{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:7PKT22RKQZKLF2VUTNV75UVTH3","short_pith_number":"pith:7PKT22RK","canonical_record":{"source":{"id":"2606.08626","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2026-06-07T13:35:36Z","cross_cats_sorted":["math.MG"],"title_canon_sha256":"fa2aaa0fa52443b9f6b6ca59e7efb0302fe59e6e8b1dc9d60dae31218a3e74c7","abstract_canon_sha256":"94ee85bfb3fc42de0e54ae1507660d24c2c047cdbed0afd1feb852d681e4f410"},"schema_version":"1.0"},"canonical_sha256":"fbd53d6a2a8654b2eab49b6bfed2b33ee592ff99888f5dbe8764a545fad3ca58","source":{"kind":"arxiv","id":"2606.08626","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2606.08626","created_at":"2026-06-09T01:05:41Z"},{"alias_kind":"arxiv_version","alias_value":"2606.08626v1","created_at":"2026-06-09T01:05:41Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.08626","created_at":"2026-06-09T01:05:41Z"},{"alias_kind":"pith_short_12","alias_value":"7PKT22RKQZKL","created_at":"2026-06-09T01:05:41Z"},{"alias_kind":"pith_short_16","alias_value":"7PKT22RKQZKLF2VU","created_at":"2026-06-09T01:05:41Z"},{"alias_kind":"pith_short_8","alias_value":"7PKT22RK","created_at":"2026-06-09T01:05:41Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:7PKT22RKQZKLF2VUTNV75UVTH3","target":"record","payload":{"canonical_record":{"source":{"id":"2606.08626","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2026-06-07T13:35:36Z","cross_cats_sorted":["math.MG"],"title_canon_sha256":"fa2aaa0fa52443b9f6b6ca59e7efb0302fe59e6e8b1dc9d60dae31218a3e74c7","abstract_canon_sha256":"94ee85bfb3fc42de0e54ae1507660d24c2c047cdbed0afd1feb852d681e4f410"},"schema_version":"1.0"},"canonical_sha256":"fbd53d6a2a8654b2eab49b6bfed2b33ee592ff99888f5dbe8764a545fad3ca58","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-09T01:05:41.806253Z","signature_b64":"+RyBZ7Paje07S+zBAR6yR4Eyk/qPNugWUYnGuIPTbEodBCv0J/OQkCt94wihHI5lZvUCeUuha7bkHHbJyzBzDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"fbd53d6a2a8654b2eab49b6bfed2b33ee592ff99888f5dbe8764a545fad3ca58","last_reissued_at":"2026-06-09T01:05:41.805862Z","signature_status":"signed_v1","first_computed_at":"2026-06-09T01:05:41.805862Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2606.08626","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":"2026-06-09T01:05:41Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"mBD2RapCUQvm64kTzVAqWlfL2ExVVSg/EODzEuCIP1sNr+JUybJ4zH6n74j14dvFSike5EExu0iqUkdjVbdVDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-24T21:34:21.699283Z"},"content_sha256":"71e5127531c522c3c0b4b5e24700d6d2c6cd9d39da5d3e7ce1fe94d00d199a55","schema_version":"1.0","event_id":"sha256:71e5127531c522c3c0b4b5e24700d6d2c6cd9d39da5d3e7ce1fe94d00d199a55"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:7PKT22RKQZKLF2VUTNV75UVTH3","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Magnetic Brunn-Minkowski inequalities","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MG"],"primary_cat":"math.DG","authors_text":"Rotem Assouline","submitted_at":"2026-06-07T13:35:36Z","abstract_excerpt":"We study Minkowski averages on Riemannian manifolds in which the interpolation is by action-minimizing magnetic geodesics with respect to a given magnetic potential. We establish equivalence between Brunn-Minkowski inequalities for this operation and lower bounds on a magnetic Ricci curvature. We then discuss various examples, including natural magnetic fields on K\\\"ahler and Sasakian manifolds, and prove a sharp, undistorted Brunn-Minkowski inequality for contact magnetic geodesics on the Heisenberg group. We also observe that closed magnetic potentials from different cohomology classes may g"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.08626","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/2606.08626/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":"2026-06-09T01:05:41Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Qqefcu3Thgngpm6RUSN8FXyPDdwglv7ZBuNdOMwggvWCGbWVxz7Wq7AR4DQrd+yOAc1QmtgCoQDTFctPXIDcDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-24T21:34:21.699671Z"},"content_sha256":"64bfa04465b6317474256b2ab977596fd5bea2beac74270ff815567abf4a118f","schema_version":"1.0","event_id":"sha256:64bfa04465b6317474256b2ab977596fd5bea2beac74270ff815567abf4a118f"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/7PKT22RKQZKLF2VUTNV75UVTH3/bundle.json","state_url":"https://pith.science/pith/7PKT22RKQZKLF2VUTNV75UVTH3/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/7PKT22RKQZKLF2VUTNV75UVTH3/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":"2026-06-24T21:34:21Z","links":{"resolver":"https://pith.science/pith/7PKT22RKQZKLF2VUTNV75UVTH3","bundle":"https://pith.science/pith/7PKT22RKQZKLF2VUTNV75UVTH3/bundle.json","state":"https://pith.science/pith/7PKT22RKQZKLF2VUTNV75UVTH3/state.json","well_known_bundle":"https://pith.science/.well-known/pith/7PKT22RKQZKLF2VUTNV75UVTH3/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:7PKT22RKQZKLF2VUTNV75UVTH3","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":"94ee85bfb3fc42de0e54ae1507660d24c2c047cdbed0afd1feb852d681e4f410","cross_cats_sorted":["math.MG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2026-06-07T13:35:36Z","title_canon_sha256":"fa2aaa0fa52443b9f6b6ca59e7efb0302fe59e6e8b1dc9d60dae31218a3e74c7"},"schema_version":"1.0","source":{"id":"2606.08626","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2606.08626","created_at":"2026-06-09T01:05:41Z"},{"alias_kind":"arxiv_version","alias_value":"2606.08626v1","created_at":"2026-06-09T01:05:41Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.08626","created_at":"2026-06-09T01:05:41Z"},{"alias_kind":"pith_short_12","alias_value":"7PKT22RKQZKL","created_at":"2026-06-09T01:05:41Z"},{"alias_kind":"pith_short_16","alias_value":"7PKT22RKQZKLF2VU","created_at":"2026-06-09T01:05:41Z"},{"alias_kind":"pith_short_8","alias_value":"7PKT22RK","created_at":"2026-06-09T01:05:41Z"}],"graph_snapshots":[{"event_id":"sha256:64bfa04465b6317474256b2ab977596fd5bea2beac74270ff815567abf4a118f","target":"graph","created_at":"2026-06-09T01:05:41Z","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/2606.08626/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We study Minkowski averages on Riemannian manifolds in which the interpolation is by action-minimizing magnetic geodesics with respect to a given magnetic potential. We establish equivalence between Brunn-Minkowski inequalities for this operation and lower bounds on a magnetic Ricci curvature. We then discuss various examples, including natural magnetic fields on K\\\"ahler and Sasakian manifolds, and prove a sharp, undistorted Brunn-Minkowski inequality for contact magnetic geodesics on the Heisenberg group. We also observe that closed magnetic potentials from different cohomology classes may g","authors_text":"Rotem Assouline","cross_cats":["math.MG"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2026-06-07T13:35:36Z","title":"Magnetic Brunn-Minkowski inequalities"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.08626","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":"sha256:71e5127531c522c3c0b4b5e24700d6d2c6cd9d39da5d3e7ce1fe94d00d199a55","target":"record","created_at":"2026-06-09T01:05:41Z","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":"94ee85bfb3fc42de0e54ae1507660d24c2c047cdbed0afd1feb852d681e4f410","cross_cats_sorted":["math.MG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2026-06-07T13:35:36Z","title_canon_sha256":"fa2aaa0fa52443b9f6b6ca59e7efb0302fe59e6e8b1dc9d60dae31218a3e74c7"},"schema_version":"1.0","source":{"id":"2606.08626","kind":"arxiv","version":1}},"canonical_sha256":"fbd53d6a2a8654b2eab49b6bfed2b33ee592ff99888f5dbe8764a545fad3ca58","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"fbd53d6a2a8654b2eab49b6bfed2b33ee592ff99888f5dbe8764a545fad3ca58","first_computed_at":"2026-06-09T01:05:41.805862Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-06-09T01:05:41.805862Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"+RyBZ7Paje07S+zBAR6yR4Eyk/qPNugWUYnGuIPTbEodBCv0J/OQkCt94wihHI5lZvUCeUuha7bkHHbJyzBzDQ==","signature_status":"signed_v1","signed_at":"2026-06-09T01:05:41.806253Z","signed_message":"canonical_sha256_bytes"},"source_id":"2606.08626","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:71e5127531c522c3c0b4b5e24700d6d2c6cd9d39da5d3e7ce1fe94d00d199a55","sha256:64bfa04465b6317474256b2ab977596fd5bea2beac74270ff815567abf4a118f"],"state_sha256":"f9ab8939a45b019b144dcb803de9e0d156f42fa4e65c368e133206f30cde47d6"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"dETcYOc7Yxm6QYtAwRXQIaQozUKKJF++c+68Bp+DMW8qOC9txkY+GXX/UnuQMeeAnjkuuW2M9nYI9zNhdZrDCA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-24T21:34:21.701759Z","bundle_sha256":"bc2988a9b5661ea74ecb0d3028e863a5cb38d881c5172953fb7bb0fd244955d1"}}