{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:KEKPP6HGLNYIB2K5Z7SUUHOMTF","short_pith_number":"pith:KEKPP6HG","canonical_record":{"source":{"id":"2607.27121","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2026-07-29T16:47:29Z","cross_cats_sorted":["math.FA","math.MG"],"title_canon_sha256":"40d136f3a23b46cf978cfa91aea49310232d7293f8254968b3342d1060518839","abstract_canon_sha256":"06bafed73a6127615076a5355360ad66f5e5bda497f8b5fa6264a55f172826f9"},"schema_version":"1.0"},"canonical_sha256":"5114f7f8e65b7080e95dcfe54a1dcc9953622d1d7d282f1d3eda9eca82716bda","source":{"kind":"arxiv","id":"2607.27121","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2607.27121","created_at":"2026-07-30T01:23:38Z"},{"alias_kind":"arxiv_version","alias_value":"2607.27121v1","created_at":"2026-07-30T01:23:38Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.27121","created_at":"2026-07-30T01:23:38Z"},{"alias_kind":"pith_short_12","alias_value":"KEKPP6HGLNYI","created_at":"2026-07-30T01:23:38Z"},{"alias_kind":"pith_short_16","alias_value":"KEKPP6HGLNYIB2K5","created_at":"2026-07-30T01:23:38Z"},{"alias_kind":"pith_short_8","alias_value":"KEKPP6HG","created_at":"2026-07-30T01:23:38Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:KEKPP6HGLNYIB2K5Z7SUUHOMTF","target":"record","payload":{"canonical_record":{"source":{"id":"2607.27121","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2026-07-29T16:47:29Z","cross_cats_sorted":["math.FA","math.MG"],"title_canon_sha256":"40d136f3a23b46cf978cfa91aea49310232d7293f8254968b3342d1060518839","abstract_canon_sha256":"06bafed73a6127615076a5355360ad66f5e5bda497f8b5fa6264a55f172826f9"},"schema_version":"1.0"},"canonical_sha256":"5114f7f8e65b7080e95dcfe54a1dcc9953622d1d7d282f1d3eda9eca82716bda","receipt":{"kind":"pith_receipt","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5114f7f8e65b7080e95dcfe54a1dcc9953622d1d7d282f1d3eda9eca82716bda","last_reissued_at":"2026-07-30T01:23:38.257804Z","signature_status":"unsigned_v0","first_computed_at":"2026-07-30T01:23:38.257804Z"},"source_kind":"arxiv","source_id":"2607.27121","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":"2026-07-30T01:23:38Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"AXYXjL1kOxcFxoGXukkV7Oh9/ala6i+xvhKMDHGn6eXmgCMlznQWhX46IBthvhIeda2BTimazgRTD+BB2/oWAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-13T06:19:33.746751Z"},"content_sha256":"2e4af59eed1045a2a3c727cccc93ce94fa363029a545f47f1a6dca86546d9d73","schema_version":"1.0","event_id":"sha256:2e4af59eed1045a2a3c727cccc93ce94fa363029a545f47f1a6dca86546d9d73"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:KEKPP6HGLNYIB2K5Z7SUUHOMTF","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"A 'global' Perspective on the Differential Geometry of Wasserstein Spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.FA","math.MG"],"primary_cat":"math.DG","authors_text":"Andrea Pinamonti, Lorenzo Dello Schiavo","submitted_at":"2026-07-29T16:47:29Z","abstract_excerpt":"We develop a global differential calculus on the $L^2$-Wasserstein space over a closed Riemannian manifold, based on derivations of cylinder functions rather than on the standard pointwise approach. Within this framework we define some fundamental geometric tools. In particular, we show that the Levi-Civita connection on the base manifold lifts to the unique torsion-free connection compatible with the 'extended' Otto metric. The corresponding Riemann tensor is exactly the lift of the base Riemann tensor, which shows that - in this framework - the correction terms of the classical gradient form"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.27121","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/2607.27121/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":"2026-07-30T01:23:38Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"UvBWoPujh0Ehd8I2TPpOctJLz4imdjSMpIsDO+ipeklSbHcJy4O/eXPALc0abr20ALPNVIQC5Gse2WIxQ3kjBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-13T06:19:33.747327Z"},"content_sha256":"08b8abb1896f6a633e343ea4dcd10a24ef9e3f015ffdcc20795369f18a3ee0b4","schema_version":"1.0","event_id":"sha256:08b8abb1896f6a633e343ea4dcd10a24ef9e3f015ffdcc20795369f18a3ee0b4"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/KEKPP6HGLNYIB2K5Z7SUUHOMTF/bundle.json","state_url":"https://pith.science/pith/KEKPP6HGLNYIB2K5Z7SUUHOMTF/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/KEKPP6HGLNYIB2K5Z7SUUHOMTF/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":"2026-08-13T06:19:33Z","links":{"resolver":"https://pith.science/pith/KEKPP6HGLNYIB2K5Z7SUUHOMTF","bundle":"https://pith.science/pith/KEKPP6HGLNYIB2K5Z7SUUHOMTF/bundle.json","state":"https://pith.science/pith/KEKPP6HGLNYIB2K5Z7SUUHOMTF/state.json","well_known_bundle":"https://pith.science/.well-known/pith/KEKPP6HGLNYIB2K5Z7SUUHOMTF/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:KEKPP6HGLNYIB2K5Z7SUUHOMTF","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":"06bafed73a6127615076a5355360ad66f5e5bda497f8b5fa6264a55f172826f9","cross_cats_sorted":["math.FA","math.MG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2026-07-29T16:47:29Z","title_canon_sha256":"40d136f3a23b46cf978cfa91aea49310232d7293f8254968b3342d1060518839"},"schema_version":"1.0","source":{"id":"2607.27121","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2607.27121","created_at":"2026-07-30T01:23:38Z"},{"alias_kind":"arxiv_version","alias_value":"2607.27121v1","created_at":"2026-07-30T01:23:38Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.27121","created_at":"2026-07-30T01:23:38Z"},{"alias_kind":"pith_short_12","alias_value":"KEKPP6HGLNYI","created_at":"2026-07-30T01:23:38Z"},{"alias_kind":"pith_short_16","alias_value":"KEKPP6HGLNYIB2K5","created_at":"2026-07-30T01:23:38Z"},{"alias_kind":"pith_short_8","alias_value":"KEKPP6HG","created_at":"2026-07-30T01:23:38Z"}],"graph_snapshots":[{"event_id":"sha256:08b8abb1896f6a633e343ea4dcd10a24ef9e3f015ffdcc20795369f18a3ee0b4","target":"graph","created_at":"2026-07-30T01:23:38Z","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/2607.27121/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We develop a global differential calculus on the $L^2$-Wasserstein space over a closed Riemannian manifold, based on derivations of cylinder functions rather than on the standard pointwise approach. Within this framework we define some fundamental geometric tools. In particular, we show that the Levi-Civita connection on the base manifold lifts to the unique torsion-free connection compatible with the 'extended' Otto metric. The corresponding Riemann tensor is exactly the lift of the base Riemann tensor, which shows that - in this framework - the correction terms of the classical gradient form","authors_text":"Andrea Pinamonti, Lorenzo Dello Schiavo","cross_cats":["math.FA","math.MG"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2026-07-29T16:47:29Z","title":"A 'global' Perspective on the Differential Geometry of Wasserstein Spaces"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.27121","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":"sha256:2e4af59eed1045a2a3c727cccc93ce94fa363029a545f47f1a6dca86546d9d73","target":"record","created_at":"2026-07-30T01:23:38Z","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":"06bafed73a6127615076a5355360ad66f5e5bda497f8b5fa6264a55f172826f9","cross_cats_sorted":["math.FA","math.MG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2026-07-29T16:47:29Z","title_canon_sha256":"40d136f3a23b46cf978cfa91aea49310232d7293f8254968b3342d1060518839"},"schema_version":"1.0","source":{"id":"2607.27121","kind":"arxiv","version":1}},"canonical_sha256":"5114f7f8e65b7080e95dcfe54a1dcc9953622d1d7d282f1d3eda9eca82716bda","receipt":{"builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"5114f7f8e65b7080e95dcfe54a1dcc9953622d1d7d282f1d3eda9eca82716bda","first_computed_at":"2026-07-30T01:23:38.257804Z","kind":"pith_receipt","last_reissued_at":"2026-07-30T01:23:38.257804Z","receipt_version":"0.3","signature_status":"unsigned_v0"},"source_id":"2607.27121","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:2e4af59eed1045a2a3c727cccc93ce94fa363029a545f47f1a6dca86546d9d73","sha256:08b8abb1896f6a633e343ea4dcd10a24ef9e3f015ffdcc20795369f18a3ee0b4"],"state_sha256":"615877121804c938b839599ef9cedbf1caa65880c542db6ba182963f7fa33b16"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"PD8n45cLvsZ7Iokvj/EErGpYLRd+LOm475rXzyTIRkoLL7UN3eOlu6zZORQrdmFV1yxqC0sa2tl0h7FruFqFAg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-13T06:19:33.751352Z","bundle_sha256":"88633837aa9f5503ab85bb1948f7aa80fb284a77d98e19a03923210073171cef"}}