{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2014:AF4WW3F2JFTB2AXBZP5K53LYXG","short_pith_number":"pith:AF4WW3F2","schema_version":"1.0","canonical_sha256":"01796b6cba49661d02e1cbfaaeed78b9b6b1f0285046ac5a7a4cf31059e23900","source":{"kind":"arxiv","id":"1401.4681","version":1},"attestation_state":"computed","paper":{"title":"Solving Kepler's equation via Smale's $\\alpha$-theory","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MP"],"primary_cat":"math-ph","authors_text":"Jorge Ortigas-Galindo, Martin Avendano, Ver\\'onica Mart\\'in-Molina","submitted_at":"2014-01-19T15:33:27Z","abstract_excerpt":"We obtain an approximate solution $\\tilde{E}=\\tilde{E}(e,M)$ of Kepler's equation $E-e\\sin(E)=M$ for any $e\\in[0,1)$ and $M\\in[0,\\pi]$. Our solution is guaranteed, via Smale's $\\alpha$-theory, to converge to the actual solution $E$ through Newton's method at quadratic speed, i.e. the $n$-th iteration produces a value $E_n$ such that $|E_n-E|\\leq (\\frac12)^{2^n-1}|\\tilde{E}-E|$. The formula provided for $\\tilde{E}$ is a piecewise rational function with conditions defined by polynomial inequalities, except for a small region near $e=1$ and $M=0$, where a single cubic root is used. We also show t"},"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":"1401.4681","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2014-01-19T15:33:27Z","cross_cats_sorted":["math.MP"],"title_canon_sha256":"605c52357f32460668d8a1361f66c06c5601657ae04fa88800c31613249818a7","abstract_canon_sha256":"7f8530c083d0ca2133bd53d9cb2a15c3458646d64adef02ef681e609e712d069"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:51:14.901838Z","signature_b64":"5X0u/sPOwmyrso8dYR78CwWvomxb7uOdjEKiD7okazTdB6U24W3HxvDp4dGoSSqoYkFcPOPJ2J4krS4lXwbYAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"01796b6cba49661d02e1cbfaaeed78b9b6b1f0285046ac5a7a4cf31059e23900","last_reissued_at":"2026-05-18T02:51:14.901304Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:51:14.901304Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Solving Kepler's equation via Smale's $\\alpha$-theory","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MP"],"primary_cat":"math-ph","authors_text":"Jorge Ortigas-Galindo, Martin Avendano, Ver\\'onica Mart\\'in-Molina","submitted_at":"2014-01-19T15:33:27Z","abstract_excerpt":"We obtain an approximate solution $\\tilde{E}=\\tilde{E}(e,M)$ of Kepler's equation $E-e\\sin(E)=M$ for any $e\\in[0,1)$ and $M\\in[0,\\pi]$. Our solution is guaranteed, via Smale's $\\alpha$-theory, to converge to the actual solution $E$ through Newton's method at quadratic speed, i.e. the $n$-th iteration produces a value $E_n$ such that $|E_n-E|\\leq (\\frac12)^{2^n-1}|\\tilde{E}-E|$. The formula provided for $\\tilde{E}$ is a piecewise rational function with conditions defined by polynomial inequalities, except for a small region near $e=1$ and $M=0$, where a single cubic root is used. We also show t"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1401.4681","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":""},"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":"1401.4681","created_at":"2026-05-18T02:51:14.901400+00:00"},{"alias_kind":"arxiv_version","alias_value":"1401.4681v1","created_at":"2026-05-18T02:51:14.901400+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1401.4681","created_at":"2026-05-18T02:51:14.901400+00:00"},{"alias_kind":"pith_short_12","alias_value":"AF4WW3F2JFTB","created_at":"2026-05-18T12:28:19.803747+00:00"},{"alias_kind":"pith_short_16","alias_value":"AF4WW3F2JFTB2AXB","created_at":"2026-05-18T12:28:19.803747+00:00"},{"alias_kind":"pith_short_8","alias_value":"AF4WW3F2","created_at":"2026-05-18T12:28:19.803747+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/AF4WW3F2JFTB2AXBZP5K53LYXG","json":"https://pith.science/pith/AF4WW3F2JFTB2AXBZP5K53LYXG.json","graph_json":"https://pith.science/api/pith-number/AF4WW3F2JFTB2AXBZP5K53LYXG/graph.json","events_json":"https://pith.science/api/pith-number/AF4WW3F2JFTB2AXBZP5K53LYXG/events.json","paper":"https://pith.science/paper/AF4WW3F2"},"agent_actions":{"view_html":"https://pith.science/pith/AF4WW3F2JFTB2AXBZP5K53LYXG","download_json":"https://pith.science/pith/AF4WW3F2JFTB2AXBZP5K53LYXG.json","view_paper":"https://pith.science/paper/AF4WW3F2","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1401.4681&json=true","fetch_graph":"https://pith.science/api/pith-number/AF4WW3F2JFTB2AXBZP5K53LYXG/graph.json","fetch_events":"https://pith.science/api/pith-number/AF4WW3F2JFTB2AXBZP5K53LYXG/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/AF4WW3F2JFTB2AXBZP5K53LYXG/action/timestamp_anchor","attest_storage":"https://pith.science/pith/AF4WW3F2JFTB2AXBZP5K53LYXG/action/storage_attestation","attest_author":"https://pith.science/pith/AF4WW3F2JFTB2AXBZP5K53LYXG/action/author_attestation","sign_citation":"https://pith.science/pith/AF4WW3F2JFTB2AXBZP5K53LYXG/action/citation_signature","submit_replication":"https://pith.science/pith/AF4WW3F2JFTB2AXBZP5K53LYXG/action/replication_record"}},"created_at":"2026-05-18T02:51:14.901400+00:00","updated_at":"2026-05-18T02:51:14.901400+00:00"}