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Then \\[ \\sqrt{C_P(\\mu_t)} \\leq \\frac{1-t}{\\sqrt{\\kappa_0}} + \\frac{t}{\\sqrt{\\kappa_1}}. \\] This estimate is optimal for every $t,\\kappa_0,\\kappa_1$, holds for all test functions without symmetry assumptions, and remains valid for extended-valued potentials.\n  We also characterize equality at an interior time: it holds "},"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":"2607.10769","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","primary_cat":"math.PR","submitted_at":"2026-07-12T13:51:42Z","cross_cats_sorted":["math.FA","math.MG","math.SP"],"title_canon_sha256":"448401d893c30fed152dabf371826b2d5bb0f06d3737e583fbf01aacaaf2552c","abstract_canon_sha256":"0176cb936cc12f84c0c72213f4c581911a9a7acb8a6286d0bc49635cc92202b5"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-14T01:21:39.192269Z","signature_b64":"iaBAUj+XchLVh0sfRWYqNRFhyzedCngo7swJf6LHWLaRXqEDTTK6YuVHl6pwC192oTezspuJ2pRhIYVwTrg4BA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"28483bb650489bc645fb0c17f820c9ad9f0bf47bf2ab7bc6128d628fa2c52896","last_reissued_at":"2026-07-14T01:21:39.191426Z","signature_status":"signed_v1","first_computed_at":"2026-07-14T01:21:39.191426Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Sharp Poincar\\'e Interpolation Along Wasserstein Geodesics","license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","headline":"","cross_cats":["math.FA","math.MG","math.SP"],"primary_cat":"math.PR","authors_text":"Bang-Xian Han, Zhuo-Nan Zhu","submitted_at":"2026-07-12T13:51:42Z","abstract_excerpt":"We prove a sharp interpolation inequality for the Poincar\\'e constant along quadratic Wasserstein geodesics. Let $\\mu_i$, $i=0,1$, be $\\kappa_i$-strongly log-concave probability measures on $\\mathbb R^n$, and let $(\\mu_t)_{t\\in[0,1]}$ be their optimal displacement interpolation. Then \\[ \\sqrt{C_P(\\mu_t)} \\leq \\frac{1-t}{\\sqrt{\\kappa_0}} + \\frac{t}{\\sqrt{\\kappa_1}}. \\] This estimate is optimal for every $t,\\kappa_0,\\kappa_1$, holds for all test functions without symmetry assumptions, and remains valid for extended-valued potentials.\n  We also characterize equality at an interior time: it holds "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.10769","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/2607.10769/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":"2607.10769","created_at":"2026-07-14T01:21:39.191861+00:00"},{"alias_kind":"arxiv_version","alias_value":"2607.10769v1","created_at":"2026-07-14T01:21:39.191861+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.10769","created_at":"2026-07-14T01:21:39.191861+00:00"},{"alias_kind":"pith_short_12","alias_value":"FBEDXNSQJCN4","created_at":"2026-07-14T01:21:39.191861+00:00"},{"alias_kind":"pith_short_16","alias_value":"FBEDXNSQJCN4MRP3","created_at":"2026-07-14T01:21:39.191861+00:00"},{"alias_kind":"pith_short_8","alias_value":"FBEDXNSQ","created_at":"2026-07-14T01:21:39.191861+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/FBEDXNSQJCN4MRP3BQL7QIGJVW","json":"https://pith.science/pith/FBEDXNSQJCN4MRP3BQL7QIGJVW.json","graph_json":"https://pith.science/api/pith-number/FBEDXNSQJCN4MRP3BQL7QIGJVW/graph.json","events_json":"https://pith.science/api/pith-number/FBEDXNSQJCN4MRP3BQL7QIGJVW/events.json","paper":"https://pith.science/paper/FBEDXNSQ"},"agent_actions":{"view_html":"https://pith.science/pith/FBEDXNSQJCN4MRP3BQL7QIGJVW","download_json":"https://pith.science/pith/FBEDXNSQJCN4MRP3BQL7QIGJVW.json","view_paper":"https://pith.science/paper/FBEDXNSQ","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2607.10769&json=true","fetch_graph":"https://pith.science/api/pith-number/FBEDXNSQJCN4MRP3BQL7QIGJVW/graph.json","fetch_events":"https://pith.science/api/pith-number/FBEDXNSQJCN4MRP3BQL7QIGJVW/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/FBEDXNSQJCN4MRP3BQL7QIGJVW/action/timestamp_anchor","attest_storage":"https://pith.science/pith/FBEDXNSQJCN4MRP3BQL7QIGJVW/action/storage_attestation","attest_author":"https://pith.science/pith/FBEDXNSQJCN4MRP3BQL7QIGJVW/action/author_attestation","sign_citation":"https://pith.science/pith/FBEDXNSQJCN4MRP3BQL7QIGJVW/action/citation_signature","submit_replication":"https://pith.science/pith/FBEDXNSQJCN4MRP3BQL7QIGJVW/action/replication_record"}},"created_at":"2026-07-14T01:21:39.191861+00:00","updated_at":"2026-07-14T01:21:39.191861+00:00"}