{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2024:DXKZV27SJLJS4VX3KOALTP24HY","short_pith_number":"pith:DXKZV27S","canonical_record":{"source":{"id":"2401.08000","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2024-01-15T22:58:37Z","cross_cats_sorted":["math.LO","math.OA"],"title_canon_sha256":"f395b726e15e97fc52e6ee82687df7df54c3a7b4a72c1925bc9da2517c492813","abstract_canon_sha256":"52dcc3ab34629be05a1f6c6250d1bb7b45632c8f15684de2c1b649b143024b45"},"schema_version":"1.0"},"canonical_sha256":"1dd59aebf24ad32e56fb5380b9bf5c3e23e823e199ae679d0f959b16291c1cbe","source":{"kind":"arxiv","id":"2401.08000","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2401.08000","created_at":"2026-07-05T07:34:05Z"},{"alias_kind":"arxiv_version","alias_value":"2401.08000v1","created_at":"2026-07-05T07:34:05Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2401.08000","created_at":"2026-07-05T07:34:05Z"},{"alias_kind":"pith_short_12","alias_value":"DXKZV27SJLJS","created_at":"2026-07-05T07:34:05Z"},{"alias_kind":"pith_short_16","alias_value":"DXKZV27SJLJS4VX3","created_at":"2026-07-05T07:34:05Z"},{"alias_kind":"pith_short_8","alias_value":"DXKZV27S","created_at":"2026-07-05T07:34:05Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2024:DXKZV27SJLJS4VX3KOALTP24HY","target":"record","payload":{"canonical_record":{"source":{"id":"2401.08000","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2024-01-15T22:58:37Z","cross_cats_sorted":["math.LO","math.OA"],"title_canon_sha256":"f395b726e15e97fc52e6ee82687df7df54c3a7b4a72c1925bc9da2517c492813","abstract_canon_sha256":"52dcc3ab34629be05a1f6c6250d1bb7b45632c8f15684de2c1b649b143024b45"},"schema_version":"1.0"},"canonical_sha256":"1dd59aebf24ad32e56fb5380b9bf5c3e23e823e199ae679d0f959b16291c1cbe","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T07:34:05.830052Z","signature_b64":"DazNm+hjoQuzSvOOU4dsCc9xJIGZ8POlhQ6P5bu3tx9L8PGCZpX4H4l1wPtFqxDf/1XXqLqa4K2TLaHO5a6EAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1dd59aebf24ad32e56fb5380b9bf5c3e23e823e199ae679d0f959b16291c1cbe","last_reissued_at":"2026-07-05T07:34:05.829682Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T07:34:05.829682Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2401.08000","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":"2026-07-05T07:34:05Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"gWyO99HIzX/nW0hCgU8n/3LMl70vcUPZt1u7aCvVeAyY/HMvxeV2WFw2cRrs6k91JG11bZpYnWr1D7tV2452Cg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-19T07:55:24.378774Z"},"content_sha256":"23e5bba6578cf7feaae5910a0ebcfa3c160216df3159cb27df5f27c8b082752f","schema_version":"1.0","event_id":"sha256:23e5bba6578cf7feaae5910a0ebcfa3c160216df3159cb27df5f27c8b082752f"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2024:DXKZV27SJLJS4VX3KOALTP24HY","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Ultracoproducts and weak containment for flows of topological groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.LO","math.OA"],"primary_cat":"math.DS","authors_text":"Andy Zucker","submitted_at":"2024-01-15T22:58:37Z","abstract_excerpt":"We develop the theory of ultracoproducts and weak containment for flows of arbitrary topological groups. This provides a nice complement to corresponding theories for p.m.p. actions and unitary representations of locally compact groups. For the class of locally Roelcke precompact groups, the theory is especially rich, allowing us to define for certain families of $G$-flows a suitable compact space of weak types. When $G$ is locally compact, all $G$-flows belong to one such family, yielding a single compact space describing all weak types of $G$-flows."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2401.08000","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/2401.08000/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":"2026-07-05T07:34:05Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"rbSt9fzEpue8LN+AgxzMXQSm8TrOWuClDqsFww5gfOktvn47pKqev3AfRGgDu6/9ch3PSji4RRZsCizwb/DGCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-19T07:55:24.379300Z"},"content_sha256":"f103a5e4c784bd392a2bd33d91a7f0206e39cce2ed320f87b7886840b07940ef","schema_version":"1.0","event_id":"sha256:f103a5e4c784bd392a2bd33d91a7f0206e39cce2ed320f87b7886840b07940ef"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/DXKZV27SJLJS4VX3KOALTP24HY/bundle.json","state_url":"https://pith.science/pith/DXKZV27SJLJS4VX3KOALTP24HY/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/DXKZV27SJLJS4VX3KOALTP24HY/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":"2026-08-19T07:55:24Z","links":{"resolver":"https://pith.science/pith/DXKZV27SJLJS4VX3KOALTP24HY","bundle":"https://pith.science/pith/DXKZV27SJLJS4VX3KOALTP24HY/bundle.json","state":"https://pith.science/pith/DXKZV27SJLJS4VX3KOALTP24HY/state.json","well_known_bundle":"https://pith.science/.well-known/pith/DXKZV27SJLJS4VX3KOALTP24HY/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2024:DXKZV27SJLJS4VX3KOALTP24HY","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":"52dcc3ab34629be05a1f6c6250d1bb7b45632c8f15684de2c1b649b143024b45","cross_cats_sorted":["math.LO","math.OA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2024-01-15T22:58:37Z","title_canon_sha256":"f395b726e15e97fc52e6ee82687df7df54c3a7b4a72c1925bc9da2517c492813"},"schema_version":"1.0","source":{"id":"2401.08000","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2401.08000","created_at":"2026-07-05T07:34:05Z"},{"alias_kind":"arxiv_version","alias_value":"2401.08000v1","created_at":"2026-07-05T07:34:05Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2401.08000","created_at":"2026-07-05T07:34:05Z"},{"alias_kind":"pith_short_12","alias_value":"DXKZV27SJLJS","created_at":"2026-07-05T07:34:05Z"},{"alias_kind":"pith_short_16","alias_value":"DXKZV27SJLJS4VX3","created_at":"2026-07-05T07:34:05Z"},{"alias_kind":"pith_short_8","alias_value":"DXKZV27S","created_at":"2026-07-05T07:34:05Z"}],"graph_snapshots":[{"event_id":"sha256:f103a5e4c784bd392a2bd33d91a7f0206e39cce2ed320f87b7886840b07940ef","target":"graph","created_at":"2026-07-05T07:34:05Z","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/2401.08000/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We develop the theory of ultracoproducts and weak containment for flows of arbitrary topological groups. This provides a nice complement to corresponding theories for p.m.p. actions and unitary representations of locally compact groups. For the class of locally Roelcke precompact groups, the theory is especially rich, allowing us to define for certain families of $G$-flows a suitable compact space of weak types. When $G$ is locally compact, all $G$-flows belong to one such family, yielding a single compact space describing all weak types of $G$-flows.","authors_text":"Andy Zucker","cross_cats":["math.LO","math.OA"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2024-01-15T22:58:37Z","title":"Ultracoproducts and weak containment for flows of topological groups"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2401.08000","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":"sha256:23e5bba6578cf7feaae5910a0ebcfa3c160216df3159cb27df5f27c8b082752f","target":"record","created_at":"2026-07-05T07:34:05Z","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":"52dcc3ab34629be05a1f6c6250d1bb7b45632c8f15684de2c1b649b143024b45","cross_cats_sorted":["math.LO","math.OA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2024-01-15T22:58:37Z","title_canon_sha256":"f395b726e15e97fc52e6ee82687df7df54c3a7b4a72c1925bc9da2517c492813"},"schema_version":"1.0","source":{"id":"2401.08000","kind":"arxiv","version":1}},"canonical_sha256":"1dd59aebf24ad32e56fb5380b9bf5c3e23e823e199ae679d0f959b16291c1cbe","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"1dd59aebf24ad32e56fb5380b9bf5c3e23e823e199ae679d0f959b16291c1cbe","first_computed_at":"2026-07-05T07:34:05.829682Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T07:34:05.829682Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"DazNm+hjoQuzSvOOU4dsCc9xJIGZ8POlhQ6P5bu3tx9L8PGCZpX4H4l1wPtFqxDf/1XXqLqa4K2TLaHO5a6EAg==","signature_status":"signed_v1","signed_at":"2026-07-05T07:34:05.830052Z","signed_message":"canonical_sha256_bytes"},"source_id":"2401.08000","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:23e5bba6578cf7feaae5910a0ebcfa3c160216df3159cb27df5f27c8b082752f","sha256:f103a5e4c784bd392a2bd33d91a7f0206e39cce2ed320f87b7886840b07940ef"],"state_sha256":"27e92f8b7cf1a374b17efa0538689c5f4f1917262e00a9eeac6259caa1fca889"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Lss47G6gnDvRE7PGQm8AjMnj6eMqJOtRssxfyN26Tcf8b1yhSgue2okh2wyw2gOlzgY2pqOwzuBKZNxEyqfqDg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-19T07:55:24.382950Z","bundle_sha256":"048d1b4aa8af5e0d09d19f02117a608e54d142a61fe130efbafdce1cdadb75c7"}}