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We assume that $A(y_1,y_2,\\cdots, y_n)$ is 1-periodic in each $y_i \\in \\mathbb{R}^d$ and real analytic. Classically, the method of reiterated homogenization has been applied to study this multiscale elliptic operator, which leads to a convergence rate limited by t"},"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":"2509.09410","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AP","submitted_at":"2025-09-11T12:44:40Z","cross_cats_sorted":[],"title_canon_sha256":"0e3c77abca92a461e54935bb5014c4239d12c5f46401a90229fc505dd17f16f3","abstract_canon_sha256":"505ff3e4d6eb890585757753df1e54b7276f1a2320ec7c9c676fb1415e13705c"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T12:09:35.638596Z","signature_b64":"3yRt5SYXNg28nfBz2r8fCa4CSMJhA5DyvgIqMSmKckJKlPitFrGLG4WeTi6ocJpm/Wb7KNSvnyBsRZy6+5PrDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6ce6d73101b7be9db1d579d035db84ea9acc11b590a0426eb176bdf3ad52ab99","last_reissued_at":"2026-07-05T12:09:35.638031Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T12:09:35.638031Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Optimal convergence rates in multiscale elliptic homogenization","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Jinping Zhuge, Weisheng Niu, Yao Xu","submitted_at":"2025-09-11T12:44:40Z","abstract_excerpt":"This paper is devoted to the quantitative homogenization of multiscale elliptic operator $-\\nabla\\cdot A_\\varepsilon \\nabla$, where $A_\\varepsilon(x) = A(x/\\varepsilon_1, x/\\varepsilon_2,\\cdots, x/\\varepsilon_n)$, $\\varepsilon = (\\varepsilon_1, \\varepsilon_2,\\cdots, \\varepsilon_n) \\in (0,1]^n$ and $\\varepsilon_i > \\varepsilon_{i+1}$. We assume that $A(y_1,y_2,\\cdots, y_n)$ is 1-periodic in each $y_i \\in \\mathbb{R}^d$ and real analytic. Classically, the method of reiterated homogenization has been applied to study this multiscale elliptic operator, which leads to a convergence rate limited by t"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2509.09410","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/2509.09410/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":"2509.09410","created_at":"2026-07-05T12:09:35.638098+00:00"},{"alias_kind":"arxiv_version","alias_value":"2509.09410v1","created_at":"2026-07-05T12:09:35.638098+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2509.09410","created_at":"2026-07-05T12:09:35.638098+00:00"},{"alias_kind":"pith_short_12","alias_value":"NTTNOMIBW67J","created_at":"2026-07-05T12:09:35.638098+00:00"},{"alias_kind":"pith_short_16","alias_value":"NTTNOMIBW67J3MOV","created_at":"2026-07-05T12:09:35.638098+00:00"},{"alias_kind":"pith_short_8","alias_value":"NTTNOMIB","created_at":"2026-07-05T12:09:35.638098+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/NTTNOMIBW67J3MOVPHIDLW4E5K","json":"https://pith.science/pith/NTTNOMIBW67J3MOVPHIDLW4E5K.json","graph_json":"https://pith.science/api/pith-number/NTTNOMIBW67J3MOVPHIDLW4E5K/graph.json","events_json":"https://pith.science/api/pith-number/NTTNOMIBW67J3MOVPHIDLW4E5K/events.json","paper":"https://pith.science/paper/NTTNOMIB"},"agent_actions":{"view_html":"https://pith.science/pith/NTTNOMIBW67J3MOVPHIDLW4E5K","download_json":"https://pith.science/pith/NTTNOMIBW67J3MOVPHIDLW4E5K.json","view_paper":"https://pith.science/paper/NTTNOMIB","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2509.09410&json=true","fetch_graph":"https://pith.science/api/pith-number/NTTNOMIBW67J3MOVPHIDLW4E5K/graph.json","fetch_events":"https://pith.science/api/pith-number/NTTNOMIBW67J3MOVPHIDLW4E5K/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/NTTNOMIBW67J3MOVPHIDLW4E5K/action/timestamp_anchor","attest_storage":"https://pith.science/pith/NTTNOMIBW67J3MOVPHIDLW4E5K/action/storage_attestation","attest_author":"https://pith.science/pith/NTTNOMIBW67J3MOVPHIDLW4E5K/action/author_attestation","sign_citation":"https://pith.science/pith/NTTNOMIBW67J3MOVPHIDLW4E5K/action/citation_signature","submit_replication":"https://pith.science/pith/NTTNOMIBW67J3MOVPHIDLW4E5K/action/replication_record"}},"created_at":"2026-07-05T12:09:35.638098+00:00","updated_at":"2026-07-05T12:09:35.638098+00:00"}