{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:2ZLRS2OHP46UQQ4TIKRSHFE2W7","short_pith_number":"pith:2ZLRS2OH","canonical_record":{"source":{"id":"2605.30756","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2026-05-29T02:37:15Z","cross_cats_sorted":[],"title_canon_sha256":"9da88a6bd501185739392dd52b6f7cec9d65b5cf3365762a0636b2295da82e9b","abstract_canon_sha256":"323ad1073be9bf3afcb0bc11575d26d4233f125e554e21d19ce895c1718457f4"},"schema_version":"1.0"},"canonical_sha256":"d6571969c77f3d48439342a323949ab7ee2b0a1cd464b91047cddfa17d345bd7","source":{"kind":"arxiv","id":"2605.30756","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2605.30756","created_at":"2026-06-01T01:03:14Z"},{"alias_kind":"arxiv_version","alias_value":"2605.30756v1","created_at":"2026-06-01T01:03:14Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2605.30756","created_at":"2026-06-01T01:03:14Z"},{"alias_kind":"pith_short_12","alias_value":"2ZLRS2OHP46U","created_at":"2026-06-01T01:03:14Z"},{"alias_kind":"pith_short_16","alias_value":"2ZLRS2OHP46UQQ4T","created_at":"2026-06-01T01:03:14Z"},{"alias_kind":"pith_short_8","alias_value":"2ZLRS2OH","created_at":"2026-06-01T01:03:14Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:2ZLRS2OHP46UQQ4TIKRSHFE2W7","target":"record","payload":{"canonical_record":{"source":{"id":"2605.30756","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2026-05-29T02:37:15Z","cross_cats_sorted":[],"title_canon_sha256":"9da88a6bd501185739392dd52b6f7cec9d65b5cf3365762a0636b2295da82e9b","abstract_canon_sha256":"323ad1073be9bf3afcb0bc11575d26d4233f125e554e21d19ce895c1718457f4"},"schema_version":"1.0"},"canonical_sha256":"d6571969c77f3d48439342a323949ab7ee2b0a1cd464b91047cddfa17d345bd7","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-01T01:03:14.624325Z","signature_b64":"P5k3GFFPHWYGf0t3j66zOegZmRk1tAljczWM+NAnY2gIw40tOIBZTOp0+RIUpX70Dm9lqPWy+08OvEPcEJWIBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d6571969c77f3d48439342a323949ab7ee2b0a1cd464b91047cddfa17d345bd7","last_reissued_at":"2026-06-01T01:03:14.623740Z","signature_status":"signed_v1","first_computed_at":"2026-06-01T01:03:14.623740Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2605.30756","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":"2026-06-01T01:03:14Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"FFzce0YmFpVHN9qX8bhgJEJuboRizZKR9W0CWZpDwhxcvtJR7rn01TQMw6tkj7EtRwowUS6qlhBzBhE6Ci2EDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-27T22:26:41.743232Z"},"content_sha256":"45744af0fa17327779b846cbaf0edba38a525838258eabd23c89e48a7857d198","schema_version":"1.0","event_id":"sha256:45744af0fa17327779b846cbaf0edba38a525838258eabd23c89e48a7857d198"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:2ZLRS2OHP46UQQ4TIKRSHFE2W7","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"A Liouville theorem for bounded empirical-harmonic functions on $\\mathcal{P}_2(M)$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Hongwei Yuan","submitted_at":"2026-05-29T02:37:15Z","abstract_excerpt":"The classical Liouville theorem states that every bounded harmonic function on Euclidean space is constant. On complete Rie mannian manifolds, analogous conclusions hold under geometric as sumptions such as nonnegative Ricci curvature. The quadratic Wasserstein space $\\mathcal{P}_2(M)$ is often regarded (following Otto) as an infinite-dimensional Riemannian manifold; however, it has no canonical infinite-dimensional Riemannian volume and therefore no canonica Laplace--Beltrami operator. In this paper we introduce a canonical finite-dimensional trace notion of harmonicity: a continuous function"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.30756","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/2605.30756/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":"2026-06-01T01:03:14Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"bMRSUIATQ5/uX7CzMYE/vm9fRSF5d4jvYsWc6ROGI+F/fJxNZ77MIYPVfgc4WlulZ/XzszT6Csk+CNegveSyDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-27T22:26:41.743633Z"},"content_sha256":"b3177c6054072059cb2b8c99365c317345d4b7b3ebf537d51695aeee3da9cdfb","schema_version":"1.0","event_id":"sha256:b3177c6054072059cb2b8c99365c317345d4b7b3ebf537d51695aeee3da9cdfb"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/2ZLRS2OHP46UQQ4TIKRSHFE2W7/bundle.json","state_url":"https://pith.science/pith/2ZLRS2OHP46UQQ4TIKRSHFE2W7/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/2ZLRS2OHP46UQQ4TIKRSHFE2W7/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":"2026-06-27T22:26:41Z","links":{"resolver":"https://pith.science/pith/2ZLRS2OHP46UQQ4TIKRSHFE2W7","bundle":"https://pith.science/pith/2ZLRS2OHP46UQQ4TIKRSHFE2W7/bundle.json","state":"https://pith.science/pith/2ZLRS2OHP46UQQ4TIKRSHFE2W7/state.json","well_known_bundle":"https://pith.science/.well-known/pith/2ZLRS2OHP46UQQ4TIKRSHFE2W7/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:2ZLRS2OHP46UQQ4TIKRSHFE2W7","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":"323ad1073be9bf3afcb0bc11575d26d4233f125e554e21d19ce895c1718457f4","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2026-05-29T02:37:15Z","title_canon_sha256":"9da88a6bd501185739392dd52b6f7cec9d65b5cf3365762a0636b2295da82e9b"},"schema_version":"1.0","source":{"id":"2605.30756","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2605.30756","created_at":"2026-06-01T01:03:14Z"},{"alias_kind":"arxiv_version","alias_value":"2605.30756v1","created_at":"2026-06-01T01:03:14Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2605.30756","created_at":"2026-06-01T01:03:14Z"},{"alias_kind":"pith_short_12","alias_value":"2ZLRS2OHP46U","created_at":"2026-06-01T01:03:14Z"},{"alias_kind":"pith_short_16","alias_value":"2ZLRS2OHP46UQQ4T","created_at":"2026-06-01T01:03:14Z"},{"alias_kind":"pith_short_8","alias_value":"2ZLRS2OH","created_at":"2026-06-01T01:03:14Z"}],"graph_snapshots":[{"event_id":"sha256:b3177c6054072059cb2b8c99365c317345d4b7b3ebf537d51695aeee3da9cdfb","target":"graph","created_at":"2026-06-01T01:03:14Z","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/2605.30756/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"The classical Liouville theorem states that every bounded harmonic function on Euclidean space is constant. On complete Rie mannian manifolds, analogous conclusions hold under geometric as sumptions such as nonnegative Ricci curvature. The quadratic Wasserstein space $\\mathcal{P}_2(M)$ is often regarded (following Otto) as an infinite-dimensional Riemannian manifold; however, it has no canonical infinite-dimensional Riemannian volume and therefore no canonica Laplace--Beltrami operator. In this paper we introduce a canonical finite-dimensional trace notion of harmonicity: a continuous function","authors_text":"Hongwei Yuan","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2026-05-29T02:37:15Z","title":"A Liouville theorem for bounded empirical-harmonic functions on $\\mathcal{P}_2(M)$"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.30756","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":"sha256:45744af0fa17327779b846cbaf0edba38a525838258eabd23c89e48a7857d198","target":"record","created_at":"2026-06-01T01:03:14Z","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":"323ad1073be9bf3afcb0bc11575d26d4233f125e554e21d19ce895c1718457f4","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2026-05-29T02:37:15Z","title_canon_sha256":"9da88a6bd501185739392dd52b6f7cec9d65b5cf3365762a0636b2295da82e9b"},"schema_version":"1.0","source":{"id":"2605.30756","kind":"arxiv","version":1}},"canonical_sha256":"d6571969c77f3d48439342a323949ab7ee2b0a1cd464b91047cddfa17d345bd7","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"d6571969c77f3d48439342a323949ab7ee2b0a1cd464b91047cddfa17d345bd7","first_computed_at":"2026-06-01T01:03:14.623740Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-06-01T01:03:14.623740Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"P5k3GFFPHWYGf0t3j66zOegZmRk1tAljczWM+NAnY2gIw40tOIBZTOp0+RIUpX70Dm9lqPWy+08OvEPcEJWIBA==","signature_status":"signed_v1","signed_at":"2026-06-01T01:03:14.624325Z","signed_message":"canonical_sha256_bytes"},"source_id":"2605.30756","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:45744af0fa17327779b846cbaf0edba38a525838258eabd23c89e48a7857d198","sha256:b3177c6054072059cb2b8c99365c317345d4b7b3ebf537d51695aeee3da9cdfb"],"state_sha256":"d696d402725860867b56dfd210d0c2d07072bf79238102b71a2bb202ea0cb895"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"PXh/KxZBA+TZqUJxiBT9lyYlUIHdAdBG6BOFXRLx3K7dwxLnOIaFM7U1E/B1OrjQ1JkYVl44geiLY71lWdCFAQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-27T22:26:41.745747Z","bundle_sha256":"387b930ba0ad9fb32d5707a9e69c3f444406777a80b4c442c98f96ad2f684a2d"}}