{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:IQOH5IWH5OVSONPNVNXQ2BXW7X","short_pith_number":"pith:IQOH5IWH","canonical_record":{"source":{"id":"2606.07478","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2026-06-05T17:32:13Z","cross_cats_sorted":[],"title_canon_sha256":"b084b2a036538ef566053cf527ae8fb055a9c4e631c1263295336c18e76d44fb","abstract_canon_sha256":"c8caa6261173a3311231395c04992abce3e5563306eebbed58ed0aec5993e299"},"schema_version":"1.0"},"canonical_sha256":"441c7ea2c7ebab2735edab6f0d06f6fdd70382a07868c22edc57e63c708135be","source":{"kind":"arxiv","id":"2606.07478","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2606.07478","created_at":"2026-06-08T01:05:29Z"},{"alias_kind":"arxiv_version","alias_value":"2606.07478v1","created_at":"2026-06-08T01:05:29Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.07478","created_at":"2026-06-08T01:05:29Z"},{"alias_kind":"pith_short_12","alias_value":"IQOH5IWH5OVS","created_at":"2026-06-08T01:05:29Z"},{"alias_kind":"pith_short_16","alias_value":"IQOH5IWH5OVSONPN","created_at":"2026-06-08T01:05:29Z"},{"alias_kind":"pith_short_8","alias_value":"IQOH5IWH","created_at":"2026-06-08T01:05:29Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:IQOH5IWH5OVSONPNVNXQ2BXW7X","target":"record","payload":{"canonical_record":{"source":{"id":"2606.07478","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2026-06-05T17:32:13Z","cross_cats_sorted":[],"title_canon_sha256":"b084b2a036538ef566053cf527ae8fb055a9c4e631c1263295336c18e76d44fb","abstract_canon_sha256":"c8caa6261173a3311231395c04992abce3e5563306eebbed58ed0aec5993e299"},"schema_version":"1.0"},"canonical_sha256":"441c7ea2c7ebab2735edab6f0d06f6fdd70382a07868c22edc57e63c708135be","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-08T01:05:29.432940Z","signature_b64":"S9izAkRuSP/Gm5i8c1UormIjYHvAai4BQpbHFiSlBdQQto7aClpyZwwfkCy4CA+s9cJJvkPr861BQe57I1+CBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"441c7ea2c7ebab2735edab6f0d06f6fdd70382a07868c22edc57e63c708135be","last_reissued_at":"2026-06-08T01:05:29.432077Z","signature_status":"signed_v1","first_computed_at":"2026-06-08T01:05:29.432077Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2606.07478","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":"2026-06-08T01:05:29Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"9Zxare9ZVD+/tmPA41PBTuZUaYjw0uamOBa9IM3WaPftYxhUKhk7DFxyDUu++PniJ/6Fn+puskNfwgTPTYJcDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-30T02:43:30.628772Z"},"content_sha256":"31feda98605ecd27293c9ad9c8ab79dbe38339938795514beb911db8ae494585","schema_version":"1.0","event_id":"sha256:31feda98605ecd27293c9ad9c8ab79dbe38339938795514beb911db8ae494585"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:IQOH5IWH5OVSONPNVNXQ2BXW7X","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Minimal Posets Realizing \\texorpdfstring{$\\mathbb{Z}_2 \\times \\mathbb{Z}_4$} as Automorphism Group","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Ponaki Das, Sainkupar Marwein Mawiong","submitted_at":"2026-06-05T17:32:13Z","abstract_excerpt":"We prove $\\beta(\\mathbb{Z}_2 \\times \\mathbb{Z}_4) = 14$, where $\\beta(G)$ denotes the minimum cardinality $|P|$ among finite posets $P$ with $\\Aut(P) \\cong G$. The lower bound is established by a complete case analysis of orbit decompositions of $P$ under faithful $G$-actions, organized by the largest orbit size. The upper bound is realized by an explicit $14$-element poset whose automorphism group is computed by a height-function argument together with a rigidity analysis of its covering relations."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.07478","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/2606.07478/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":"2026-06-08T01:05:29Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"W+ZoQhg5wdoNM5BZ1iCqTajL6N4hRfcCNAUq4aIOXEfoVw+Aurk2q+EW1J+h4It5A6CXJse7JfrSolte7vDvCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-30T02:43:30.629429Z"},"content_sha256":"c89510386a109de503bdcca241626cda1f04081b745fe1bc98f56bb06664dc91","schema_version":"1.0","event_id":"sha256:c89510386a109de503bdcca241626cda1f04081b745fe1bc98f56bb06664dc91"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/IQOH5IWH5OVSONPNVNXQ2BXW7X/bundle.json","state_url":"https://pith.science/pith/IQOH5IWH5OVSONPNVNXQ2BXW7X/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/IQOH5IWH5OVSONPNVNXQ2BXW7X/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":"2026-06-30T02:43:30Z","links":{"resolver":"https://pith.science/pith/IQOH5IWH5OVSONPNVNXQ2BXW7X","bundle":"https://pith.science/pith/IQOH5IWH5OVSONPNVNXQ2BXW7X/bundle.json","state":"https://pith.science/pith/IQOH5IWH5OVSONPNVNXQ2BXW7X/state.json","well_known_bundle":"https://pith.science/.well-known/pith/IQOH5IWH5OVSONPNVNXQ2BXW7X/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:IQOH5IWH5OVSONPNVNXQ2BXW7X","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":"c8caa6261173a3311231395c04992abce3e5563306eebbed58ed0aec5993e299","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2026-06-05T17:32:13Z","title_canon_sha256":"b084b2a036538ef566053cf527ae8fb055a9c4e631c1263295336c18e76d44fb"},"schema_version":"1.0","source":{"id":"2606.07478","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2606.07478","created_at":"2026-06-08T01:05:29Z"},{"alias_kind":"arxiv_version","alias_value":"2606.07478v1","created_at":"2026-06-08T01:05:29Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.07478","created_at":"2026-06-08T01:05:29Z"},{"alias_kind":"pith_short_12","alias_value":"IQOH5IWH5OVS","created_at":"2026-06-08T01:05:29Z"},{"alias_kind":"pith_short_16","alias_value":"IQOH5IWH5OVSONPN","created_at":"2026-06-08T01:05:29Z"},{"alias_kind":"pith_short_8","alias_value":"IQOH5IWH","created_at":"2026-06-08T01:05:29Z"}],"graph_snapshots":[{"event_id":"sha256:c89510386a109de503bdcca241626cda1f04081b745fe1bc98f56bb06664dc91","target":"graph","created_at":"2026-06-08T01:05:29Z","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/2606.07478/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We prove $\\beta(\\mathbb{Z}_2 \\times \\mathbb{Z}_4) = 14$, where $\\beta(G)$ denotes the minimum cardinality $|P|$ among finite posets $P$ with $\\Aut(P) \\cong G$. The lower bound is established by a complete case analysis of orbit decompositions of $P$ under faithful $G$-actions, organized by the largest orbit size. The upper bound is realized by an explicit $14$-element poset whose automorphism group is computed by a height-function argument together with a rigidity analysis of its covering relations.","authors_text":"Ponaki Das, Sainkupar Marwein Mawiong","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2026-06-05T17:32:13Z","title":"Minimal Posets Realizing \\texorpdfstring{$\\mathbb{Z}_2 \\times \\mathbb{Z}_4$} as Automorphism Group"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.07478","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":"sha256:31feda98605ecd27293c9ad9c8ab79dbe38339938795514beb911db8ae494585","target":"record","created_at":"2026-06-08T01:05:29Z","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":"c8caa6261173a3311231395c04992abce3e5563306eebbed58ed0aec5993e299","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2026-06-05T17:32:13Z","title_canon_sha256":"b084b2a036538ef566053cf527ae8fb055a9c4e631c1263295336c18e76d44fb"},"schema_version":"1.0","source":{"id":"2606.07478","kind":"arxiv","version":1}},"canonical_sha256":"441c7ea2c7ebab2735edab6f0d06f6fdd70382a07868c22edc57e63c708135be","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"441c7ea2c7ebab2735edab6f0d06f6fdd70382a07868c22edc57e63c708135be","first_computed_at":"2026-06-08T01:05:29.432077Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-06-08T01:05:29.432077Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"S9izAkRuSP/Gm5i8c1UormIjYHvAai4BQpbHFiSlBdQQto7aClpyZwwfkCy4CA+s9cJJvkPr861BQe57I1+CBA==","signature_status":"signed_v1","signed_at":"2026-06-08T01:05:29.432940Z","signed_message":"canonical_sha256_bytes"},"source_id":"2606.07478","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:31feda98605ecd27293c9ad9c8ab79dbe38339938795514beb911db8ae494585","sha256:c89510386a109de503bdcca241626cda1f04081b745fe1bc98f56bb06664dc91"],"state_sha256":"580cc53e3290a841458d43cd62b6907c9d4291bef2a40a797267c7a05566ceaa"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Nnwcis3rlI+mTYHqPnhjgUqhWIKtnPmeXGconDR9U8z2iR4sJJqucxVpVljR8A5UmSb6g5Uy0z2YSpzE0n8aBA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-30T02:43:30.632852Z","bundle_sha256":"a60f8967a53e370fde55a58dc2d80e990460eb26660f1adfce31ad70425c905b"}}