{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2025:TVVC6FN57N3552UQBTXKJADKCH","short_pith_number":"pith:TVVC6FN5","canonical_record":{"source":{"id":"2507.05395","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AP","submitted_at":"2025-07-07T18:29:24Z","cross_cats_sorted":["math.DG"],"title_canon_sha256":"75a155b4d8763245393d62ff3ec76e08ca948b2c6ef62358642c867aaeec903a","abstract_canon_sha256":"748c4f0394f1df3305c391c637dba8fc85aa7b11a509394f880593dfedc89d8b"},"schema_version":"1.0"},"canonical_sha256":"9d6a2f15bdfb77deea900ceea4806a11ca8b98d102350305b69406b6a485d472","source":{"kind":"arxiv","id":"2507.05395","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2507.05395","created_at":"2026-07-05T11:33:29Z"},{"alias_kind":"arxiv_version","alias_value":"2507.05395v1","created_at":"2026-07-05T11:33:29Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2507.05395","created_at":"2026-07-05T11:33:29Z"},{"alias_kind":"pith_short_12","alias_value":"TVVC6FN57N35","created_at":"2026-07-05T11:33:29Z"},{"alias_kind":"pith_short_16","alias_value":"TVVC6FN57N3552UQ","created_at":"2026-07-05T11:33:29Z"},{"alias_kind":"pith_short_8","alias_value":"TVVC6FN5","created_at":"2026-07-05T11:33:29Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2025:TVVC6FN57N3552UQBTXKJADKCH","target":"record","payload":{"canonical_record":{"source":{"id":"2507.05395","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AP","submitted_at":"2025-07-07T18:29:24Z","cross_cats_sorted":["math.DG"],"title_canon_sha256":"75a155b4d8763245393d62ff3ec76e08ca948b2c6ef62358642c867aaeec903a","abstract_canon_sha256":"748c4f0394f1df3305c391c637dba8fc85aa7b11a509394f880593dfedc89d8b"},"schema_version":"1.0"},"canonical_sha256":"9d6a2f15bdfb77deea900ceea4806a11ca8b98d102350305b69406b6a485d472","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T11:33:29.746496Z","signature_b64":"iV69kTyGJl4hT3wnNgAc8MfGPKtzD6X/vbv/KnU5FYIfyF6GyRXrVKgNVL7p29Lb17cGh5/7cyMlJJrlcjL2AA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9d6a2f15bdfb77deea900ceea4806a11ca8b98d102350305b69406b6a485d472","last_reissued_at":"2026-07-05T11:33:29.746080Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T11:33:29.746080Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2507.05395","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":"2026-07-05T11:33:29Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"8jpGctll4+y+2vFqvIyGayI9trt0G9jTWBBkJu6Y8j2iFfVLgJXHM0ACcVnq8Z1AXJ+oNK3116E2jgDDEfDwCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-09T23:51:56.263861Z"},"content_sha256":"73751f35adfce8c372d80aa7ea43236b01a65a7f7c05a9a80e2c7890c98ee19b","schema_version":"1.0","event_id":"sha256:73751f35adfce8c372d80aa7ea43236b01a65a7f7c05a9a80e2c7890c98ee19b"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2025:TVVC6FN57N3552UQBTXKJADKCH","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Boundary regularity of optimal transport maps on convex domains","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.DG"],"primary_cat":"math.AP","authors_text":"Freid Tong, Tristan C. Collins","submitted_at":"2025-07-07T18:29:24Z","abstract_excerpt":"We study the regularity of optimal transport maps between convex domains with quadratic cost. For nondegenerate $C^{\\alpha}$-densities, we prove $C^{1, 1-\\varepsilon}$-regularity of the potentials up to the boundary. If in addition the boundary is $C^{1, \\alpha}$, we improve this to $C^{2, \\alpha}$-regularity. We also investigate pointwise $C^{1, 1}$-regularity at boundary points. We obtain a complete characterization of pointwise $C^{1, 1}$-regularity for planar polytopes in terms of the geometry of tangent cones. Furthermore, we study the regularity of optimal transport maps with degenerate "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2507.05395","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/2507.05395/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":"2026-07-05T11:33:29Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"+wZ+S7xBisfzpoMi9nwlHCEUUqLLnLJqrcIkvPgtijkJuHmT1QLUJW71NVZX8+HSJrkxUqaOPW0e63xisplSAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-09T23:51:56.264455Z"},"content_sha256":"867cf6979045b3c808b84685aa320792265597e0550a676c5adbbef59ffb508e","schema_version":"1.0","event_id":"sha256:867cf6979045b3c808b84685aa320792265597e0550a676c5adbbef59ffb508e"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/TVVC6FN57N3552UQBTXKJADKCH/bundle.json","state_url":"https://pith.science/pith/TVVC6FN57N3552UQBTXKJADKCH/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/TVVC6FN57N3552UQBTXKJADKCH/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":"2026-08-09T23:51:56Z","links":{"resolver":"https://pith.science/pith/TVVC6FN57N3552UQBTXKJADKCH","bundle":"https://pith.science/pith/TVVC6FN57N3552UQBTXKJADKCH/bundle.json","state":"https://pith.science/pith/TVVC6FN57N3552UQBTXKJADKCH/state.json","well_known_bundle":"https://pith.science/.well-known/pith/TVVC6FN57N3552UQBTXKJADKCH/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2025:TVVC6FN57N3552UQBTXKJADKCH","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":"748c4f0394f1df3305c391c637dba8fc85aa7b11a509394f880593dfedc89d8b","cross_cats_sorted":["math.DG"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AP","submitted_at":"2025-07-07T18:29:24Z","title_canon_sha256":"75a155b4d8763245393d62ff3ec76e08ca948b2c6ef62358642c867aaeec903a"},"schema_version":"1.0","source":{"id":"2507.05395","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2507.05395","created_at":"2026-07-05T11:33:29Z"},{"alias_kind":"arxiv_version","alias_value":"2507.05395v1","created_at":"2026-07-05T11:33:29Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2507.05395","created_at":"2026-07-05T11:33:29Z"},{"alias_kind":"pith_short_12","alias_value":"TVVC6FN57N35","created_at":"2026-07-05T11:33:29Z"},{"alias_kind":"pith_short_16","alias_value":"TVVC6FN57N3552UQ","created_at":"2026-07-05T11:33:29Z"},{"alias_kind":"pith_short_8","alias_value":"TVVC6FN5","created_at":"2026-07-05T11:33:29Z"}],"graph_snapshots":[{"event_id":"sha256:867cf6979045b3c808b84685aa320792265597e0550a676c5adbbef59ffb508e","target":"graph","created_at":"2026-07-05T11:33:29Z","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/2507.05395/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We study the regularity of optimal transport maps between convex domains with quadratic cost. For nondegenerate $C^{\\alpha}$-densities, we prove $C^{1, 1-\\varepsilon}$-regularity of the potentials up to the boundary. If in addition the boundary is $C^{1, \\alpha}$, we improve this to $C^{2, \\alpha}$-regularity. We also investigate pointwise $C^{1, 1}$-regularity at boundary points. We obtain a complete characterization of pointwise $C^{1, 1}$-regularity for planar polytopes in terms of the geometry of tangent cones. Furthermore, we study the regularity of optimal transport maps with degenerate ","authors_text":"Freid Tong, Tristan C. Collins","cross_cats":["math.DG"],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AP","submitted_at":"2025-07-07T18:29:24Z","title":"Boundary regularity of optimal transport maps on convex domains"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2507.05395","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":"sha256:73751f35adfce8c372d80aa7ea43236b01a65a7f7c05a9a80e2c7890c98ee19b","target":"record","created_at":"2026-07-05T11:33:29Z","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":"748c4f0394f1df3305c391c637dba8fc85aa7b11a509394f880593dfedc89d8b","cross_cats_sorted":["math.DG"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AP","submitted_at":"2025-07-07T18:29:24Z","title_canon_sha256":"75a155b4d8763245393d62ff3ec76e08ca948b2c6ef62358642c867aaeec903a"},"schema_version":"1.0","source":{"id":"2507.05395","kind":"arxiv","version":1}},"canonical_sha256":"9d6a2f15bdfb77deea900ceea4806a11ca8b98d102350305b69406b6a485d472","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"9d6a2f15bdfb77deea900ceea4806a11ca8b98d102350305b69406b6a485d472","first_computed_at":"2026-07-05T11:33:29.746080Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T11:33:29.746080Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"iV69kTyGJl4hT3wnNgAc8MfGPKtzD6X/vbv/KnU5FYIfyF6GyRXrVKgNVL7p29Lb17cGh5/7cyMlJJrlcjL2AA==","signature_status":"signed_v1","signed_at":"2026-07-05T11:33:29.746496Z","signed_message":"canonical_sha256_bytes"},"source_id":"2507.05395","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:73751f35adfce8c372d80aa7ea43236b01a65a7f7c05a9a80e2c7890c98ee19b","sha256:867cf6979045b3c808b84685aa320792265597e0550a676c5adbbef59ffb508e"],"state_sha256":"c867c571fd915a4aede66c19aa12312717bacecbc73303a3a19c05de0b1aa667"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"awdEga/l8jctyRiTmCp0fsf9EVCxF1TUsKnO75PQrA7eI3XiXU0lEPG4Wb7tMhtZ6xaMsVRzDd8X8sU3dTfaBQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-09T23:51:56.269493Z","bundle_sha256":"942ee0381d7280753d59c0ea9cea4b1b96233d99c67d5241d3350d362623e3ec"}}