{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2009:OALTJ4T5ZPAJFRAFNJKOZTM7YM","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":"0a7395b35d643507a3dbda4f103cf7bef8ffd8dd659fd1641dcf1be81b8ce617","cross_cats_sorted":["astro-ph.CO","gr-qc","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2009-04-15T04:13:08Z","title_canon_sha256":"acaa09eeb6de403570d7b5b89a06ccba02012c2d42674e9b82ba9acf8741b135"},"schema_version":"1.0","source":{"id":"0904.2236","kind":"arxiv","version":4}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0904.2236","created_at":"2026-05-18T02:14:00Z"},{"alias_kind":"arxiv_version","alias_value":"0904.2236v4","created_at":"2026-05-18T02:14:00Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0904.2236","created_at":"2026-05-18T02:14:00Z"},{"alias_kind":"pith_short_12","alias_value":"OALTJ4T5ZPAJ","created_at":"2026-05-18T12:26:01Z"},{"alias_kind":"pith_short_16","alias_value":"OALTJ4T5ZPAJFRAF","created_at":"2026-05-18T12:26:01Z"},{"alias_kind":"pith_short_8","alias_value":"OALTJ4T5","created_at":"2026-05-18T12:26:01Z"}],"graph_snapshots":[{"event_id":"sha256:54b780b897c532ab67b625ec0541853b96a71e98efb78de5d421fd5e84ed1d37","target":"graph","created_at":"2026-05-18T02:14:00Z","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 prove a theorem about magnification relations for all generic general caustic singularities up to codimension five: folds, cusps, swallowtail, elliptic umbilic, hyperbolic umbilic, butterfly, parabolic umbilic, wigwam, symbolic umbilic, 2nd elliptic umbilic, and 2nd hyperbolic umbilic. Specifically, we prove that for a generic family of general mappings between planes exhibiting any of these singularities, and for a point in the target lying anywhere in the region giving rise to the maximum number of real pre-images (lensed images), the total signed magnification of the pre-images will alwa","authors_text":"Amir B. Aazami, Arlie O. Petters","cross_cats":["astro-ph.CO","gr-qc","math.MP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2009-04-15T04:13:08Z","title":"A Universal Magnification Theorem II. Generic Caustics up to Codimension Five"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0904.2236","kind":"arxiv","version":4},"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:afb41532c6e85698e98194be5db8f1630dd24e9d434eca8a916e600d2bf4d602","target":"record","created_at":"2026-05-18T02:14:00Z","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":"0a7395b35d643507a3dbda4f103cf7bef8ffd8dd659fd1641dcf1be81b8ce617","cross_cats_sorted":["astro-ph.CO","gr-qc","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2009-04-15T04:13:08Z","title_canon_sha256":"acaa09eeb6de403570d7b5b89a06ccba02012c2d42674e9b82ba9acf8741b135"},"schema_version":"1.0","source":{"id":"0904.2236","kind":"arxiv","version":4}},"canonical_sha256":"701734f27dcbc092c4056a54eccd9fc3394145da9ccdf9a2dfbe832995544a61","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"701734f27dcbc092c4056a54eccd9fc3394145da9ccdf9a2dfbe832995544a61","first_computed_at":"2026-05-18T02:14:00.371758Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:14:00.371758Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"5aDVOAuFsCLyb657bPRkM6gtj/QBUL5WutfWvJeXV6iuekHdq1VDIcMvytk0tneol0NILgk8oJwmMNIxjj0oAg==","signature_status":"signed_v1","signed_at":"2026-05-18T02:14:00.372447Z","signed_message":"canonical_sha256_bytes"},"source_id":"0904.2236","source_kind":"arxiv","source_version":4}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:afb41532c6e85698e98194be5db8f1630dd24e9d434eca8a916e600d2bf4d602","sha256:54b780b897c532ab67b625ec0541853b96a71e98efb78de5d421fd5e84ed1d37"],"state_sha256":"169858bb431fd568544cffadbe2d06c2e3de3deba11fe8b9bd32c44c65d57b9a"}