{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2025:B7ZK7PQIEVRZJBXU7VXXFFL23L","short_pith_number":"pith:B7ZK7PQI","canonical_record":{"source":{"id":"2512.21278","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.LO","submitted_at":"2025-12-24T16:49:10Z","cross_cats_sorted":[],"title_canon_sha256":"2ce97db9fb49d32319413249063dc43f82ff5e802447607d5c4b17326cdef280","abstract_canon_sha256":"80703580dacbdf132997305a1bc5a6c49725d5030f38b55dba47a24d58acc605"},"schema_version":"1.0"},"canonical_sha256":"0ff2afbe0825639486f4fd6f72957adae5e4c96e5aba60a4436ebcf19f9ad397","source":{"kind":"arxiv","id":"2512.21278","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2512.21278","created_at":"2026-07-29T01:24:27Z"},{"alias_kind":"arxiv_version","alias_value":"2512.21278v2","created_at":"2026-07-29T01:24:27Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2512.21278","created_at":"2026-07-29T01:24:27Z"},{"alias_kind":"pith_short_12","alias_value":"B7ZK7PQIEVRZ","created_at":"2026-07-29T01:24:27Z"},{"alias_kind":"pith_short_16","alias_value":"B7ZK7PQIEVRZJBXU","created_at":"2026-07-29T01:24:27Z"},{"alias_kind":"pith_short_8","alias_value":"B7ZK7PQI","created_at":"2026-07-29T01:24:27Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2025:B7ZK7PQIEVRZJBXU7VXXFFL23L","target":"record","payload":{"canonical_record":{"source":{"id":"2512.21278","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.LO","submitted_at":"2025-12-24T16:49:10Z","cross_cats_sorted":[],"title_canon_sha256":"2ce97db9fb49d32319413249063dc43f82ff5e802447607d5c4b17326cdef280","abstract_canon_sha256":"80703580dacbdf132997305a1bc5a6c49725d5030f38b55dba47a24d58acc605"},"schema_version":"1.0"},"canonical_sha256":"0ff2afbe0825639486f4fd6f72957adae5e4c96e5aba60a4436ebcf19f9ad397","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-29T01:24:27.309837Z","signature_b64":"TOfyak+50gSbMS3v7xNCdiL4o7IICRxMuM8MMh6t0Kxk2SV089IU3wvPa/QCTB4ooNcGWSMsGOyjgj7LMr6EBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0ff2afbe0825639486f4fd6f72957adae5e4c96e5aba60a4436ebcf19f9ad397","last_reissued_at":"2026-07-29T01:24:27.308870Z","signature_status":"signed_v1","first_computed_at":"2026-07-29T01:24:27.308870Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2512.21278","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":"2026-07-29T01:24:27Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"PcW+n72Om6tDllfs9Xxs4f9PbhiLYL5BUrYnKUKyKxaM/W+t1UO51hrKojUa1seNM6+UaHU6qNY/GaGIrX94AQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-07T08:26:14.682933Z"},"content_sha256":"088fc1cc265189c8193793cf204bdcf3f4e8ca85dcd59be1afcb64e236a7ac62","schema_version":"1.0","event_id":"sha256:088fc1cc265189c8193793cf204bdcf3f4e8ca85dcd59be1afcb64e236a7ac62"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2025:B7ZK7PQIEVRZJBXU7VXXFFL23L","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Taking model-complete cores","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.LO","authors_text":"Bertalan Bodor, Manuel Bodirsky, Paolo Marimon","submitted_at":"2025-12-24T16:49:10Z","abstract_excerpt":"A first-order theory $T$ is a model-complete core theory if every first-order formula is equivalent modulo $T$ to an existential positive formula; a core companion of a theory $T$ is a model-complete core theory $S$ such that every model of $T$ maps homomorphically to a model of $S$ and vice-versa. Whilst core companions may not exist in general, if they exist, they are unique. Moreover, $\\omega$-categorical theories always have a core companion, which is also $\\omega$-categorical. We show that many model-theoretic properties, such as stability, $\\mathrm{NIP}$, simplicity, and $\\mathrm{NSOP}_k"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2512.21278","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2512.21278/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":"2026-07-29T01:24:27Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"0pegDiUVuV1ZtN4QCWy1MD0j6rd0QygUxr03eb2rfZCN5xE77D1Y3lfXZIdsZ7kyXWGgO28rs2n+MnIAmC2eAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-07T08:26:14.684089Z"},"content_sha256":"a9a9011e44e9188d7cb0ae7d9c3f3f8540f70962bc5dfba0fc02250a0454f5d9","schema_version":"1.0","event_id":"sha256:a9a9011e44e9188d7cb0ae7d9c3f3f8540f70962bc5dfba0fc02250a0454f5d9"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/B7ZK7PQIEVRZJBXU7VXXFFL23L/bundle.json","state_url":"https://pith.science/pith/B7ZK7PQIEVRZJBXU7VXXFFL23L/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/B7ZK7PQIEVRZJBXU7VXXFFL23L/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":"2026-08-07T08:26:14Z","links":{"resolver":"https://pith.science/pith/B7ZK7PQIEVRZJBXU7VXXFFL23L","bundle":"https://pith.science/pith/B7ZK7PQIEVRZJBXU7VXXFFL23L/bundle.json","state":"https://pith.science/pith/B7ZK7PQIEVRZJBXU7VXXFFL23L/state.json","well_known_bundle":"https://pith.science/.well-known/pith/B7ZK7PQIEVRZJBXU7VXXFFL23L/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2025:B7ZK7PQIEVRZJBXU7VXXFFL23L","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":"80703580dacbdf132997305a1bc5a6c49725d5030f38b55dba47a24d58acc605","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.LO","submitted_at":"2025-12-24T16:49:10Z","title_canon_sha256":"2ce97db9fb49d32319413249063dc43f82ff5e802447607d5c4b17326cdef280"},"schema_version":"1.0","source":{"id":"2512.21278","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2512.21278","created_at":"2026-07-29T01:24:27Z"},{"alias_kind":"arxiv_version","alias_value":"2512.21278v2","created_at":"2026-07-29T01:24:27Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2512.21278","created_at":"2026-07-29T01:24:27Z"},{"alias_kind":"pith_short_12","alias_value":"B7ZK7PQIEVRZ","created_at":"2026-07-29T01:24:27Z"},{"alias_kind":"pith_short_16","alias_value":"B7ZK7PQIEVRZJBXU","created_at":"2026-07-29T01:24:27Z"},{"alias_kind":"pith_short_8","alias_value":"B7ZK7PQI","created_at":"2026-07-29T01:24:27Z"}],"graph_snapshots":[{"event_id":"sha256:a9a9011e44e9188d7cb0ae7d9c3f3f8540f70962bc5dfba0fc02250a0454f5d9","target":"graph","created_at":"2026-07-29T01:24:27Z","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/2512.21278/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"A first-order theory $T$ is a model-complete core theory if every first-order formula is equivalent modulo $T$ to an existential positive formula; a core companion of a theory $T$ is a model-complete core theory $S$ such that every model of $T$ maps homomorphically to a model of $S$ and vice-versa. Whilst core companions may not exist in general, if they exist, they are unique. Moreover, $\\omega$-categorical theories always have a core companion, which is also $\\omega$-categorical. We show that many model-theoretic properties, such as stability, $\\mathrm{NIP}$, simplicity, and $\\mathrm{NSOP}_k","authors_text":"Bertalan Bodor, Manuel Bodirsky, Paolo Marimon","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.LO","submitted_at":"2025-12-24T16:49:10Z","title":"Taking model-complete cores"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2512.21278","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":"sha256:088fc1cc265189c8193793cf204bdcf3f4e8ca85dcd59be1afcb64e236a7ac62","target":"record","created_at":"2026-07-29T01:24:27Z","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":"80703580dacbdf132997305a1bc5a6c49725d5030f38b55dba47a24d58acc605","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.LO","submitted_at":"2025-12-24T16:49:10Z","title_canon_sha256":"2ce97db9fb49d32319413249063dc43f82ff5e802447607d5c4b17326cdef280"},"schema_version":"1.0","source":{"id":"2512.21278","kind":"arxiv","version":2}},"canonical_sha256":"0ff2afbe0825639486f4fd6f72957adae5e4c96e5aba60a4436ebcf19f9ad397","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"0ff2afbe0825639486f4fd6f72957adae5e4c96e5aba60a4436ebcf19f9ad397","first_computed_at":"2026-07-29T01:24:27.308870Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-29T01:24:27.308870Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"TOfyak+50gSbMS3v7xNCdiL4o7IICRxMuM8MMh6t0Kxk2SV089IU3wvPa/QCTB4ooNcGWSMsGOyjgj7LMr6EBA==","signature_status":"signed_v1","signed_at":"2026-07-29T01:24:27.309837Z","signed_message":"canonical_sha256_bytes"},"source_id":"2512.21278","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:088fc1cc265189c8193793cf204bdcf3f4e8ca85dcd59be1afcb64e236a7ac62","sha256:a9a9011e44e9188d7cb0ae7d9c3f3f8540f70962bc5dfba0fc02250a0454f5d9"],"state_sha256":"d1a189546c8a49ecf09559fb7fd2f00521ab8808ad1b5ae701dda6d600775fa4"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"dZTTMmekSlrH57Sw2NC/jc+YfNE3WbiHudVSAf8mRYW4uXN6h8oDu90Gt0HeDjYigSbsN7TMfAaD6ujs/8+XAg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-07T08:26:14.694213Z","bundle_sha256":"a725162b3b04188ad46eac43cb6829408c39b6b9fd2768fa4ba7ca28906cf8df"}}