{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2025:6LZNJ4IWAROV47EVEL6ILCSAXA","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":"94bbbb7bc03d30cff076b421b3b300b2734dc8eb6cf592ce0ed3f38cdcaeae39","cross_cats_sorted":["math.PR","stat.ML","stat.TH"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.ST","submitted_at":"2025-07-08T17:59:02Z","title_canon_sha256":"547f7f7d6f376aa35de7da9fb53b9769261afcdf974a99de855797333eeff82b"},"schema_version":"1.0","source":{"id":"2507.06226","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2507.06226","created_at":"2026-07-05T11:33:54Z"},{"alias_kind":"arxiv_version","alias_value":"2507.06226v1","created_at":"2026-07-05T11:33:54Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2507.06226","created_at":"2026-07-05T11:33:54Z"},{"alias_kind":"pith_short_12","alias_value":"6LZNJ4IWAROV","created_at":"2026-07-05T11:33:54Z"},{"alias_kind":"pith_short_16","alias_value":"6LZNJ4IWAROV47EV","created_at":"2026-07-05T11:33:54Z"},{"alias_kind":"pith_short_8","alias_value":"6LZNJ4IW","created_at":"2026-07-05T11:33:54Z"}],"graph_snapshots":[{"event_id":"sha256:d995f96d909c4d95f06b3bb678ecef0590ce267250bbe8ee93ffa5666917b773","target":"graph","created_at":"2026-07-05T11:33:54Z","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/2507.06226/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"A celebrated result of Pollard proves asymptotic consistency for $k$-means clustering when the population distribution has finite variance. In this work, we point out that the population-level $k$-means clustering problem is, in fact, well-posed under the weaker assumption of a finite expectation, and we investigate whether some form of asymptotic consistency holds in this setting. As we illustrate in a variety of negative results, the complete story is quite subtle; for example, the empirical $k$-means cluster centers may fail to converge even if there exists a unique set of population $k$-me","authors_text":"Adam Quinn Jaffe, Mo\\\"ise Blanchard, Nikita Zhivotovskiy","cross_cats":["math.PR","stat.ML","stat.TH"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.ST","submitted_at":"2025-07-08T17:59:02Z","title":"Consistency and Inconsistency in $K$-Means Clustering"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2507.06226","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:cef2ce0ccdfb8bdebd2199d34c9fc34d494e57a9bf11d0fe9145e657e2f0f0c4","target":"record","created_at":"2026-07-05T11:33:54Z","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":"94bbbb7bc03d30cff076b421b3b300b2734dc8eb6cf592ce0ed3f38cdcaeae39","cross_cats_sorted":["math.PR","stat.ML","stat.TH"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.ST","submitted_at":"2025-07-08T17:59:02Z","title_canon_sha256":"547f7f7d6f376aa35de7da9fb53b9769261afcdf974a99de855797333eeff82b"},"schema_version":"1.0","source":{"id":"2507.06226","kind":"arxiv","version":1}},"canonical_sha256":"f2f2d4f116045d5e7c9522fc858a40b83f9d9adec26b740081cf74ac0da74816","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"f2f2d4f116045d5e7c9522fc858a40b83f9d9adec26b740081cf74ac0da74816","first_computed_at":"2026-07-05T11:33:54.592874Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T11:33:54.592874Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"LXXrKdnqwxrCi3+kgFTt5dzgnk7LyYj6N1Q0nHbm9tAszFtABjef81Bzu+0ebYNZGVsS1XXc4TuQggAoBk+6DQ==","signature_status":"signed_v1","signed_at":"2026-07-05T11:33:54.593358Z","signed_message":"canonical_sha256_bytes"},"source_id":"2507.06226","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:cef2ce0ccdfb8bdebd2199d34c9fc34d494e57a9bf11d0fe9145e657e2f0f0c4","sha256:d995f96d909c4d95f06b3bb678ecef0590ce267250bbe8ee93ffa5666917b773"],"state_sha256":"06ab6c2c02dc2e1c36149871c71aa5ccb0cb3f62e4fe98ae232577c4b1a0ce44"}