{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2019:PA7SQF5OFC5SNENRRVC6Q4BIMK","short_pith_number":"pith:PA7SQF5O","schema_version":"1.0","canonical_sha256":"783f2817ae28bb2691b18d45e8702862b14d1ebc7fe007cfd264973f8a739b52","source":{"kind":"arxiv","id":"1908.09400","version":2},"attestation_state":"computed","paper":{"title":"Optimal Curve Straightening is $\\exists\\mathbb{R}$-Complete","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cs.CG","authors_text":"Jeff Erickson","submitted_at":"2019-08-25T22:06:22Z","abstract_excerpt":"We prove that the following problem has the same computational complexity as the existential theory of the reals: Given a generic self-intersecting closed curve $\\gamma$ in the plane and an integer $m$, is there a polygon with $m$ vertices that is isotopic to $\\gamma$? Our reduction implies implies two stronger results, as corollaries of similar results for pseudoline arrangements. First, there are isotopy classes in which every $m$-gon with integer coordinates requires $2^{\\Omega(m)}$ bits of precision. Second, for any semi-algebraic set $V$, there is an integer $m$ and a closed curve $\\gamma"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1908.09400","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CG","submitted_at":"2019-08-25T22:06:22Z","cross_cats_sorted":[],"title_canon_sha256":"27b3882b28329408cd9f6782adafcae5459458bc13f3aeccb104d81cfd279aa8","abstract_canon_sha256":"01241d8fd810da25719ad82e8b69aaeb0fe9260899ad85c19e611de6145a80fe"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-04T23:59:51.230193Z","signature_b64":"YopLLxMJ9Pk9NHWDtGE74k3VOhSVnCuiQLJIdOCkD07cHXr7A9uLRYbr0ilrMxe1Mywjxnp5sqMfom3JQC1PBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"783f2817ae28bb2691b18d45e8702862b14d1ebc7fe007cfd264973f8a739b52","last_reissued_at":"2026-07-04T23:59:51.229815Z","signature_status":"signed_v1","first_computed_at":"2026-07-04T23:59:51.229815Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Optimal Curve Straightening is $\\exists\\mathbb{R}$-Complete","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cs.CG","authors_text":"Jeff Erickson","submitted_at":"2019-08-25T22:06:22Z","abstract_excerpt":"We prove that the following problem has the same computational complexity as the existential theory of the reals: Given a generic self-intersecting closed curve $\\gamma$ in the plane and an integer $m$, is there a polygon with $m$ vertices that is isotopic to $\\gamma$? Our reduction implies implies two stronger results, as corollaries of similar results for pseudoline arrangements. First, there are isotopy classes in which every $m$-gon with integer coordinates requires $2^{\\Omega(m)}$ bits of precision. Second, for any semi-algebraic set $V$, there is an integer $m$ and a closed curve $\\gamma"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1908.09400","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/1908.09400/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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1908.09400","created_at":"2026-07-04T23:59:51.229871+00:00"},{"alias_kind":"arxiv_version","alias_value":"1908.09400v2","created_at":"2026-07-04T23:59:51.229871+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1908.09400","created_at":"2026-07-04T23:59:51.229871+00:00"},{"alias_kind":"pith_short_12","alias_value":"PA7SQF5OFC5S","created_at":"2026-07-04T23:59:51.229871+00:00"},{"alias_kind":"pith_short_16","alias_value":"PA7SQF5OFC5SNENR","created_at":"2026-07-04T23:59:51.229871+00:00"},{"alias_kind":"pith_short_8","alias_value":"PA7SQF5O","created_at":"2026-07-04T23:59:51.229871+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/PA7SQF5OFC5SNENRRVC6Q4BIMK","json":"https://pith.science/pith/PA7SQF5OFC5SNENRRVC6Q4BIMK.json","graph_json":"https://pith.science/api/pith-number/PA7SQF5OFC5SNENRRVC6Q4BIMK/graph.json","events_json":"https://pith.science/api/pith-number/PA7SQF5OFC5SNENRRVC6Q4BIMK/events.json","paper":"https://pith.science/paper/PA7SQF5O"},"agent_actions":{"view_html":"https://pith.science/pith/PA7SQF5OFC5SNENRRVC6Q4BIMK","download_json":"https://pith.science/pith/PA7SQF5OFC5SNENRRVC6Q4BIMK.json","view_paper":"https://pith.science/paper/PA7SQF5O","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1908.09400&json=true","fetch_graph":"https://pith.science/api/pith-number/PA7SQF5OFC5SNENRRVC6Q4BIMK/graph.json","fetch_events":"https://pith.science/api/pith-number/PA7SQF5OFC5SNENRRVC6Q4BIMK/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/PA7SQF5OFC5SNENRRVC6Q4BIMK/action/timestamp_anchor","attest_storage":"https://pith.science/pith/PA7SQF5OFC5SNENRRVC6Q4BIMK/action/storage_attestation","attest_author":"https://pith.science/pith/PA7SQF5OFC5SNENRRVC6Q4BIMK/action/author_attestation","sign_citation":"https://pith.science/pith/PA7SQF5OFC5SNENRRVC6Q4BIMK/action/citation_signature","submit_replication":"https://pith.science/pith/PA7SQF5OFC5SNENRRVC6Q4BIMK/action/replication_record"}},"created_at":"2026-07-04T23:59:51.229871+00:00","updated_at":"2026-07-04T23:59:51.229871+00:00"}