{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:3BZMRM5SK2LSUULTNUO7K3VSSE","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":"d2bf2905e771b58a1a038f97eb8ae55f86aa6d14746687fa3c39d20ed8defa1b","cross_cats_sorted":["cs.AI","cs.IT","math.IT"],"license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","primary_cat":"cs.IR","submitted_at":"2026-06-10T08:11:41Z","title_canon_sha256":"ceeb451d253048ce3c6af33e0573a0172792a09e87d7f09f4a3c55d387d90f83"},"schema_version":"1.0","source":{"id":"2606.11780","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2606.11780","created_at":"2026-06-11T01:10:07Z"},{"alias_kind":"arxiv_version","alias_value":"2606.11780v1","created_at":"2026-06-11T01:10:07Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.11780","created_at":"2026-06-11T01:10:07Z"},{"alias_kind":"pith_short_12","alias_value":"3BZMRM5SK2LS","created_at":"2026-06-11T01:10:07Z"},{"alias_kind":"pith_short_16","alias_value":"3BZMRM5SK2LSUULT","created_at":"2026-06-11T01:10:07Z"},{"alias_kind":"pith_short_8","alias_value":"3BZMRM5S","created_at":"2026-06-11T01:10:07Z"}],"graph_snapshots":[{"event_id":"sha256:cb54d26a5fbc836c398a656b05f386af6f9b50adfbe4c8fb70341f43642360be","target":"graph","created_at":"2026-06-11T01:10:07Z","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/2606.11780/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We establish conditions for embedding a corpus of $N$ documents as $d$-dimensional vectors such that every $k$-subset $S \\subseteq [N]$ is realizable as a result of top-$k$ retrieval by some query vector. Recent work shows that $d = O(k)$ suffices for such embeddings to exist in $\\mathbb{R}^d$, independently of $N$. We theoretically prove that this corpus-independent bound is specific to infinite precision. With $B$ bits per coordinate, perfect top-$k$ retrieval requires $Bd = \\Omega(k \\ln N)$; thus, at any fixed precision, the dimension must grow at least logarithmically with $N$. Specializin","authors_text":"Koki Okajima, Tsukasa Yoshida","cross_cats":["cs.AI","cs.IT","math.IT"],"headline":"","license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","primary_cat":"cs.IR","submitted_at":"2026-06-10T08:11:41Z","title":"What Limits Does Quantization Place on Dense Top-$k$ Retrieval? A Theoretical Study"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.11780","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:f28ceb36fc152d2ae9d3e9bae7401c6bae1c7faf51ab164043833ddf5dec0ae7","target":"record","created_at":"2026-06-11T01:10:07Z","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":"d2bf2905e771b58a1a038f97eb8ae55f86aa6d14746687fa3c39d20ed8defa1b","cross_cats_sorted":["cs.AI","cs.IT","math.IT"],"license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","primary_cat":"cs.IR","submitted_at":"2026-06-10T08:11:41Z","title_canon_sha256":"ceeb451d253048ce3c6af33e0573a0172792a09e87d7f09f4a3c55d387d90f83"},"schema_version":"1.0","source":{"id":"2606.11780","kind":"arxiv","version":1}},"canonical_sha256":"d872c8b3b256972a51736d1df56eb291335e2cc38fb2b028d985788bbc024390","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"d872c8b3b256972a51736d1df56eb291335e2cc38fb2b028d985788bbc024390","first_computed_at":"2026-06-11T01:10:07.426637Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-06-11T01:10:07.426637Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"HHRocTRiBu4udTRYyaG55OAuKOd9TTYuCk8b+pUWpUm40BL05CB9YtX6UmgfUnxdM14B29SpqoMUh+wy2+c6AQ==","signature_status":"signed_v1","signed_at":"2026-06-11T01:10:07.427490Z","signed_message":"canonical_sha256_bytes"},"source_id":"2606.11780","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:f28ceb36fc152d2ae9d3e9bae7401c6bae1c7faf51ab164043833ddf5dec0ae7","sha256:cb54d26a5fbc836c398a656b05f386af6f9b50adfbe4c8fb70341f43642360be"],"state_sha256":"cc1c527fc46c419b3e6d01091db35360c909033a9e4b6da89df468d551e1bf3e"}