{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2006:6WYZMEWW2P7N5RH3FXK6UFEKR3","short_pith_number":"pith:6WYZMEWW","canonical_record":{"source":{"id":"math/0602381","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.PR","submitted_at":"2006-02-17T16:27:19Z","cross_cats_sorted":[],"title_canon_sha256":"e22bd361d84ceb552533fbde8528825935b0584ee15dd9d3dd0ed86862444156","abstract_canon_sha256":"ec438c2129128d0ae28d1b2060425f83dffb8509be7ce650eb9f3174e63efa91"},"schema_version":"1.0"},"canonical_sha256":"f5b19612d6d3fedec4fb2dd5ea148a8ee9c7c20b3c4f6a055454c0f4f96186ea","source":{"kind":"arxiv","id":"math/0602381","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/0602381","created_at":"2026-05-18T01:08:50Z"},{"alias_kind":"arxiv_version","alias_value":"math/0602381v1","created_at":"2026-05-18T01:08:50Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0602381","created_at":"2026-05-18T01:08:50Z"},{"alias_kind":"pith_short_12","alias_value":"6WYZMEWW2P7N","created_at":"2026-05-18T12:25:53Z"},{"alias_kind":"pith_short_16","alias_value":"6WYZMEWW2P7N5RH3","created_at":"2026-05-18T12:25:53Z"},{"alias_kind":"pith_short_8","alias_value":"6WYZMEWW","created_at":"2026-05-18T12:25:53Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2006:6WYZMEWW2P7N5RH3FXK6UFEKR3","target":"record","payload":{"canonical_record":{"source":{"id":"math/0602381","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.PR","submitted_at":"2006-02-17T16:27:19Z","cross_cats_sorted":[],"title_canon_sha256":"e22bd361d84ceb552533fbde8528825935b0584ee15dd9d3dd0ed86862444156","abstract_canon_sha256":"ec438c2129128d0ae28d1b2060425f83dffb8509be7ce650eb9f3174e63efa91"},"schema_version":"1.0"},"canonical_sha256":"f5b19612d6d3fedec4fb2dd5ea148a8ee9c7c20b3c4f6a055454c0f4f96186ea","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:08:50.541614Z","signature_b64":"89AP/WewPEr1xZCHniWImUQTaaMIGaZMYemh7UOcjqd4WSOGZFOh6TWA9U128YZPxgTOBuyyjQzHXRDiE5eaAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f5b19612d6d3fedec4fb2dd5ea148a8ee9c7c20b3c4f6a055454c0f4f96186ea","last_reissued_at":"2026-05-18T01:08:50.541105Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:08:50.541105Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"math/0602381","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:08:50Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"oDU4U3nxd0J9TgIK2MS5zTaSza+kShyLqlhUOMifIXOlCeUxJ2IZD9RRgOR2Wj5QvDRrPubUqEy6AmtMux7EAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-24T06:40:26.106183Z"},"content_sha256":"b51db7bd24bff7833f98a6df24cda81d9e4a3318b0c066c0085743b9f873f2df","schema_version":"1.0","event_id":"sha256:b51db7bd24bff7833f98a6df24cda81d9e4a3318b0c066c0085743b9f873f2df"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2006:6WYZMEWW2P7N5RH3FXK6UFEKR3","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Distortion mismatch in the quantization of probability measures","license":"","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Gilles Pag\\`es (PMA), Harald Luschgy, Siegried Graf (Universit\\\"at Passau)","submitted_at":"2006-02-17T16:27:19Z","abstract_excerpt":"We elucidate the asymptotics of the L^s-quantization error induced by a sequence of L^r-optimal n-quantizers of a probability distribution P on R^d when s>r. In particular we show that under natural assumptions, the optimal rate is preserved as long as s<r+d (and for every s in the case of a compactly supported distribution). We derive some applications of these results to the error bounds for quantization based quadrature formulae in numerical integration on R^d and on the Wiener space."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0602381","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":""},"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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:08:50Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"6M+oes18H6MNrX7CG8iJZ5v/zplUKlC3dS92FeHdBCXIe3ZCXwbZVeiIgyFP3om+z7Qm7fvQ1YvpsBZhVN40AQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-24T06:40:26.106567Z"},"content_sha256":"6800f0a6b6e57877e7c9616785f81817f2c534f122a194dc5865adab2ac9dd23","schema_version":"1.0","event_id":"sha256:6800f0a6b6e57877e7c9616785f81817f2c534f122a194dc5865adab2ac9dd23"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/6WYZMEWW2P7N5RH3FXK6UFEKR3/bundle.json","state_url":"https://pith.science/pith/6WYZMEWW2P7N5RH3FXK6UFEKR3/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/6WYZMEWW2P7N5RH3FXK6UFEKR3/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-24T06:40:26Z","links":{"resolver":"https://pith.science/pith/6WYZMEWW2P7N5RH3FXK6UFEKR3","bundle":"https://pith.science/pith/6WYZMEWW2P7N5RH3FXK6UFEKR3/bundle.json","state":"https://pith.science/pith/6WYZMEWW2P7N5RH3FXK6UFEKR3/state.json","well_known_bundle":"https://pith.science/.well-known/pith/6WYZMEWW2P7N5RH3FXK6UFEKR3/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2006:6WYZMEWW2P7N5RH3FXK6UFEKR3","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":"ec438c2129128d0ae28d1b2060425f83dffb8509be7ce650eb9f3174e63efa91","cross_cats_sorted":[],"license":"","primary_cat":"math.PR","submitted_at":"2006-02-17T16:27:19Z","title_canon_sha256":"e22bd361d84ceb552533fbde8528825935b0584ee15dd9d3dd0ed86862444156"},"schema_version":"1.0","source":{"id":"math/0602381","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/0602381","created_at":"2026-05-18T01:08:50Z"},{"alias_kind":"arxiv_version","alias_value":"math/0602381v1","created_at":"2026-05-18T01:08:50Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0602381","created_at":"2026-05-18T01:08:50Z"},{"alias_kind":"pith_short_12","alias_value":"6WYZMEWW2P7N","created_at":"2026-05-18T12:25:53Z"},{"alias_kind":"pith_short_16","alias_value":"6WYZMEWW2P7N5RH3","created_at":"2026-05-18T12:25:53Z"},{"alias_kind":"pith_short_8","alias_value":"6WYZMEWW","created_at":"2026-05-18T12:25:53Z"}],"graph_snapshots":[{"event_id":"sha256:6800f0a6b6e57877e7c9616785f81817f2c534f122a194dc5865adab2ac9dd23","target":"graph","created_at":"2026-05-18T01:08:50Z","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 elucidate the asymptotics of the L^s-quantization error induced by a sequence of L^r-optimal n-quantizers of a probability distribution P on R^d when s>r. In particular we show that under natural assumptions, the optimal rate is preserved as long as s<r+d (and for every s in the case of a compactly supported distribution). We derive some applications of these results to the error bounds for quantization based quadrature formulae in numerical integration on R^d and on the Wiener space.","authors_text":"Gilles Pag\\`es (PMA), Harald Luschgy, Siegried Graf (Universit\\\"at Passau)","cross_cats":[],"headline":"","license":"","primary_cat":"math.PR","submitted_at":"2006-02-17T16:27:19Z","title":"Distortion mismatch in the quantization of probability measures"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0602381","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:b51db7bd24bff7833f98a6df24cda81d9e4a3318b0c066c0085743b9f873f2df","target":"record","created_at":"2026-05-18T01:08:50Z","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":"ec438c2129128d0ae28d1b2060425f83dffb8509be7ce650eb9f3174e63efa91","cross_cats_sorted":[],"license":"","primary_cat":"math.PR","submitted_at":"2006-02-17T16:27:19Z","title_canon_sha256":"e22bd361d84ceb552533fbde8528825935b0584ee15dd9d3dd0ed86862444156"},"schema_version":"1.0","source":{"id":"math/0602381","kind":"arxiv","version":1}},"canonical_sha256":"f5b19612d6d3fedec4fb2dd5ea148a8ee9c7c20b3c4f6a055454c0f4f96186ea","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"f5b19612d6d3fedec4fb2dd5ea148a8ee9c7c20b3c4f6a055454c0f4f96186ea","first_computed_at":"2026-05-18T01:08:50.541105Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:08:50.541105Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"89AP/WewPEr1xZCHniWImUQTaaMIGaZMYemh7UOcjqd4WSOGZFOh6TWA9U128YZPxgTOBuyyjQzHXRDiE5eaAw==","signature_status":"signed_v1","signed_at":"2026-05-18T01:08:50.541614Z","signed_message":"canonical_sha256_bytes"},"source_id":"math/0602381","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:b51db7bd24bff7833f98a6df24cda81d9e4a3318b0c066c0085743b9f873f2df","sha256:6800f0a6b6e57877e7c9616785f81817f2c534f122a194dc5865adab2ac9dd23"],"state_sha256":"6fddca78efb4645d7f65fb76ccf1f5b674c64dd7723f487d6ebfdf81de43ec08"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Ob+8IIcvurttM8w8x2vBrdvS//qMTLyXflFVBozbwuPbgOGB7Zt6JIIz/geWPEQsj/MnOAD+GNu6ObUw4ug/Dw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-24T06:40:26.108672Z","bundle_sha256":"ede071ae51915bf209c2649dcfbd2176cb66ca030398ecf46550e7c7a971de9e"}}