{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:5UN26B5YKCMEAACSGAAHLSJSKL","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":"ba1b605ec928ff1a83b6a8540d52e7e494d8506441539a41212e97dc9a49a819","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.PR","submitted_at":"2026-08-12T04:09:06Z","title_canon_sha256":"dbb21da7650a09d58d66e8b2fd93f47c2dbc0979eb408df071302b1cfca218e7"},"schema_version":"1.0","source":{"id":"2608.12415","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2608.12415","created_at":"2026-08-14T00:42:48Z"},{"alias_kind":"arxiv_version","alias_value":"2608.12415v1","created_at":"2026-08-14T00:42:48Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2608.12415","created_at":"2026-08-14T00:42:48Z"},{"alias_kind":"pith_short_12","alias_value":"5UN26B5YKCME","created_at":"2026-08-14T00:42:48Z"},{"alias_kind":"pith_short_16","alias_value":"5UN26B5YKCMEAACS","created_at":"2026-08-14T00:42:48Z"},{"alias_kind":"pith_short_8","alias_value":"5UN26B5Y","created_at":"2026-08-14T00:42:48Z"}],"graph_snapshots":[{"event_id":"sha256:3cc43bd523710663e8182cf340000dc1de2a94a5cb7205df409ecdf0dc3fe1a5","target":"graph","created_at":"2026-08-14T00:42:48Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2608.12415/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We resolve the ellipsoid fitting conjecture of Saunderson, Chandrasekaran, Parrilo, and Willsky up to a vanishing factor. Concretely, for $m$ independent Gaussian points in dimension $d$, we show that with high probability,\n  for $m \\leq (1-o_d(1)) \\cdot d^2/4$, there exists a centered ellipsoid passing through all $m$ points;\n  for $m\\geq (1+o_d(1) )\\cdot d^2/4$, no such ellipsoid exists.\n  This confirms that the ellipsoid fitting problem has a sharp phase transition at $d^2/4$.","authors_text":"Aaron Potechin, Jeff Xu, Madhur Tulsiani, Sofia de la Cerda","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.PR","submitted_at":"2026-08-12T04:09:06Z","title":"Sharp Phase Transition for Ellipsoid Fitting"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2608.12415","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:e547bcc4fc482ffa4a3751525cfd6145c43d3025541c0b000a2797ac1ef12ddb","target":"record","created_at":"2026-08-14T00:42:48Z","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":"ba1b605ec928ff1a83b6a8540d52e7e494d8506441539a41212e97dc9a49a819","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.PR","submitted_at":"2026-08-12T04:09:06Z","title_canon_sha256":"dbb21da7650a09d58d66e8b2fd93f47c2dbc0979eb408df071302b1cfca218e7"},"schema_version":"1.0","source":{"id":"2608.12415","kind":"arxiv","version":1}},"canonical_sha256":"ed1baf07b85098400052300075c93252c8e3a69dfe8d5ec9c8f1857b88d6fd4e","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"ed1baf07b85098400052300075c93252c8e3a69dfe8d5ec9c8f1857b88d6fd4e","first_computed_at":"2026-08-14T00:42:48.685453Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-08-14T00:42:48.685453Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"pvF89pKIbXygcOBDhvL7hMcLQUZLlQ9XhM24daiCnGFIgHUsUfMJspGu+keLtAEWDw0QVUN2eeFVHD4HiTB1Dg==","signature_status":"signed_v1","signed_at":"2026-08-14T00:42:48.695968Z","signed_message":"canonical_sha256_bytes"},"source_id":"2608.12415","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:e547bcc4fc482ffa4a3751525cfd6145c43d3025541c0b000a2797ac1ef12ddb","sha256:3cc43bd523710663e8182cf340000dc1de2a94a5cb7205df409ecdf0dc3fe1a5"],"state_sha256":"7e21a3785ad31fda11649f7229936775b4368f5309097311ca6afb1657510136"}