d191daa0baed9014a3bc80a098e457783c76e98f60bab04ef84b5276f239ad","abstract_canon_sha256":"752b472e0a1fba3026e687436644814241fec0875c8415d50344d9f1a2a5b430"},"schema_version":"1.0"},"canonical_sha256":"eabd42fca8c5138a3bf717fc67e7dd8671af728fc5e7f26e757207e3daea5053","source":{"kind":"arxiv","id":"2606.06503","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2606.06503","created_at":"2026-06-08T00:03:39Z"},{"alias_kind":"arxiv_version","alias_value":"2606.06503v1","created_at":"2026-06-08T00:03:39Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.06503","created_at":"2026-06-08T00:03:39Z"},{"alias_kind":"pith_short_12","alias_value":"5K6UF7FIYUJY","created_at":"2026-06-08T00:03:39Z"},{"alias_kind":"pith_short_16","alias_value":"5K6UF7FIYUJYUO7X","created_at":"2026-06-08T00:03:39Z"},{"alias_kind":"pith_short_8","alias_value":"5K6UF7FI","created_at":"2026-06-08T00:03:39Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:5K6UF7FIYUJYUO7XC76GPZ65QZ","target":"record","payload":{"canonical_record":{"source":{"id":"2606.06503","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GM","submitted_at":"2026-05-22T20:16:21Z","cross_cats_sorted":[],"title_canon_sha256":"18d191daa0baed9014a3bc80a098e457783c76e98f60bab04ef84b5276f239ad","abstract_canon_sha256":"752b472e0a1fba3026e687436644814241fec0875c8415d50344d9f1a2a5b430"},"schema_version":"1.0"},"canonical_sha256":"eabd42fca8c5138a3bf717fc67e7dd8671af728fc5e7f26e757207e3daea5053","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-08T00:03:39.721369Z","signature_b64":"dDxAZSWYcoEPCDHwIo42oQMSyO4DsHXuYWRtSGFl5fIdyWBRRoEFZImrllOB/Rob/WUQTidx7dcI1+dhUMehCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"eabd42fca8c5138a3bf717fc67e7dd8671af728fc5e7f26e757207e3daea5053","last_reissued_at":"2026-06-08T00:03:39.720297Z","signature_status":"signed_v1","first_computed_at":"2026-06-08T00:03:39.720297Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2606.06503","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":"2026-06-08T00:03:39Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"WA5ARrv+ZrC10+mxc1jdLxLgf85p3hNi90lafEU9KXzcEdXvpxW6XpjsFHFLLIr3PfQjYC3aSFifRgUKXl5dAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-20T23:23:05.982161Z"},"content_sha256":"1032975d4a8c5dd5cf9914cc0cdb59559c4f24eac8c6514cab57f3d762bfa943","schema_version":"1.0","event_id":"sha256:1032975d4a8c5dd5cf9914cc0cdb59559c4f24eac8c6514cab57f3d762bfa943"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:5K6UF7FIYUJYUO7XC76GPZ65QZ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Variations on Majorization of Vectors and Connections to Determinantal Inequalities","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GM","authors_text":"Hristo Sendov, Samir Mondal, Shaun Fallat","submitted_at":"2026-05-22T20:16:21Z","abstract_excerpt":"Majorization is a fundamental tool for comparing vectors, with connections to convexity, doubly stochastic matrices, eigenvalues, singular values, and zeros of polynomials. In matrix analysis, it plays a central role in the study of eigenvalue inequalities, particularly those arising from classical determinantal inequalities such as those attributed to Hadamard and Fischer in the context of positive semidefinite matrices. A result of Fischer and Holbrook shows that equality in the Hardy--Littlewood--P\\'olya theorem for non-affine convex functions is closely linked to block structure in the ass"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.06503","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/2606.06503/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":"2026-06-08T00:03:39Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"2KOB9NoFdH7gaVFiYCYdIDpx+2YdZ1C/m0uD2J86wWQntvW38jFCYE9QVNBKnidveoIZC81g6UH5lEFy9X5sBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-20T23:23:05.982553Z"},"content_sha256":"303264e8cecf8ed563125dd19c824e2f906115e22eda7ab8f2c3292ce0253f08","schema_version":"1.0","event_id":"sha256:303264e8cecf8ed563125dd19c824e2f906115e22eda7ab8f2c3292ce0253f08"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/5K6UF7FIYUJYUO7XC76GPZ65QZ/bundle.json","state_url":"https://pith.science/pith/5K6UF7FIYUJYUO7XC76GPZ65QZ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/5K6UF7FIYUJYUO7XC76GPZ65QZ/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":"2026-06-20T23:23:05Z","links":{"resolver":"https://pith.science/pith/5K6UF7FIYUJYUO7XC76GPZ65QZ","bundle":"https://pith.science/pith/5K6UF7FIYUJYUO7XC76GPZ65QZ/bundle.json","state":"https://pith.science/pith/5K6UF7FIYUJYUO7XC76GPZ65QZ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/5K6UF7FIYUJYUO7XC76GPZ65QZ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:5K6UF7FIYUJYUO7XC76GPZ65QZ","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":"752b472e0a1fba3026e687436644814241fec0875c8415d50344d9f1a2a5b430","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GM","submitted_at":"2026-05-22T20:16:21Z","title_canon_sha256":"18d191daa0baed9014a3bc80a098e457783c76e98f60bab04ef84b5276f239ad"},"schema_version":"1.0","source":{"id":"2606.06503","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2606.06503","created_at":"2026-06-08T00:03:39Z"},{"alias_kind":"arxiv_version","alias_value":"2606.06503v1","created_at":"2026-06-08T00:03:39Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.06503","created_at":"2026-06-08T00:03:39Z"},{"alias_kind":"pith_short_12","alias_value":"5K6UF7FIYUJY","created_at":"2026-06-08T00:03:39Z"},{"alias_kind":"pith_short_16","alias_value":"5K6UF7FIYUJYUO7X","created_at":"2026-06-08T00:03:39Z"},{"alias_kind":"pith_short_8","alias_value":"5K6UF7FI","created_at":"2026-06-08T00:03:39Z"}],"graph_snapshots":[{"event_id":"sha256:303264e8cecf8ed563125dd19c824e2f906115e22eda7ab8f2c3292ce0253f08","target":"graph","created_at":"2026-06-08T00:03:39Z","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/2606.06503/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Majorization is a fundamental tool for comparing vectors, with connections to convexity, doubly stochastic matrices, eigenvalues, singular values, and zeros of polynomials. In matrix analysis, it plays a central role in the study of eigenvalue inequalities, particularly those arising from classical determinantal inequalities such as those attributed to Hadamard and Fischer in the context of positive semidefinite matrices. A result of Fischer and Holbrook shows that equality in the Hardy--Littlewood--P\\'olya theorem for non-affine convex functions is closely linked to block structure in the ass","authors_text":"Hristo Sendov, Samir Mondal, Shaun Fallat","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GM","submitted_at":"2026-05-22T20:16:21Z","title":"Variations on Majorization of Vectors and Connections to Determinantal Inequalities"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.06503","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":"sha256:1032975d4a8c5dd5cf9914cc0cdb59559c4f24eac8c6514cab57f3d762bfa943","target":"record","created_at":"2026-06-08T00:03:39Z","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":"752b472e0a1fba3026e687436644814241fec0875c8415d50344d9f1a2a5b430","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GM","submitted_at":"2026-05-22T20:16:21Z","title_canon_sha256":"18d191daa0baed9014a3bc80a098e457783c76e98f60bab04ef84b5276f239ad"},"schema_version":"1.0","source":{"id":"2606.06503","kind":"arxiv","version":1}},"canonical_sha256":"eabd42fca8c5138a3bf717fc67e7dd8671af728fc5e7f26e757207e3daea5053","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"eabd42fca8c5138a3bf717fc67e7dd8671af728fc5e7f26e757207e3daea5053","first_computed_at":"2026-06-08T00:03:39.720297Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-06-08T00:03:39.720297Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"dDxAZSWYcoEPCDHwIo42oQMSyO4DsHXuYWRtSGFl5fIdyWBRRoEFZImrllOB/Rob/WUQTidx7dcI1+dhUMehCg==","signature_status":"signed_v1","signed_at":"2026-06-08T00:03:39.721369Z","signed_message":"canonical_sha256_bytes"},"source_id":"2606.06503","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:1032975d4a8c5dd5cf9914cc0cdb59559c4f24eac8c6514cab57f3d762bfa943","sha256:303264e8cecf8ed563125dd19c824e2f906115e22eda7ab8f2c3292ce0253f08"],"state_sha256":"dff9f046b80df13aa297f8a7c0aa4fbd6a37c3c767f5e9897f23a176c7ac7657"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"sw2dRk4YqNWAQZnET8i5YneZzax5GD4L2Tgyz8qN7p2LA506zxY3N1OGUhnfH5aMMrkgezB57/jtZwpCgS2oDQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-20T23:23:05.984606Z","bundle_sha256":"557b2379f79509e3728aa8b617ae1f809e9a856b6ebd8efb4537d60fc803b064"}}