{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2026:NCNPYYKALJBMGK5HX4R5KDR3EE","short_pith_number":"pith:NCNPYYKA","schema_version":"1.0","canonical_sha256":"689afc61405a42c32ba7bf23d50e3b21378ddc45d9032a4f38f9b52ba91f1761","source":{"kind":"arxiv","id":"2608.01776","version":1},"attestation_state":"computed","paper":{"title":"At least seven modes in a heteroscedastic three-component bivariate Gaussian mixture","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["stat.TH"],"primary_cat":"math.ST","authors_text":"Akifumi Okuno, Hirotaka Matsumoto, Yutaro Kabata","submitted_at":"2026-08-03T06:48:40Z","abstract_excerpt":"A Gaussian mixture density can have more modes than components. It has been conjectured that the maximum number of modes of a $d$-variate $k$-component Gaussian mixture density is $\\binom{d+k-1}{d}$, which equals six for $(d,k)=(2,3)$. We construct an explicit family of equally weighted heteroscedastic three-component bivariate Gaussian mixture densities with at least seven distinct nondegenerate modes, showing that this conjectured upper bound fails for $(d,k)=(2,3)$. To the best of our knowledge, this provides the first counterexample to the conjecture across all pairs $(d,k)$."},"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":"2608.01776","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.ST","submitted_at":"2026-08-03T06:48:40Z","cross_cats_sorted":["stat.TH"],"title_canon_sha256":"04abd063bf50701cf46ffbdf84ba9dd37e507c9a4d2cff9b4d3df74ded5e65b7","abstract_canon_sha256":"51c97dfefa7718e77568a64c839d2af0c256401bfe17b70aa961c52b57842388"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-08-04T02:06:56.925233Z","signature_b64":"aYFTl0aqM+3XvRKQeye+qfZImlgvEn3BD0vsLCX/oJ5hAqT+QR3qFRdKb2riI8fOR0RmJp5+yixeeE+NEX1tDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"689afc61405a42c32ba7bf23d50e3b21378ddc45d9032a4f38f9b52ba91f1761","last_reissued_at":"2026-08-04T02:06:56.923647Z","signature_status":"signed_v1","first_computed_at":"2026-08-04T02:06:56.923647Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"At least seven modes in a heteroscedastic three-component bivariate Gaussian mixture","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["stat.TH"],"primary_cat":"math.ST","authors_text":"Akifumi Okuno, Hirotaka Matsumoto, Yutaro Kabata","submitted_at":"2026-08-03T06:48:40Z","abstract_excerpt":"A Gaussian mixture density can have more modes than components. It has been conjectured that the maximum number of modes of a $d$-variate $k$-component Gaussian mixture density is $\\binom{d+k-1}{d}$, which equals six for $(d,k)=(2,3)$. We construct an explicit family of equally weighted heteroscedastic three-component bivariate Gaussian mixture densities with at least seven distinct nondegenerate modes, showing that this conjectured upper bound fails for $(d,k)=(2,3)$. To the best of our knowledge, this provides the first counterexample to the conjecture across all pairs $(d,k)$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2608.01776","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2608.01776/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":"2608.01776","created_at":"2026-08-04T02:06:56.925097+00:00"},{"alias_kind":"arxiv_version","alias_value":"2608.01776v1","created_at":"2026-08-04T02:06:56.925097+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2608.01776","created_at":"2026-08-04T02:06:56.925097+00:00"},{"alias_kind":"pith_short_12","alias_value":"NCNPYYKALJBM","created_at":"2026-08-04T02:06:56.925097+00:00"},{"alias_kind":"pith_short_16","alias_value":"NCNPYYKALJBMGK5H","created_at":"2026-08-04T02:06:56.925097+00:00"},{"alias_kind":"pith_short_8","alias_value":"NCNPYYKA","created_at":"2026-08-04T02:06:56.925097+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/NCNPYYKALJBMGK5HX4R5KDR3EE","json":"https://pith.science/pith/NCNPYYKALJBMGK5HX4R5KDR3EE.json","graph_json":"https://pith.science/api/pith-number/NCNPYYKALJBMGK5HX4R5KDR3EE/graph.json","events_json":"https://pith.science/api/pith-number/NCNPYYKALJBMGK5HX4R5KDR3EE/events.json","paper":"https://pith.science/paper/NCNPYYKA"},"agent_actions":{"view_html":"https://pith.science/pith/NCNPYYKALJBMGK5HX4R5KDR3EE","download_json":"https://pith.science/pith/NCNPYYKALJBMGK5HX4R5KDR3EE.json","view_paper":"https://pith.science/paper/NCNPYYKA","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2608.01776&json=true","fetch_graph":"https://pith.science/api/pith-number/NCNPYYKALJBMGK5HX4R5KDR3EE/graph.json","fetch_events":"https://pith.science/api/pith-number/NCNPYYKALJBMGK5HX4R5KDR3EE/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/NCNPYYKALJBMGK5HX4R5KDR3EE/action/timestamp_anchor","attest_storage":"https://pith.science/pith/NCNPYYKALJBMGK5HX4R5KDR3EE/action/storage_attestation","attest_author":"https://pith.science/pith/NCNPYYKALJBMGK5HX4R5KDR3EE/action/author_attestation","sign_citation":"https://pith.science/pith/NCNPYYKALJBMGK5HX4R5KDR3EE/action/citation_signature","submit_replication":"https://pith.science/pith/NCNPYYKALJBMGK5HX4R5KDR3EE/action/replication_record"}},"created_at":"2026-08-04T02:06:56.925097+00:00","updated_at":"2026-08-04T02:06:56.925097+00:00"}