{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2025:AFP56EA5ZESF4IVIGX2EBSGD6N","short_pith_number":"pith:AFP56EA5","canonical_record":{"source":{"id":"2509.22491","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AG","submitted_at":"2025-09-26T15:35:52Z","cross_cats_sorted":[],"title_canon_sha256":"da90c06d37ba864a2e506e1624b4402a742667ced931530657298c2b2833495e","abstract_canon_sha256":"7e8c22232a2d187241fb6aad857d09808edd9908ce8a735170724132e5d54faf"},"schema_version":"1.0"},"canonical_sha256":"015fdf101dc9245e22a835f440c8c3f3766bffdc8fb598344331d33211a93be1","source":{"kind":"arxiv","id":"2509.22491","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2509.22491","created_at":"2026-08-13T01:29:36Z"},{"alias_kind":"arxiv_version","alias_value":"2509.22491v2","created_at":"2026-08-13T01:29:36Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2509.22491","created_at":"2026-08-13T01:29:36Z"},{"alias_kind":"pith_short_12","alias_value":"AFP56EA5ZESF","created_at":"2026-08-13T01:29:36Z"},{"alias_kind":"pith_short_16","alias_value":"AFP56EA5ZESF4IVI","created_at":"2026-08-13T01:29:36Z"},{"alias_kind":"pith_short_8","alias_value":"AFP56EA5","created_at":"2026-08-13T01:29:36Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2025:AFP56EA5ZESF4IVIGX2EBSGD6N","target":"record","payload":{"canonical_record":{"source":{"id":"2509.22491","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AG","submitted_at":"2025-09-26T15:35:52Z","cross_cats_sorted":[],"title_canon_sha256":"da90c06d37ba864a2e506e1624b4402a742667ced931530657298c2b2833495e","abstract_canon_sha256":"7e8c22232a2d187241fb6aad857d09808edd9908ce8a735170724132e5d54faf"},"schema_version":"1.0"},"canonical_sha256":"015fdf101dc9245e22a835f440c8c3f3766bffdc8fb598344331d33211a93be1","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-08-13T01:29:36.754202Z","signature_b64":"LVowKbu4jHlQ9pXL0Do+jctWGXyIuDvRrlX0wdqZdcqv4268PCxy9XDVzZSZPRAIx1ydtsxllyy003VL+kGKCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"015fdf101dc9245e22a835f440c8c3f3766bffdc8fb598344331d33211a93be1","last_reissued_at":"2026-08-13T01:29:36.751799Z","signature_status":"signed_v1","first_computed_at":"2026-08-13T01:29:36.751799Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2509.22491","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":"2026-08-13T01:29:36Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Utanvi4hcCTLPQpxyy0b+gZIuUSEMHyPCeMBsZhY6iD6YL8FO/2BG8/nTw9/Lr24VBjMVcnRxNg49WjgyK4aBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-13T21:58:44.803118Z"},"content_sha256":"86e82aa4142916d873a86abc2cb5ec93fe78f614715e92278b0987b430d67bc9","schema_version":"1.0","event_id":"sha256:86e82aa4142916d873a86abc2cb5ec93fe78f614715e92278b0987b430d67bc9"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2025:AFP56EA5ZESF4IVIGX2EBSGD6N","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Counting Fourier-Mukai partners of cubic fourfolds","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.AG","authors_text":"Christian B\\\"ohning, Hans-Christian Graf von Bothmer, Lisa Marquand","submitted_at":"2025-09-26T15:35:52Z","abstract_excerpt":"We develop an algorithm to count the number of virtual Fourier-Mukai partners for a given cubic fourfold, with initial input the primitive algebraic lattice and transcendental Hodge structure. Under some mild assumptions, we prove that our virtual count is enough to recover the actual count of Fourier-Mukai partners. We apply our algorithm to examples of cubics with a symplectic automorphism. In particular, we prove that a general cubic with a symplectic involution has 1120 non-trivial Fourier-Mukai partners, each admitting an Eckardt involution, and all birational. As a corollary, we prove th"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2509.22491","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/2509.22491/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":"2026-08-13T01:29:36Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"V0s78htKhgohxTczB/gnbIt0fYCWoBU1C9fJiO4uMWBCJwVihDPCGVagDAEI6QqbrPENXabhz9IY+XJnreKfDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-13T21:58:44.804064Z"},"content_sha256":"efd7c2ffcdb4205f2c5329795613055219cc612b01cef8c4b35629295112650c","schema_version":"1.0","event_id":"sha256:efd7c2ffcdb4205f2c5329795613055219cc612b01cef8c4b35629295112650c"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/AFP56EA5ZESF4IVIGX2EBSGD6N/bundle.json","state_url":"https://pith.science/pith/AFP56EA5ZESF4IVIGX2EBSGD6N/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/AFP56EA5ZESF4IVIGX2EBSGD6N/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":"2026-08-13T21:58:44Z","links":{"resolver":"https://pith.science/pith/AFP56EA5ZESF4IVIGX2EBSGD6N","bundle":"https://pith.science/pith/AFP56EA5ZESF4IVIGX2EBSGD6N/bundle.json","state":"https://pith.science/pith/AFP56EA5ZESF4IVIGX2EBSGD6N/state.json","well_known_bundle":"https://pith.science/.well-known/pith/AFP56EA5ZESF4IVIGX2EBSGD6N/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2025:AFP56EA5ZESF4IVIGX2EBSGD6N","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":"7e8c22232a2d187241fb6aad857d09808edd9908ce8a735170724132e5d54faf","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AG","submitted_at":"2025-09-26T15:35:52Z","title_canon_sha256":"da90c06d37ba864a2e506e1624b4402a742667ced931530657298c2b2833495e"},"schema_version":"1.0","source":{"id":"2509.22491","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2509.22491","created_at":"2026-08-13T01:29:36Z"},{"alias_kind":"arxiv_version","alias_value":"2509.22491v2","created_at":"2026-08-13T01:29:36Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2509.22491","created_at":"2026-08-13T01:29:36Z"},{"alias_kind":"pith_short_12","alias_value":"AFP56EA5ZESF","created_at":"2026-08-13T01:29:36Z"},{"alias_kind":"pith_short_16","alias_value":"AFP56EA5ZESF4IVI","created_at":"2026-08-13T01:29:36Z"},{"alias_kind":"pith_short_8","alias_value":"AFP56EA5","created_at":"2026-08-13T01:29:36Z"}],"graph_snapshots":[{"event_id":"sha256:efd7c2ffcdb4205f2c5329795613055219cc612b01cef8c4b35629295112650c","target":"graph","created_at":"2026-08-13T01:29:36Z","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/2509.22491/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We develop an algorithm to count the number of virtual Fourier-Mukai partners for a given cubic fourfold, with initial input the primitive algebraic lattice and transcendental Hodge structure. Under some mild assumptions, we prove that our virtual count is enough to recover the actual count of Fourier-Mukai partners. We apply our algorithm to examples of cubics with a symplectic automorphism. In particular, we prove that a general cubic with a symplectic involution has 1120 non-trivial Fourier-Mukai partners, each admitting an Eckardt involution, and all birational. As a corollary, we prove th","authors_text":"Christian B\\\"ohning, Hans-Christian Graf von Bothmer, Lisa Marquand","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AG","submitted_at":"2025-09-26T15:35:52Z","title":"Counting Fourier-Mukai partners of cubic fourfolds"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2509.22491","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":"sha256:86e82aa4142916d873a86abc2cb5ec93fe78f614715e92278b0987b430d67bc9","target":"record","created_at":"2026-08-13T01:29:36Z","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":"7e8c22232a2d187241fb6aad857d09808edd9908ce8a735170724132e5d54faf","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AG","submitted_at":"2025-09-26T15:35:52Z","title_canon_sha256":"da90c06d37ba864a2e506e1624b4402a742667ced931530657298c2b2833495e"},"schema_version":"1.0","source":{"id":"2509.22491","kind":"arxiv","version":2}},"canonical_sha256":"015fdf101dc9245e22a835f440c8c3f3766bffdc8fb598344331d33211a93be1","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"015fdf101dc9245e22a835f440c8c3f3766bffdc8fb598344331d33211a93be1","first_computed_at":"2026-08-13T01:29:36.751799Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-08-13T01:29:36.751799Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"LVowKbu4jHlQ9pXL0Do+jctWGXyIuDvRrlX0wdqZdcqv4268PCxy9XDVzZSZPRAIx1ydtsxllyy003VL+kGKCw==","signature_status":"signed_v1","signed_at":"2026-08-13T01:29:36.754202Z","signed_message":"canonical_sha256_bytes"},"source_id":"2509.22491","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:86e82aa4142916d873a86abc2cb5ec93fe78f614715e92278b0987b430d67bc9","sha256:efd7c2ffcdb4205f2c5329795613055219cc612b01cef8c4b35629295112650c"],"state_sha256":"32c213f55bd244fe836e21c80993fc8e2a9d29ec27206ea633b6f9e69abca10c"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"QCKS0bQnn+GwgDYOvGiMK6DvuBJD2S+VdMWY6eQ+bbW+274CgmUIQaJxdL1pQq2eXIDtTUDOTu9yDiNdHPzmDQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-13T21:58:44.828910Z","bundle_sha256":"c45dbd40693dbeabf1f59bc46b1b94e35e663d8d773b27611ec87e8b75ec071f"}}