{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2008:C2UYEZKUPSFIEEDPM7PPGBRZ37","short_pith_number":"pith:C2UYEZKU","canonical_record":{"source":{"id":"0804.2819","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2008-04-17T14:05:41Z","cross_cats_sorted":["math.PR"],"title_canon_sha256":"cb902331b755d1a61f493160379ae1407d7dbcff07b1e40d65b53deae8b392ea","abstract_canon_sha256":"aeaf6f68f94dc5ee23008c7d9401d235b13ba0413f10ee98578710fdc59e26fa"},"schema_version":"1.0"},"canonical_sha256":"16a98265547c8a82106f67def30639dfc771f092d348a1717508c7505e22e8b2","source":{"kind":"arxiv","id":"0804.2819","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0804.2819","created_at":"2026-05-18T02:15:53Z"},{"alias_kind":"arxiv_version","alias_value":"0804.2819v1","created_at":"2026-05-18T02:15:53Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0804.2819","created_at":"2026-05-18T02:15:53Z"},{"alias_kind":"pith_short_12","alias_value":"C2UYEZKUPSFI","created_at":"2026-05-18T12:25:57Z"},{"alias_kind":"pith_short_16","alias_value":"C2UYEZKUPSFIEEDP","created_at":"2026-05-18T12:25:57Z"},{"alias_kind":"pith_short_8","alias_value":"C2UYEZKU","created_at":"2026-05-18T12:25:57Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2008:C2UYEZKUPSFIEEDPM7PPGBRZ37","target":"record","payload":{"canonical_record":{"source":{"id":"0804.2819","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2008-04-17T14:05:41Z","cross_cats_sorted":["math.PR"],"title_canon_sha256":"cb902331b755d1a61f493160379ae1407d7dbcff07b1e40d65b53deae8b392ea","abstract_canon_sha256":"aeaf6f68f94dc5ee23008c7d9401d235b13ba0413f10ee98578710fdc59e26fa"},"schema_version":"1.0"},"canonical_sha256":"16a98265547c8a82106f67def30639dfc771f092d348a1717508c7505e22e8b2","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:15:53.523561Z","signature_b64":"Ta5jqCwPmnqktXPncIx+Xvn5n0356pHuUf9H+s4GIs12oIduPvZt8E1J9KfsH1Z+T2YOvusAfoQLutYM0PRWAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"16a98265547c8a82106f67def30639dfc771f092d348a1717508c7505e22e8b2","last_reissued_at":"2026-05-18T02:15:53.522878Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:15:53.522878Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"0804.2819","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":"2026-05-18T02:15:53Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"4tNFl2sLYbPo5bPMQp1BTPU2SumFxF7EfE7iomc6YDn4pxB3WpGlBJBjTdPWC/C4xrSu2h73JOkH/bz02UjBAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-22T22:23:19.793750Z"},"content_sha256":"d8c31ecc754dd64ec6cb02e9a06e575bfd5a8e511ca5d0bcf9523a5b0dd60af6","schema_version":"1.0","event_id":"sha256:d8c31ecc754dd64ec6cb02e9a06e575bfd5a8e511ca5d0bcf9523a5b0dd60af6"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2008:C2UYEZKUPSFIEEDPM7PPGBRZ37","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Bipolarization of posets and natural interpolation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.PR"],"primary_cat":"cs.DM","authors_text":"Christophe Labreuche (TRT), Michel Grabisch (CES)","submitted_at":"2008-04-17T14:05:41Z","abstract_excerpt":"The Choquet integral w.r.t. a capacity can be seen in the finite case as a parsimonious linear interpolator between vertices of $[0,1]^n$. We take this basic fact as a starting point to define the Choquet integral in a very general way, using the geometric realization of lattices and their natural triangulation, as in the work of Koshevoy. A second aim of the paper is to define a general mechanism for the bipolarization of ordered structures. Bisets (or signed sets), as well as bisubmodular functions, bicapacities, bicooperative games, as well as the Choquet integral defined for them can be se"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0804.2819","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":""},"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":"2026-05-18T02:15:53Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"GEwByvAtOo62dJDbaKUHmkOaCT0z3cIbd5xEUHjghdYIsHjiEjNwFoBdPzv64UBN6p1aB+ihOMqRqdiMjstmAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-22T22:23:19.794098Z"},"content_sha256":"f6f9b59ed015dd7e52310b66442c3f0d6a29ffc3a3456bac4a6337dfb22a1e4e","schema_version":"1.0","event_id":"sha256:f6f9b59ed015dd7e52310b66442c3f0d6a29ffc3a3456bac4a6337dfb22a1e4e"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/C2UYEZKUPSFIEEDPM7PPGBRZ37/bundle.json","state_url":"https://pith.science/pith/C2UYEZKUPSFIEEDPM7PPGBRZ37/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/C2UYEZKUPSFIEEDPM7PPGBRZ37/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":"2026-06-22T22:23:19Z","links":{"resolver":"https://pith.science/pith/C2UYEZKUPSFIEEDPM7PPGBRZ37","bundle":"https://pith.science/pith/C2UYEZKUPSFIEEDPM7PPGBRZ37/bundle.json","state":"https://pith.science/pith/C2UYEZKUPSFIEEDPM7PPGBRZ37/state.json","well_known_bundle":"https://pith.science/.well-known/pith/C2UYEZKUPSFIEEDPM7PPGBRZ37/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2008:C2UYEZKUPSFIEEDPM7PPGBRZ37","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":"aeaf6f68f94dc5ee23008c7d9401d235b13ba0413f10ee98578710fdc59e26fa","cross_cats_sorted":["math.PR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2008-04-17T14:05:41Z","title_canon_sha256":"cb902331b755d1a61f493160379ae1407d7dbcff07b1e40d65b53deae8b392ea"},"schema_version":"1.0","source":{"id":"0804.2819","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0804.2819","created_at":"2026-05-18T02:15:53Z"},{"alias_kind":"arxiv_version","alias_value":"0804.2819v1","created_at":"2026-05-18T02:15:53Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0804.2819","created_at":"2026-05-18T02:15:53Z"},{"alias_kind":"pith_short_12","alias_value":"C2UYEZKUPSFI","created_at":"2026-05-18T12:25:57Z"},{"alias_kind":"pith_short_16","alias_value":"C2UYEZKUPSFIEEDP","created_at":"2026-05-18T12:25:57Z"},{"alias_kind":"pith_short_8","alias_value":"C2UYEZKU","created_at":"2026-05-18T12:25:57Z"}],"graph_snapshots":[{"event_id":"sha256:f6f9b59ed015dd7e52310b66442c3f0d6a29ffc3a3456bac4a6337dfb22a1e4e","target":"graph","created_at":"2026-05-18T02:15:53Z","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 Choquet integral w.r.t. a capacity can be seen in the finite case as a parsimonious linear interpolator between vertices of $[0,1]^n$. We take this basic fact as a starting point to define the Choquet integral in a very general way, using the geometric realization of lattices and their natural triangulation, as in the work of Koshevoy. A second aim of the paper is to define a general mechanism for the bipolarization of ordered structures. Bisets (or signed sets), as well as bisubmodular functions, bicapacities, bicooperative games, as well as the Choquet integral defined for them can be se","authors_text":"Christophe Labreuche (TRT), Michel Grabisch (CES)","cross_cats":["math.PR"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2008-04-17T14:05:41Z","title":"Bipolarization of posets and natural interpolation"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0804.2819","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:d8c31ecc754dd64ec6cb02e9a06e575bfd5a8e511ca5d0bcf9523a5b0dd60af6","target":"record","created_at":"2026-05-18T02:15:53Z","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":"aeaf6f68f94dc5ee23008c7d9401d235b13ba0413f10ee98578710fdc59e26fa","cross_cats_sorted":["math.PR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2008-04-17T14:05:41Z","title_canon_sha256":"cb902331b755d1a61f493160379ae1407d7dbcff07b1e40d65b53deae8b392ea"},"schema_version":"1.0","source":{"id":"0804.2819","kind":"arxiv","version":1}},"canonical_sha256":"16a98265547c8a82106f67def30639dfc771f092d348a1717508c7505e22e8b2","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"16a98265547c8a82106f67def30639dfc771f092d348a1717508c7505e22e8b2","first_computed_at":"2026-05-18T02:15:53.522878Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:15:53.522878Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Ta5jqCwPmnqktXPncIx+Xvn5n0356pHuUf9H+s4GIs12oIduPvZt8E1J9KfsH1Z+T2YOvusAfoQLutYM0PRWAg==","signature_status":"signed_v1","signed_at":"2026-05-18T02:15:53.523561Z","signed_message":"canonical_sha256_bytes"},"source_id":"0804.2819","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:d8c31ecc754dd64ec6cb02e9a06e575bfd5a8e511ca5d0bcf9523a5b0dd60af6","sha256:f6f9b59ed015dd7e52310b66442c3f0d6a29ffc3a3456bac4a6337dfb22a1e4e"],"state_sha256":"b7073bc0f74da40d642cbe84512d6dca91787fbc618cb2cdbf943041d2f46676"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"pW6mKfeYMEkXwJ/vsfah9jI01ADK8201eUH/oQSRW0bb63woHytHyaxMmUp46XN2XgspANb+SY0MJEQWc+yhDQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-22T22:23:19.796117Z","bundle_sha256":"24f87df2d18a2f531dc85edc98ec0865964ac1501ca523d9820c282e2e8815e9"}}