{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2024:BA7X3QE3AHSEVYUXVJGU6LL6QK","short_pith_number":"pith:BA7X3QE3","canonical_record":{"source":{"id":"2412.11295","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CT","submitted_at":"2024-12-15T20:05:47Z","cross_cats_sorted":["cs.LO","math.LO"],"title_canon_sha256":"3fbf599fc28a6a468ca0127196ed87591bb1784e20aa97ea043beb36539a526e","abstract_canon_sha256":"9f1b97c1e6a8abd1f06218a6390f6fd72e94ed41622f2aa763a8cbdd57ac872d"},"schema_version":"1.0"},"canonical_sha256":"083f7dc09b01e44ae297aa4d4f2d7e82aab8baa1973decad3b252291d55fd19d","source":{"kind":"arxiv","id":"2412.11295","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2412.11295","created_at":"2026-07-05T09:49:30Z"},{"alias_kind":"arxiv_version","alias_value":"2412.11295v1","created_at":"2026-07-05T09:49:30Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2412.11295","created_at":"2026-07-05T09:49:30Z"},{"alias_kind":"pith_short_12","alias_value":"BA7X3QE3AHSE","created_at":"2026-07-05T09:49:30Z"},{"alias_kind":"pith_short_16","alias_value":"BA7X3QE3AHSEVYUX","created_at":"2026-07-05T09:49:30Z"},{"alias_kind":"pith_short_8","alias_value":"BA7X3QE3","created_at":"2026-07-05T09:49:30Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2024:BA7X3QE3AHSEVYUXVJGU6LL6QK","target":"record","payload":{"canonical_record":{"source":{"id":"2412.11295","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CT","submitted_at":"2024-12-15T20:05:47Z","cross_cats_sorted":["cs.LO","math.LO"],"title_canon_sha256":"3fbf599fc28a6a468ca0127196ed87591bb1784e20aa97ea043beb36539a526e","abstract_canon_sha256":"9f1b97c1e6a8abd1f06218a6390f6fd72e94ed41622f2aa763a8cbdd57ac872d"},"schema_version":"1.0"},"canonical_sha256":"083f7dc09b01e44ae297aa4d4f2d7e82aab8baa1973decad3b252291d55fd19d","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T09:49:30.182389Z","signature_b64":"8oYwNLiIugJSRVWOVfyrwatdWdxxNSBRTLRFwHlLUbvG9iOPftXS42BEvWty5m1Myu3lkrpUKQJ99aU2P+tiBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"083f7dc09b01e44ae297aa4d4f2d7e82aab8baa1973decad3b252291d55fd19d","last_reissued_at":"2026-07-05T09:49:30.181898Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T09:49:30.181898Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2412.11295","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":"2026-07-05T09:49:30Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"l74/LCRVrCWrJNpSKFclt/dol/VpU2ePXI06obP+VfRFxXBjy3pa3EYHCYBOmqM4CBO4Gc3ASU7VTns0oF+SBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-12T08:00:17.549810Z"},"content_sha256":"5ebdbc67e26d9898698a5963c8d94ea1febea454d6bed17698ca32ac1769d712","schema_version":"1.0","event_id":"sha256:5ebdbc67e26d9898698a5963c8d94ea1febea454d6bed17698ca32ac1769d712"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2024:BA7X3QE3AHSEVYUXVJGU6LL6QK","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"The Relational Quotient Completion","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["cs.LO","math.LO"],"primary_cat":"math.CT","authors_text":"Fabio Pasquali, Francesco Dagnino","submitted_at":"2024-12-15T20:05:47Z","abstract_excerpt":"Taking a quotient roughly means changing the notion of equality on a given object, set or type. In a quantitative setting, equality naturally generalises to a distance, measuring how much elements are similar instead of just stating their equivalence. Hence, quotients can be understood quantitatively as a change of distance.\n  In this paper, we show how, combining Lawvere's doctrines and the calculus of relations, one can unify quantitative and usual quotients in a common picture. More in detail, we introduce relational doctrines as a functorial description of (the core of) the calculus of rel"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2412.11295","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/2412.11295/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":"2026-07-05T09:49:30Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"e1C5tVES9wlnH/GgAmZw7RZZyBBqJgXQBV9Q4R9auTKVpKGbduT2S7Fa32PN5IlyUKkUyUZx3rdVJfyDrM9zBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-12T08:00:17.550346Z"},"content_sha256":"220f5b6cb8a6998a189f2bbc9c6879de50f264384c8dd893b41c0b70e7c0b403","schema_version":"1.0","event_id":"sha256:220f5b6cb8a6998a189f2bbc9c6879de50f264384c8dd893b41c0b70e7c0b403"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/BA7X3QE3AHSEVYUXVJGU6LL6QK/bundle.json","state_url":"https://pith.science/pith/BA7X3QE3AHSEVYUXVJGU6LL6QK/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/BA7X3QE3AHSEVYUXVJGU6LL6QK/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":"2026-08-12T08:00:17Z","links":{"resolver":"https://pith.science/pith/BA7X3QE3AHSEVYUXVJGU6LL6QK","bundle":"https://pith.science/pith/BA7X3QE3AHSEVYUXVJGU6LL6QK/bundle.json","state":"https://pith.science/pith/BA7X3QE3AHSEVYUXVJGU6LL6QK/state.json","well_known_bundle":"https://pith.science/.well-known/pith/BA7X3QE3AHSEVYUXVJGU6LL6QK/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2024:BA7X3QE3AHSEVYUXVJGU6LL6QK","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":"9f1b97c1e6a8abd1f06218a6390f6fd72e94ed41622f2aa763a8cbdd57ac872d","cross_cats_sorted":["cs.LO","math.LO"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CT","submitted_at":"2024-12-15T20:05:47Z","title_canon_sha256":"3fbf599fc28a6a468ca0127196ed87591bb1784e20aa97ea043beb36539a526e"},"schema_version":"1.0","source":{"id":"2412.11295","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2412.11295","created_at":"2026-07-05T09:49:30Z"},{"alias_kind":"arxiv_version","alias_value":"2412.11295v1","created_at":"2026-07-05T09:49:30Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2412.11295","created_at":"2026-07-05T09:49:30Z"},{"alias_kind":"pith_short_12","alias_value":"BA7X3QE3AHSE","created_at":"2026-07-05T09:49:30Z"},{"alias_kind":"pith_short_16","alias_value":"BA7X3QE3AHSEVYUX","created_at":"2026-07-05T09:49:30Z"},{"alias_kind":"pith_short_8","alias_value":"BA7X3QE3","created_at":"2026-07-05T09:49:30Z"}],"graph_snapshots":[{"event_id":"sha256:220f5b6cb8a6998a189f2bbc9c6879de50f264384c8dd893b41c0b70e7c0b403","target":"graph","created_at":"2026-07-05T09:49:30Z","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/2412.11295/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Taking a quotient roughly means changing the notion of equality on a given object, set or type. In a quantitative setting, equality naturally generalises to a distance, measuring how much elements are similar instead of just stating their equivalence. Hence, quotients can be understood quantitatively as a change of distance.\n  In this paper, we show how, combining Lawvere's doctrines and the calculus of relations, one can unify quantitative and usual quotients in a common picture. More in detail, we introduce relational doctrines as a functorial description of (the core of) the calculus of rel","authors_text":"Fabio Pasquali, Francesco Dagnino","cross_cats":["cs.LO","math.LO"],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CT","submitted_at":"2024-12-15T20:05:47Z","title":"The Relational Quotient Completion"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2412.11295","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":"sha256:5ebdbc67e26d9898698a5963c8d94ea1febea454d6bed17698ca32ac1769d712","target":"record","created_at":"2026-07-05T09:49:30Z","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":"9f1b97c1e6a8abd1f06218a6390f6fd72e94ed41622f2aa763a8cbdd57ac872d","cross_cats_sorted":["cs.LO","math.LO"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CT","submitted_at":"2024-12-15T20:05:47Z","title_canon_sha256":"3fbf599fc28a6a468ca0127196ed87591bb1784e20aa97ea043beb36539a526e"},"schema_version":"1.0","source":{"id":"2412.11295","kind":"arxiv","version":1}},"canonical_sha256":"083f7dc09b01e44ae297aa4d4f2d7e82aab8baa1973decad3b252291d55fd19d","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"083f7dc09b01e44ae297aa4d4f2d7e82aab8baa1973decad3b252291d55fd19d","first_computed_at":"2026-07-05T09:49:30.181898Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T09:49:30.181898Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"8oYwNLiIugJSRVWOVfyrwatdWdxxNSBRTLRFwHlLUbvG9iOPftXS42BEvWty5m1Myu3lkrpUKQJ99aU2P+tiBg==","signature_status":"signed_v1","signed_at":"2026-07-05T09:49:30.182389Z","signed_message":"canonical_sha256_bytes"},"source_id":"2412.11295","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:5ebdbc67e26d9898698a5963c8d94ea1febea454d6bed17698ca32ac1769d712","sha256:220f5b6cb8a6998a189f2bbc9c6879de50f264384c8dd893b41c0b70e7c0b403"],"state_sha256":"754d5389d091c9b327f2035e10c42018ca576026617cbe86fdf9d8059c31bedf"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"h3K7fSSMn3fGPWrZwIQ2qoG+VczbJ4K4uEvd3lMMkqn7s4jXyg1QYyXMYM8WTLUAcL2gaE2yj6mhkXPHCPrHCg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-12T08:00:17.555719Z","bundle_sha256":"c9229cf24f21f170e9e009bdb4f61d3ca55ecd5c82fc3486142699bcf14084d8"}}