{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2024:BWC3OBV3I3AAMA7YN673NMX2RQ","short_pith_number":"pith:BWC3OBV3","canonical_record":{"source":{"id":"2404.04730","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CT","submitted_at":"2024-04-06T21:20:52Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"09c780baea200ba616489966c6a33dfa982539bcb29b18f3b5221fc7408cddfd","abstract_canon_sha256":"5e8315bf982c77032de2705d68bfe9b3a5c5f19a0290b4cafefe14fabf1ecfbc"},"schema_version":"1.0"},"canonical_sha256":"0d85b706bb46c00603f86fbfb6b2fa8c2b8a4b7536d4627b4e1e235a9ef01d80","source":{"kind":"arxiv","id":"2404.04730","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2404.04730","created_at":"2026-07-05T08:05:20Z"},{"alias_kind":"arxiv_version","alias_value":"2404.04730v1","created_at":"2026-07-05T08:05:20Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2404.04730","created_at":"2026-07-05T08:05:20Z"},{"alias_kind":"pith_short_12","alias_value":"BWC3OBV3I3AA","created_at":"2026-07-05T08:05:20Z"},{"alias_kind":"pith_short_16","alias_value":"BWC3OBV3I3AAMA7Y","created_at":"2026-07-05T08:05:20Z"},{"alias_kind":"pith_short_8","alias_value":"BWC3OBV3","created_at":"2026-07-05T08:05:20Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2024:BWC3OBV3I3AAMA7YN673NMX2RQ","target":"record","payload":{"canonical_record":{"source":{"id":"2404.04730","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CT","submitted_at":"2024-04-06T21:20:52Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"09c780baea200ba616489966c6a33dfa982539bcb29b18f3b5221fc7408cddfd","abstract_canon_sha256":"5e8315bf982c77032de2705d68bfe9b3a5c5f19a0290b4cafefe14fabf1ecfbc"},"schema_version":"1.0"},"canonical_sha256":"0d85b706bb46c00603f86fbfb6b2fa8c2b8a4b7536d4627b4e1e235a9ef01d80","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T08:05:20.887220Z","signature_b64":"mYXAMtmSkb2YHHRAdbBlsgDUjlmNIjnky0Ikgm8Cmg378FY1EuNF8Iu/OhIT9rMYTSBpv5cqDbT2kQqsIuFADQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0d85b706bb46c00603f86fbfb6b2fa8c2b8a4b7536d4627b4e1e235a9ef01d80","last_reissued_at":"2026-07-05T08:05:20.886803Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T08:05:20.886803Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2404.04730","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":"2026-07-05T08:05:20Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"QKflxijqzqkDqQ/KWErjYIPmBBVUOY9sSrHexYJpvnyXcIyG38OtLYrpKXUnx/DVNFDps0s0gVkpLCNrWmPmAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-14T07:44:37.780784Z"},"content_sha256":"3b755c5f5e4062a0f4d1344ea8943d4f224dfd2aee52bee7a886929ef01d8735","schema_version":"1.0","event_id":"sha256:3b755c5f5e4062a0f4d1344ea8943d4f224dfd2aee52bee7a886929ef01d8735"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2024:BWC3OBV3I3AAMA7YN673NMX2RQ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Projective Geometries and Simple Pointed Matroids as $\\mathbb{F}_1$-modules","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.CT","authors_text":"Jonathan Beardsley, So Nakamura","submitted_at":"2024-04-06T21:20:52Z","abstract_excerpt":"We describe a fully faithful embedding of projective geometries, given in terms of closure operators, into $\\mathbb{F}_1$-modules, in the sense of Connes and Consani. This factors through a faithful functor out of simple pointed matroids. This follows from our construction of a fully faithful embedding of weakly unital, commutative hypermagmas into $\\fun$-modules. This embedding is of independent interest as it generalizes the classical Eilenberg-MacLane embedding for commutative monoids and recovers Segal's nerve construction for commutative partial monoids. For this reason, we spend some tim"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2404.04730","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/2404.04730/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":"2026-07-05T08:05:20Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"KEOyR7vnY7nL3TyiHeEMS1f4oj7S8dDX4UXHCby99QnYiRsCtTlTr82gogXMwSvN0X7XUE3jte9q0JgMsn76DQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-14T07:44:37.781613Z"},"content_sha256":"34ca2964b63bcd84cec7523742cf74f6329c06a6723d763d76aa408e87d34caf","schema_version":"1.0","event_id":"sha256:34ca2964b63bcd84cec7523742cf74f6329c06a6723d763d76aa408e87d34caf"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/BWC3OBV3I3AAMA7YN673NMX2RQ/bundle.json","state_url":"https://pith.science/pith/BWC3OBV3I3AAMA7YN673NMX2RQ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/BWC3OBV3I3AAMA7YN673NMX2RQ/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":"2026-08-14T07:44:37Z","links":{"resolver":"https://pith.science/pith/BWC3OBV3I3AAMA7YN673NMX2RQ","bundle":"https://pith.science/pith/BWC3OBV3I3AAMA7YN673NMX2RQ/bundle.json","state":"https://pith.science/pith/BWC3OBV3I3AAMA7YN673NMX2RQ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/BWC3OBV3I3AAMA7YN673NMX2RQ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2024:BWC3OBV3I3AAMA7YN673NMX2RQ","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":"5e8315bf982c77032de2705d68bfe9b3a5c5f19a0290b4cafefe14fabf1ecfbc","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CT","submitted_at":"2024-04-06T21:20:52Z","title_canon_sha256":"09c780baea200ba616489966c6a33dfa982539bcb29b18f3b5221fc7408cddfd"},"schema_version":"1.0","source":{"id":"2404.04730","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2404.04730","created_at":"2026-07-05T08:05:20Z"},{"alias_kind":"arxiv_version","alias_value":"2404.04730v1","created_at":"2026-07-05T08:05:20Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2404.04730","created_at":"2026-07-05T08:05:20Z"},{"alias_kind":"pith_short_12","alias_value":"BWC3OBV3I3AA","created_at":"2026-07-05T08:05:20Z"},{"alias_kind":"pith_short_16","alias_value":"BWC3OBV3I3AAMA7Y","created_at":"2026-07-05T08:05:20Z"},{"alias_kind":"pith_short_8","alias_value":"BWC3OBV3","created_at":"2026-07-05T08:05:20Z"}],"graph_snapshots":[{"event_id":"sha256:34ca2964b63bcd84cec7523742cf74f6329c06a6723d763d76aa408e87d34caf","target":"graph","created_at":"2026-07-05T08:05:20Z","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/2404.04730/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We describe a fully faithful embedding of projective geometries, given in terms of closure operators, into $\\mathbb{F}_1$-modules, in the sense of Connes and Consani. This factors through a faithful functor out of simple pointed matroids. This follows from our construction of a fully faithful embedding of weakly unital, commutative hypermagmas into $\\fun$-modules. This embedding is of independent interest as it generalizes the classical Eilenberg-MacLane embedding for commutative monoids and recovers Segal's nerve construction for commutative partial monoids. For this reason, we spend some tim","authors_text":"Jonathan Beardsley, So Nakamura","cross_cats":["math.CO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CT","submitted_at":"2024-04-06T21:20:52Z","title":"Projective Geometries and Simple Pointed Matroids as $\\mathbb{F}_1$-modules"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2404.04730","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":"sha256:3b755c5f5e4062a0f4d1344ea8943d4f224dfd2aee52bee7a886929ef01d8735","target":"record","created_at":"2026-07-05T08:05:20Z","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":"5e8315bf982c77032de2705d68bfe9b3a5c5f19a0290b4cafefe14fabf1ecfbc","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CT","submitted_at":"2024-04-06T21:20:52Z","title_canon_sha256":"09c780baea200ba616489966c6a33dfa982539bcb29b18f3b5221fc7408cddfd"},"schema_version":"1.0","source":{"id":"2404.04730","kind":"arxiv","version":1}},"canonical_sha256":"0d85b706bb46c00603f86fbfb6b2fa8c2b8a4b7536d4627b4e1e235a9ef01d80","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"0d85b706bb46c00603f86fbfb6b2fa8c2b8a4b7536d4627b4e1e235a9ef01d80","first_computed_at":"2026-07-05T08:05:20.886803Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T08:05:20.886803Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"mYXAMtmSkb2YHHRAdbBlsgDUjlmNIjnky0Ikgm8Cmg378FY1EuNF8Iu/OhIT9rMYTSBpv5cqDbT2kQqsIuFADQ==","signature_status":"signed_v1","signed_at":"2026-07-05T08:05:20.887220Z","signed_message":"canonical_sha256_bytes"},"source_id":"2404.04730","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:3b755c5f5e4062a0f4d1344ea8943d4f224dfd2aee52bee7a886929ef01d8735","sha256:34ca2964b63bcd84cec7523742cf74f6329c06a6723d763d76aa408e87d34caf"],"state_sha256":"c620041e8f8623c4dfa89feeeededc23563fc12b73f247cf15703187b26e6092"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"katvvyZNclRp4lK7xnpnqCPfX/Oh9AaEolzAMzOyaTyanYJzNXjiqV7Kebe9KeNPzjiM3xXgRbV/pQD9bosYBQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-14T07:44:37.788474Z","bundle_sha256":"b40d7b2ccfe438a522c92fff40d830b64a5bf2158cf22479a2adb52423105dcd"}}