{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:5VE2ZROLXEKKON6N3Z7TNMGK4H","short_pith_number":"pith:5VE2ZROL","canonical_record":{"source":{"id":"2606.07325","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.ST","submitted_at":"2026-06-05T14:43:10Z","cross_cats_sorted":["cs.AI","cs.IT","math.IT","stat.TH"],"title_canon_sha256":"c9aff5a066589f59d904d8ccaacfee2983507ce3cffd9e4ec3d74c4df3a94c81","abstract_canon_sha256":"19fa89d75a8d0b49efa6d65ae378ba57726441e4b6a934b4aa9961f223b3df15"},"schema_version":"1.0"},"canonical_sha256":"ed49acc5cbb914a737cdde7f36b0cae1d8f6b22253d294261a40dd773a6d55c7","source":{"kind":"arxiv","id":"2606.07325","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2606.07325","created_at":"2026-06-08T01:05:19Z"},{"alias_kind":"arxiv_version","alias_value":"2606.07325v1","created_at":"2026-06-08T01:05:19Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.07325","created_at":"2026-06-08T01:05:19Z"},{"alias_kind":"pith_short_12","alias_value":"5VE2ZROLXEKK","created_at":"2026-06-08T01:05:19Z"},{"alias_kind":"pith_short_16","alias_value":"5VE2ZROLXEKKON6N","created_at":"2026-06-08T01:05:19Z"},{"alias_kind":"pith_short_8","alias_value":"5VE2ZROL","created_at":"2026-06-08T01:05:19Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:5VE2ZROLXEKKON6N3Z7TNMGK4H","target":"record","payload":{"canonical_record":{"source":{"id":"2606.07325","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.ST","submitted_at":"2026-06-05T14:43:10Z","cross_cats_sorted":["cs.AI","cs.IT","math.IT","stat.TH"],"title_canon_sha256":"c9aff5a066589f59d904d8ccaacfee2983507ce3cffd9e4ec3d74c4df3a94c81","abstract_canon_sha256":"19fa89d75a8d0b49efa6d65ae378ba57726441e4b6a934b4aa9961f223b3df15"},"schema_version":"1.0"},"canonical_sha256":"ed49acc5cbb914a737cdde7f36b0cae1d8f6b22253d294261a40dd773a6d55c7","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-08T01:05:19.632839Z","signature_b64":"/XQHZ7sjHMg2G9j+SLxH+nYJVQxXcG53+3QaDnmpDpLc1C9brgP1aM61AMlvdg1Ia8vtJCl4lSDsR4HnGgkUBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ed49acc5cbb914a737cdde7f36b0cae1d8f6b22253d294261a40dd773a6d55c7","last_reissued_at":"2026-06-08T01:05:19.631996Z","signature_status":"signed_v1","first_computed_at":"2026-06-08T01:05:19.631996Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2606.07325","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":"2026-06-08T01:05:19Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"wBFaI/BktQThuj0UjxM/ECD1QaYoOTCv4h5cWOQaPGLhOtcPe8jzYmLwVWuuopcenxoIiT6wDpnALXoRy7JOCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-29T21:58:57.854668Z"},"content_sha256":"f2ccfbf12a9afe0eb56fcd1bc0342626a73136ce2e28655c66036aea1ead1613","schema_version":"1.0","event_id":"sha256:f2ccfbf12a9afe0eb56fcd1bc0342626a73136ce2e28655c66036aea1ead1613"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:5VE2ZROLXEKKON6N3Z7TNMGK4H","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"A Temporal Spatial Minimax Rate for Smoothly-Varying Distributions in Wasserstein Space","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["cs.AI","cs.IT","math.IT","stat.TH"],"primary_cat":"math.ST","authors_text":"Munsik Kim","submitted_at":"2026-06-05T14:43:10Z","abstract_excerpt":"We study the minimax rate of estimating a future value $\\mu_{t_n+h}$ of a curve $t\\mapsto\\mu_t$ in the $2$-Wasserstein space $\\mathcal{P}_2(\\mathbb{R}^d)$ from finitely many noisy snapshots of its past, under an adiabatic bound $\\|\\nabla_t^k v\\|\\le\\varepsilon$ on the $k$-th covariant derivative of the velocity field. Our central result is a unified temporal-spatial minimax lower bound: over regular, locally transport-rich subclasses, every estimator incurs $W_2$-risk with $M$-exponent $\\gamma_d(k+1)/(k+1+\\gamma_d)$, $\\gamma_d=\\min(1/d,1/2)$ ($M$ the total sample size). It follows from a tempor"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.07325","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/2606.07325/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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-06-08T01:05:19Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"vUDwW7Vq5ayNwgvaw9k9y7eTDksjESHd6jh+da8yeUY6ZjZUyMZqUznG32FrB44Qyw2GF/HjTOa4QKNC+1pmAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-29T21:58:57.855067Z"},"content_sha256":"afecb1794016abfdc6d5d1cd447ec4c00ec70e2e8b1d1f9686e96fe9e57963b3","schema_version":"1.0","event_id":"sha256:afecb1794016abfdc6d5d1cd447ec4c00ec70e2e8b1d1f9686e96fe9e57963b3"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/5VE2ZROLXEKKON6N3Z7TNMGK4H/bundle.json","state_url":"https://pith.science/pith/5VE2ZROLXEKKON6N3Z7TNMGK4H/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/5VE2ZROLXEKKON6N3Z7TNMGK4H/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":"2026-06-29T21:58:57Z","links":{"resolver":"https://pith.science/pith/5VE2ZROLXEKKON6N3Z7TNMGK4H","bundle":"https://pith.science/pith/5VE2ZROLXEKKON6N3Z7TNMGK4H/bundle.json","state":"https://pith.science/pith/5VE2ZROLXEKKON6N3Z7TNMGK4H/state.json","well_known_bundle":"https://pith.science/.well-known/pith/5VE2ZROLXEKKON6N3Z7TNMGK4H/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:5VE2ZROLXEKKON6N3Z7TNMGK4H","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":"19fa89d75a8d0b49efa6d65ae378ba57726441e4b6a934b4aa9961f223b3df15","cross_cats_sorted":["cs.AI","cs.IT","math.IT","stat.TH"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.ST","submitted_at":"2026-06-05T14:43:10Z","title_canon_sha256":"c9aff5a066589f59d904d8ccaacfee2983507ce3cffd9e4ec3d74c4df3a94c81"},"schema_version":"1.0","source":{"id":"2606.07325","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2606.07325","created_at":"2026-06-08T01:05:19Z"},{"alias_kind":"arxiv_version","alias_value":"2606.07325v1","created_at":"2026-06-08T01:05:19Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.07325","created_at":"2026-06-08T01:05:19Z"},{"alias_kind":"pith_short_12","alias_value":"5VE2ZROLXEKK","created_at":"2026-06-08T01:05:19Z"},{"alias_kind":"pith_short_16","alias_value":"5VE2ZROLXEKKON6N","created_at":"2026-06-08T01:05:19Z"},{"alias_kind":"pith_short_8","alias_value":"5VE2ZROL","created_at":"2026-06-08T01:05:19Z"}],"graph_snapshots":[{"event_id":"sha256:afecb1794016abfdc6d5d1cd447ec4c00ec70e2e8b1d1f9686e96fe9e57963b3","target":"graph","created_at":"2026-06-08T01:05:19Z","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/2606.07325/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We study the minimax rate of estimating a future value $\\mu_{t_n+h}$ of a curve $t\\mapsto\\mu_t$ in the $2$-Wasserstein space $\\mathcal{P}_2(\\mathbb{R}^d)$ from finitely many noisy snapshots of its past, under an adiabatic bound $\\|\\nabla_t^k v\\|\\le\\varepsilon$ on the $k$-th covariant derivative of the velocity field. Our central result is a unified temporal-spatial minimax lower bound: over regular, locally transport-rich subclasses, every estimator incurs $W_2$-risk with $M$-exponent $\\gamma_d(k+1)/(k+1+\\gamma_d)$, $\\gamma_d=\\min(1/d,1/2)$ ($M$ the total sample size). It follows from a tempor","authors_text":"Munsik Kim","cross_cats":["cs.AI","cs.IT","math.IT","stat.TH"],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.ST","submitted_at":"2026-06-05T14:43:10Z","title":"A Temporal Spatial Minimax Rate for Smoothly-Varying Distributions in Wasserstein Space"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.07325","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:f2ccfbf12a9afe0eb56fcd1bc0342626a73136ce2e28655c66036aea1ead1613","target":"record","created_at":"2026-06-08T01:05:19Z","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":"19fa89d75a8d0b49efa6d65ae378ba57726441e4b6a934b4aa9961f223b3df15","cross_cats_sorted":["cs.AI","cs.IT","math.IT","stat.TH"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.ST","submitted_at":"2026-06-05T14:43:10Z","title_canon_sha256":"c9aff5a066589f59d904d8ccaacfee2983507ce3cffd9e4ec3d74c4df3a94c81"},"schema_version":"1.0","source":{"id":"2606.07325","kind":"arxiv","version":1}},"canonical_sha256":"ed49acc5cbb914a737cdde7f36b0cae1d8f6b22253d294261a40dd773a6d55c7","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"ed49acc5cbb914a737cdde7f36b0cae1d8f6b22253d294261a40dd773a6d55c7","first_computed_at":"2026-06-08T01:05:19.631996Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-06-08T01:05:19.631996Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"/XQHZ7sjHMg2G9j+SLxH+nYJVQxXcG53+3QaDnmpDpLc1C9brgP1aM61AMlvdg1Ia8vtJCl4lSDsR4HnGgkUBQ==","signature_status":"signed_v1","signed_at":"2026-06-08T01:05:19.632839Z","signed_message":"canonical_sha256_bytes"},"source_id":"2606.07325","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:f2ccfbf12a9afe0eb56fcd1bc0342626a73136ce2e28655c66036aea1ead1613","sha256:afecb1794016abfdc6d5d1cd447ec4c00ec70e2e8b1d1f9686e96fe9e57963b3"],"state_sha256":"2841cdeaec0ff131d151e629282ad28c75a1bd9df86004c3faca5d25bd33d5e5"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"RJegSPXmsf+KWbxRTGVIA6w56y72M5XMlqoMbeaLO0vPmM7LxnTZsh6mm4xyWFWsZwXwsy6S3XkblCaBrFVMDw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-29T21:58:57.857173Z","bundle_sha256":"ded4dc06fedf4760197f6c9a0883674553b96d6f49c83165b51a924df1cc53c6"}}