{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:PPUPWQHKJQGXLEKTOLBF5LIJVS","short_pith_number":"pith:PPUPWQHK","canonical_record":{"source":{"id":"2607.13209","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.ST","submitted_at":"2026-07-14T19:01:12Z","cross_cats_sorted":["stat.TH"],"title_canon_sha256":"463b09b61143d27ab3be65f3b1e0783c16cb4a1136e8bb7ecda99cc03b6845bf","abstract_canon_sha256":"c78fcf06fee9fc38a1b70123b1298a690448b52b03d7235eb7b3d439269389c6"},"schema_version":"1.0"},"canonical_sha256":"7be8fb40ea4c0d75915372c25ead09ac9403792f6231dbe59858bd64ccbd91ab","source":{"kind":"arxiv","id":"2607.13209","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2607.13209","created_at":"2026-07-16T00:22:02Z"},{"alias_kind":"arxiv_version","alias_value":"2607.13209v1","created_at":"2026-07-16T00:22:02Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.13209","created_at":"2026-07-16T00:22:02Z"},{"alias_kind":"pith_short_12","alias_value":"PPUPWQHKJQGX","created_at":"2026-07-16T00:22:02Z"},{"alias_kind":"pith_short_16","alias_value":"PPUPWQHKJQGXLEKT","created_at":"2026-07-16T00:22:02Z"},{"alias_kind":"pith_short_8","alias_value":"PPUPWQHK","created_at":"2026-07-16T00:22:02Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:PPUPWQHKJQGXLEKTOLBF5LIJVS","target":"record","payload":{"canonical_record":{"source":{"id":"2607.13209","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.ST","submitted_at":"2026-07-14T19:01:12Z","cross_cats_sorted":["stat.TH"],"title_canon_sha256":"463b09b61143d27ab3be65f3b1e0783c16cb4a1136e8bb7ecda99cc03b6845bf","abstract_canon_sha256":"c78fcf06fee9fc38a1b70123b1298a690448b52b03d7235eb7b3d439269389c6"},"schema_version":"1.0"},"canonical_sha256":"7be8fb40ea4c0d75915372c25ead09ac9403792f6231dbe59858bd64ccbd91ab","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-16T00:22:02.942459Z","signature_b64":"t+4Tr1Id+YvJOiaTOA+1KhaC+0vmupxkdsJxwWCGC8J2A6BrsxIwENkeX7hCtDks4Th2Jlw9T6Svzx4oVTG0AQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7be8fb40ea4c0d75915372c25ead09ac9403792f6231dbe59858bd64ccbd91ab","last_reissued_at":"2026-07-16T00:22:02.941599Z","signature_status":"signed_v1","first_computed_at":"2026-07-16T00:22:02.941599Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2607.13209","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":"2026-07-16T00:22:02Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"zsqaJIa5dos/+HZeAE+UkIXcoLmKgQuqrsB5qu5EmytyRfD+4OGv1c+PC8x3+0lh/AdQXeymUn4gAnRmYlXYCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-06T11:51:27.897213Z"},"content_sha256":"109b3ad9329e72ade98c3e0ce41378c019df6cdb87f8a52b9cb75e3af7eb4f6f","schema_version":"1.0","event_id":"sha256:109b3ad9329e72ade98c3e0ce41378c019df6cdb87f8a52b9cb75e3af7eb4f6f"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:PPUPWQHKJQGXLEKTOLBF5LIJVS","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"A unified approach for testing in Hilbert spaces on incomplete data","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["stat.TH"],"primary_cat":"math.ST","authors_text":"Daniel Gaigall, Philipp W\\\"ubbolding","submitted_at":"2026-07-14T19:01:12Z","abstract_excerpt":"We consider statistical testing on the basis of incomplete observations with values in a separable Hilbert space, where the dimension is possibly large or even infinite. The general Hilbert space setting allows various data types as they arise in modern applications, in particular high dimensional and functional data. Possible Hilbert space testing problems are goodness-of-fit, symmetry, homogeneity and independence. We present an approach for modeling incomplete data that covers several problems in practice, e.g., ultra high dimensional random vectors with missing entries or partially observe"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.13209","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/2607.13209/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":"2026-07-16T00:22:02Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"FTS4AdEshSgazZFU1bQJ8mbquI/x8DDJbB+/HDYlvM30Jc6xYYbjMnXOqpbQsdr5U6xBILi4E4aG2zOBkeq/AA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-06T11:51:27.897758Z"},"content_sha256":"3f0a3a8e29b6446348299d6e3bef3b9017b3b14f32167c1d7a1d77178ae324fc","schema_version":"1.0","event_id":"sha256:3f0a3a8e29b6446348299d6e3bef3b9017b3b14f32167c1d7a1d77178ae324fc"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/PPUPWQHKJQGXLEKTOLBF5LIJVS/bundle.json","state_url":"https://pith.science/pith/PPUPWQHKJQGXLEKTOLBF5LIJVS/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/PPUPWQHKJQGXLEKTOLBF5LIJVS/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":"2026-08-06T11:51:27Z","links":{"resolver":"https://pith.science/pith/PPUPWQHKJQGXLEKTOLBF5LIJVS","bundle":"https://pith.science/pith/PPUPWQHKJQGXLEKTOLBF5LIJVS/bundle.json","state":"https://pith.science/pith/PPUPWQHKJQGXLEKTOLBF5LIJVS/state.json","well_known_bundle":"https://pith.science/.well-known/pith/PPUPWQHKJQGXLEKTOLBF5LIJVS/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:PPUPWQHKJQGXLEKTOLBF5LIJVS","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":"c78fcf06fee9fc38a1b70123b1298a690448b52b03d7235eb7b3d439269389c6","cross_cats_sorted":["stat.TH"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.ST","submitted_at":"2026-07-14T19:01:12Z","title_canon_sha256":"463b09b61143d27ab3be65f3b1e0783c16cb4a1136e8bb7ecda99cc03b6845bf"},"schema_version":"1.0","source":{"id":"2607.13209","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2607.13209","created_at":"2026-07-16T00:22:02Z"},{"alias_kind":"arxiv_version","alias_value":"2607.13209v1","created_at":"2026-07-16T00:22:02Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.13209","created_at":"2026-07-16T00:22:02Z"},{"alias_kind":"pith_short_12","alias_value":"PPUPWQHKJQGX","created_at":"2026-07-16T00:22:02Z"},{"alias_kind":"pith_short_16","alias_value":"PPUPWQHKJQGXLEKT","created_at":"2026-07-16T00:22:02Z"},{"alias_kind":"pith_short_8","alias_value":"PPUPWQHK","created_at":"2026-07-16T00:22:02Z"}],"graph_snapshots":[{"event_id":"sha256:3f0a3a8e29b6446348299d6e3bef3b9017b3b14f32167c1d7a1d77178ae324fc","target":"graph","created_at":"2026-07-16T00:22:02Z","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/2607.13209/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We consider statistical testing on the basis of incomplete observations with values in a separable Hilbert space, where the dimension is possibly large or even infinite. The general Hilbert space setting allows various data types as they arise in modern applications, in particular high dimensional and functional data. Possible Hilbert space testing problems are goodness-of-fit, symmetry, homogeneity and independence. We present an approach for modeling incomplete data that covers several problems in practice, e.g., ultra high dimensional random vectors with missing entries or partially observe","authors_text":"Daniel Gaigall, Philipp W\\\"ubbolding","cross_cats":["stat.TH"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.ST","submitted_at":"2026-07-14T19:01:12Z","title":"A unified approach for testing in Hilbert spaces on incomplete data"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.13209","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":"sha256:109b3ad9329e72ade98c3e0ce41378c019df6cdb87f8a52b9cb75e3af7eb4f6f","target":"record","created_at":"2026-07-16T00:22:02Z","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":"c78fcf06fee9fc38a1b70123b1298a690448b52b03d7235eb7b3d439269389c6","cross_cats_sorted":["stat.TH"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.ST","submitted_at":"2026-07-14T19:01:12Z","title_canon_sha256":"463b09b61143d27ab3be65f3b1e0783c16cb4a1136e8bb7ecda99cc03b6845bf"},"schema_version":"1.0","source":{"id":"2607.13209","kind":"arxiv","version":1}},"canonical_sha256":"7be8fb40ea4c0d75915372c25ead09ac9403792f6231dbe59858bd64ccbd91ab","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"7be8fb40ea4c0d75915372c25ead09ac9403792f6231dbe59858bd64ccbd91ab","first_computed_at":"2026-07-16T00:22:02.941599Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-16T00:22:02.941599Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"t+4Tr1Id+YvJOiaTOA+1KhaC+0vmupxkdsJxwWCGC8J2A6BrsxIwENkeX7hCtDks4Th2Jlw9T6Svzx4oVTG0AQ==","signature_status":"signed_v1","signed_at":"2026-07-16T00:22:02.942459Z","signed_message":"canonical_sha256_bytes"},"source_id":"2607.13209","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:109b3ad9329e72ade98c3e0ce41378c019df6cdb87f8a52b9cb75e3af7eb4f6f","sha256:3f0a3a8e29b6446348299d6e3bef3b9017b3b14f32167c1d7a1d77178ae324fc"],"state_sha256":"025d187d214b31280eb3192dac09469a690b38ac6b5da761d51d5e5814ff466a"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"NK48tHHfb+3dY4dJ2SlhHPdLUME7UGbGRHYNojhOB/8FK557MBMjko2mZ2DJrLuyrAw5uESR6yFDDnjQl8naCw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-06T11:51:27.902788Z","bundle_sha256":"1653ec975fcc094f93c4df721de971919c5ec6498017830094f0e71821e6a345"}}