{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2019:2MHXV6Q6PBK53S7YCU3SBEG7M7","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":"336460c90dd02172a3382321c32ccc2faefa3dcc87fe6a4a0a274fc26fb91c68","cross_cats_sorted":["math.AG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2019-01-30T16:22:01Z","title_canon_sha256":"e348f1a5a0a68fe3bb6a902348912ddb51b6c4c4940b836927714e6e7708c9e6"},"schema_version":"1.0","source":{"id":"1901.10929","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1901.10929","created_at":"2026-05-17T23:55:05Z"},{"alias_kind":"arxiv_version","alias_value":"1901.10929v1","created_at":"2026-05-17T23:55:05Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1901.10929","created_at":"2026-05-17T23:55:05Z"},{"alias_kind":"pith_short_12","alias_value":"2MHXV6Q6PBK5","created_at":"2026-05-18T12:33:07Z"},{"alias_kind":"pith_short_16","alias_value":"2MHXV6Q6PBK53S7Y","created_at":"2026-05-18T12:33:07Z"},{"alias_kind":"pith_short_8","alias_value":"2MHXV6Q6","created_at":"2026-05-18T12:33:07Z"}],"graph_snapshots":[{"event_id":"sha256:0a775dca4c761cfec156fd4c4bd1f6260146fd4cca4078955de583bd98a80207","target":"graph","created_at":"2026-05-17T23:55:05Z","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":"By generalising the notion of a unimodular sequence, we create an expression for the winding number of certain ordered sets of lattice points. Since the winding number of the vertices of a Fano polygon is necessarily one, we use this expression as a restriction to classify all Fano polygons without T-singularities and whose basket of residual singularities is of the form $\\left\\{ \\frac{1}{r}(1,s_{1}), \\frac{1}{r}(1,s_{2}), \\ldots, \\frac{1}{r}(1,s_{k}) \\right\\}$ for $k,r \\in \\mathbb{Z}_{>0}$, and $1 \\leq s_{i} < r$ is coprime to $r$.","authors_text":"Akihiro Higashitani, Daniel Cavey","cross_cats":["math.AG"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2019-01-30T16:22:01Z","title":"Winding Number of $r$-modular sequences and Applications to the Singularity Content of a Fano Polygon"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1901.10929","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":"sha256:3b91e3db96ac5cff602235e62ce5a3babf702eaca1060db3a9f4b92bfbd502dc","target":"record","created_at":"2026-05-17T23:55:05Z","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":"336460c90dd02172a3382321c32ccc2faefa3dcc87fe6a4a0a274fc26fb91c68","cross_cats_sorted":["math.AG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2019-01-30T16:22:01Z","title_canon_sha256":"e348f1a5a0a68fe3bb6a902348912ddb51b6c4c4940b836927714e6e7708c9e6"},"schema_version":"1.0","source":{"id":"1901.10929","kind":"arxiv","version":1}},"canonical_sha256":"d30f7afa1e7855ddcbf815372090df67f3a8ae9ed3db20554bc6308a24b4aa02","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"d30f7afa1e7855ddcbf815372090df67f3a8ae9ed3db20554bc6308a24b4aa02","first_computed_at":"2026-05-17T23:55:05.708367Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:55:05.708367Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"LnWLx0BEbf/lB6mv3FRLrwj7Hj4FmVWABvJ+V5HB65vFh/TbvXdtsSQ8v9R4AFunANjev/7QN9GQ8dOv9zW2Bg==","signature_status":"signed_v1","signed_at":"2026-05-17T23:55:05.708832Z","signed_message":"canonical_sha256_bytes"},"source_id":"1901.10929","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:3b91e3db96ac5cff602235e62ce5a3babf702eaca1060db3a9f4b92bfbd502dc","sha256:0a775dca4c761cfec156fd4c4bd1f6260146fd4cca4078955de583bd98a80207"],"state_sha256":"37715024e66048be85cf77308c34c131c2dc153f3e1f72d93809b0f76a48c37e"}