{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2007:LTHQ654ZYUDZJMDVLY7KPEZATD","short_pith_number":"pith:LTHQ654Z","canonical_record":{"source":{"id":"0710.4453","kind":"arxiv","version":2},"metadata":{"license":"","primary_cat":"math.MG","submitted_at":"2007-10-24T12:47:15Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"5bce06509bd8ee32ed696ca17a8918efd7a98e75efc0b8751146cc3076e20aa1","abstract_canon_sha256":"abf05a1a001da063c71c2ac25c73ac4b66db621870f93e99749062982e7ff42d"},"schema_version":"1.0"},"canonical_sha256":"5ccf0f7799c50794b0755e3ea7932098d183ddfb2cdde644a372342cd5b88a90","source":{"kind":"arxiv","id":"0710.4453","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0710.4453","created_at":"2026-05-18T04:09:13Z"},{"alias_kind":"arxiv_version","alias_value":"0710.4453v2","created_at":"2026-05-18T04:09:13Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0710.4453","created_at":"2026-05-18T04:09:13Z"},{"alias_kind":"pith_short_12","alias_value":"LTHQ654ZYUDZ","created_at":"2026-05-18T12:25:55Z"},{"alias_kind":"pith_short_16","alias_value":"LTHQ654ZYUDZJMDV","created_at":"2026-05-18T12:25:55Z"},{"alias_kind":"pith_short_8","alias_value":"LTHQ654Z","created_at":"2026-05-18T12:25:55Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2007:LTHQ654ZYUDZJMDVLY7KPEZATD","target":"record","payload":{"canonical_record":{"source":{"id":"0710.4453","kind":"arxiv","version":2},"metadata":{"license":"","primary_cat":"math.MG","submitted_at":"2007-10-24T12:47:15Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"5bce06509bd8ee32ed696ca17a8918efd7a98e75efc0b8751146cc3076e20aa1","abstract_canon_sha256":"abf05a1a001da063c71c2ac25c73ac4b66db621870f93e99749062982e7ff42d"},"schema_version":"1.0"},"canonical_sha256":"5ccf0f7799c50794b0755e3ea7932098d183ddfb2cdde644a372342cd5b88a90","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:09:13.948335Z","signature_b64":"NZphJpo1tGWWDQxN6xC9fD2rnFoe8xameUcruFAnVGPTtgO6A2/n1PKs9mFC7TYjMgl/9S2COMq9tmlJpw0aBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5ccf0f7799c50794b0755e3ea7932098d183ddfb2cdde644a372342cd5b88a90","last_reissued_at":"2026-05-18T04:09:13.947655Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:09:13.947655Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"0710.4453","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":"2026-05-18T04:09:13Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Kg7WUn9wJJErVIOIP6WUxm2NtHBUCJdAae4By2A3ewOp0p8np0/s3M8hPjZtQ7cNv4MVlvhXR+60WTF7FEcNAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-25T23:43:30.915691Z"},"content_sha256":"14a2a832aab0e32193423e60dbab0f36a86cbdf9990dca47612eefe072a60249","schema_version":"1.0","event_id":"sha256:14a2a832aab0e32193423e60dbab0f36a86cbdf9990dca47612eefe072a60249"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2007:LTHQ654ZYUDZJMDVLY7KPEZATD","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Non-rational configurations, polytopes, and surfaces","license":"","headline":"","cross_cats":["math.CO"],"primary_cat":"math.MG","authors_text":"G\\\"unter M. Ziegler","submitted_at":"2007-10-24T12:47:15Z","abstract_excerpt":"It is an amazing and a bit counter-intuitive discovery by Micha Perles from the sixties that there are ``non-rational polytopes'': combinatorial types of convex polytopes that cannot be realized with rational vertex coordinates.\n  We describe a simple construction of non-rational polytopes that does not need duality (Perles' ``Gale diagrams''): It starts from a non-rational point configuration in the plane, and proceeds with so-called Lawrence extensions.\n  We also show that there are non-rational polyhedral surfaces in 3-space, a discovery by Ulrich Brehm from 1997. His construction also star"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0710.4453","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":""},"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":"2026-05-18T04:09:13Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"GjFJF9m2MhDkjQgCRWxJkZgextAYzUY82uSFeSZxC527roSbhv8knavhvOTz7ykRWziK8N7Kdoycq6R0vOj/Aw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-25T23:43:30.916074Z"},"content_sha256":"504ae10d29d0c31de6f353e42886350b7b9d7492dc0064625b9288c86fab9845","schema_version":"1.0","event_id":"sha256:504ae10d29d0c31de6f353e42886350b7b9d7492dc0064625b9288c86fab9845"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/LTHQ654ZYUDZJMDVLY7KPEZATD/bundle.json","state_url":"https://pith.science/pith/LTHQ654ZYUDZJMDVLY7KPEZATD/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/LTHQ654ZYUDZJMDVLY7KPEZATD/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":"2026-05-25T23:43:30Z","links":{"resolver":"https://pith.science/pith/LTHQ654ZYUDZJMDVLY7KPEZATD","bundle":"https://pith.science/pith/LTHQ654ZYUDZJMDVLY7KPEZATD/bundle.json","state":"https://pith.science/pith/LTHQ654ZYUDZJMDVLY7KPEZATD/state.json","well_known_bundle":"https://pith.science/.well-known/pith/LTHQ654ZYUDZJMDVLY7KPEZATD/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2007:LTHQ654ZYUDZJMDVLY7KPEZATD","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":"abf05a1a001da063c71c2ac25c73ac4b66db621870f93e99749062982e7ff42d","cross_cats_sorted":["math.CO"],"license":"","primary_cat":"math.MG","submitted_at":"2007-10-24T12:47:15Z","title_canon_sha256":"5bce06509bd8ee32ed696ca17a8918efd7a98e75efc0b8751146cc3076e20aa1"},"schema_version":"1.0","source":{"id":"0710.4453","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0710.4453","created_at":"2026-05-18T04:09:13Z"},{"alias_kind":"arxiv_version","alias_value":"0710.4453v2","created_at":"2026-05-18T04:09:13Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0710.4453","created_at":"2026-05-18T04:09:13Z"},{"alias_kind":"pith_short_12","alias_value":"LTHQ654ZYUDZ","created_at":"2026-05-18T12:25:55Z"},{"alias_kind":"pith_short_16","alias_value":"LTHQ654ZYUDZJMDV","created_at":"2026-05-18T12:25:55Z"},{"alias_kind":"pith_short_8","alias_value":"LTHQ654Z","created_at":"2026-05-18T12:25:55Z"}],"graph_snapshots":[{"event_id":"sha256:504ae10d29d0c31de6f353e42886350b7b9d7492dc0064625b9288c86fab9845","target":"graph","created_at":"2026-05-18T04:09:13Z","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"},"paper":{"abstract_excerpt":"It is an amazing and a bit counter-intuitive discovery by Micha Perles from the sixties that there are ``non-rational polytopes'': combinatorial types of convex polytopes that cannot be realized with rational vertex coordinates.\n  We describe a simple construction of non-rational polytopes that does not need duality (Perles' ``Gale diagrams''): It starts from a non-rational point configuration in the plane, and proceeds with so-called Lawrence extensions.\n  We also show that there are non-rational polyhedral surfaces in 3-space, a discovery by Ulrich Brehm from 1997. His construction also star","authors_text":"G\\\"unter M. Ziegler","cross_cats":["math.CO"],"headline":"","license":"","primary_cat":"math.MG","submitted_at":"2007-10-24T12:47:15Z","title":"Non-rational configurations, polytopes, and surfaces"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0710.4453","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":"sha256:14a2a832aab0e32193423e60dbab0f36a86cbdf9990dca47612eefe072a60249","target":"record","created_at":"2026-05-18T04:09:13Z","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":"abf05a1a001da063c71c2ac25c73ac4b66db621870f93e99749062982e7ff42d","cross_cats_sorted":["math.CO"],"license":"","primary_cat":"math.MG","submitted_at":"2007-10-24T12:47:15Z","title_canon_sha256":"5bce06509bd8ee32ed696ca17a8918efd7a98e75efc0b8751146cc3076e20aa1"},"schema_version":"1.0","source":{"id":"0710.4453","kind":"arxiv","version":2}},"canonical_sha256":"5ccf0f7799c50794b0755e3ea7932098d183ddfb2cdde644a372342cd5b88a90","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"5ccf0f7799c50794b0755e3ea7932098d183ddfb2cdde644a372342cd5b88a90","first_computed_at":"2026-05-18T04:09:13.947655Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:09:13.947655Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"NZphJpo1tGWWDQxN6xC9fD2rnFoe8xameUcruFAnVGPTtgO6A2/n1PKs9mFC7TYjMgl/9S2COMq9tmlJpw0aBg==","signature_status":"signed_v1","signed_at":"2026-05-18T04:09:13.948335Z","signed_message":"canonical_sha256_bytes"},"source_id":"0710.4453","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:14a2a832aab0e32193423e60dbab0f36a86cbdf9990dca47612eefe072a60249","sha256:504ae10d29d0c31de6f353e42886350b7b9d7492dc0064625b9288c86fab9845"],"state_sha256":"c23becfe3fcd8e911d479aba2eaf3c4dd314a8f5bf0ca0f9b400cf2b7e637398"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"kycDbZpZ+mipJtKXsCX+6zZmZVNbQZH5wtC6HVJsEGp1jQ2iwpHUT0HmEt2wNlZfJPv73YFEUa9D3wdzsV/qCw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-25T23:43:30.918760Z","bundle_sha256":"c7bc58ff33fa3dfb7dfa5af9e25936f3fe545ffcfbfaec505841da1368e82558"}}