{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:Z6OVDMIMWJQZ6VWNQNSJMNZ6Y3","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":"e91a98dfa174f20379e6b26b78129356c2abe0f1fb0103ccfa3178ac6307d37e","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2012-02-29T19:18:44Z","title_canon_sha256":"dad8c394c3fa94a265a52940ce560ca544609aeb67edbdab714a912d3fc033f8"},"schema_version":"1.0","source":{"id":"1202.6652","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1202.6652","created_at":"2026-05-18T04:01:11Z"},{"alias_kind":"arxiv_version","alias_value":"1202.6652v1","created_at":"2026-05-18T04:01:11Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1202.6652","created_at":"2026-05-18T04:01:11Z"},{"alias_kind":"pith_short_12","alias_value":"Z6OVDMIMWJQZ","created_at":"2026-05-18T12:27:30Z"},{"alias_kind":"pith_short_16","alias_value":"Z6OVDMIMWJQZ6VWN","created_at":"2026-05-18T12:27:30Z"},{"alias_kind":"pith_short_8","alias_value":"Z6OVDMIM","created_at":"2026-05-18T12:27:30Z"}],"graph_snapshots":[{"event_id":"sha256:b9235bee687473523b30b6886c32357cecb8e233168ff7896e34a93ddd207b6d","target":"graph","created_at":"2026-05-18T04:01:11Z","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":"We define periodic frameworks as graphs on the torus, using the language of gain graphs. We present some fundamental definitions and results about the infinitesimal rigidity of graphs on a torus of fixed size and shape, and find necessary conditions for the generic rigidity of pe- riodic frameworks on a d-dimensional fixed torus.","authors_text":"Elissa Ross","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2012-02-29T19:18:44Z","title":"The rigidity of periodic frameworks as graphs on a fixed torus"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1202.6652","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:e87120b1210c9f5217df7f6d6d25a2cd0ee6d9c80253563c4550c24f5ac12be9","target":"record","created_at":"2026-05-18T04:01:11Z","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":"e91a98dfa174f20379e6b26b78129356c2abe0f1fb0103ccfa3178ac6307d37e","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2012-02-29T19:18:44Z","title_canon_sha256":"dad8c394c3fa94a265a52940ce560ca544609aeb67edbdab714a912d3fc033f8"},"schema_version":"1.0","source":{"id":"1202.6652","kind":"arxiv","version":1}},"canonical_sha256":"cf9d51b10cb2619f56cd836496373ec6c678b0acc8991bb1454a1378aa6e1307","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"cf9d51b10cb2619f56cd836496373ec6c678b0acc8991bb1454a1378aa6e1307","first_computed_at":"2026-05-18T04:01:11.980387Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:01:11.980387Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"5rg5S4kzYq/an+mV765Pc9N/Ww1QAeA+uHa9v0eLYNgv1aEWCJp/nvgJIoL0BmrGrdQTyqXeDLjpKaknpdb8AA==","signature_status":"signed_v1","signed_at":"2026-05-18T04:01:11.980904Z","signed_message":"canonical_sha256_bytes"},"source_id":"1202.6652","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:e87120b1210c9f5217df7f6d6d25a2cd0ee6d9c80253563c4550c24f5ac12be9","sha256:b9235bee687473523b30b6886c32357cecb8e233168ff7896e34a93ddd207b6d"],"state_sha256":"45b6dcf14ef865aed719f6c94c921fd3c20d82821a92774fca2473c5c88eff35"}