{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2025:5TBHA3ISHBESBGHOEDNQ6JJHVQ","short_pith_number":"pith:5TBHA3IS","canonical_record":{"source":{"id":"2507.04691","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OA","submitted_at":"2025-07-07T06:13:56Z","cross_cats_sorted":["math.GR"],"title_canon_sha256":"15ae8d25090af4ea8f61a75e617f395a9d429972e07f536a73c75f24741c2937","abstract_canon_sha256":"b9114dbfa2b4c05f347f249d45146cc36238ff851a069ac9af1db8e3febc14a8"},"schema_version":"1.0"},"canonical_sha256":"ecc2706d1238492098ee20db0f2527ac3cf072793bdf1c03afbcee6ddb6956ed","source":{"kind":"arxiv","id":"2507.04691","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2507.04691","created_at":"2026-07-05T11:52:22Z"},{"alias_kind":"arxiv_version","alias_value":"2507.04691v2","created_at":"2026-07-05T11:52:22Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2507.04691","created_at":"2026-07-05T11:52:22Z"},{"alias_kind":"pith_short_12","alias_value":"5TBHA3ISHBES","created_at":"2026-07-05T11:52:22Z"},{"alias_kind":"pith_short_16","alias_value":"5TBHA3ISHBESBGHO","created_at":"2026-07-05T11:52:22Z"},{"alias_kind":"pith_short_8","alias_value":"5TBHA3IS","created_at":"2026-07-05T11:52:22Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2025:5TBHA3ISHBESBGHOEDNQ6JJHVQ","target":"record","payload":{"canonical_record":{"source":{"id":"2507.04691","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OA","submitted_at":"2025-07-07T06:13:56Z","cross_cats_sorted":["math.GR"],"title_canon_sha256":"15ae8d25090af4ea8f61a75e617f395a9d429972e07f536a73c75f24741c2937","abstract_canon_sha256":"b9114dbfa2b4c05f347f249d45146cc36238ff851a069ac9af1db8e3febc14a8"},"schema_version":"1.0"},"canonical_sha256":"ecc2706d1238492098ee20db0f2527ac3cf072793bdf1c03afbcee6ddb6956ed","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T11:52:22.076563Z","signature_b64":"hwQauSKLixuagayKjnkB9mWd8/TBLjBE/lO+IX5UK9TyR5UsqtJMxBFNYqlnP9QQbSYwZcvRXn6G0FoxLb4WDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ecc2706d1238492098ee20db0f2527ac3cf072793bdf1c03afbcee6ddb6956ed","last_reissued_at":"2026-07-05T11:52:22.076065Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T11:52:22.076065Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2507.04691","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T11:52:22Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"9of/ihFKlpbYc6UyvycCP+YSiKgYEW/YYf6X7plW0VCWoVAdJUzMzpBjHNDJpWUnh3wZ71ZGOYchsDkg8+7YAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-15T19:01:31.171579Z"},"content_sha256":"2d570adb76252a6ecac59ab8f5096d80bca3e7558333bd8cab51a1ce8bfa038e","schema_version":"1.0","event_id":"sha256:2d570adb76252a6ecac59ab8f5096d80bca3e7558333bd8cab51a1ce8bfa038e"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2025:5TBHA3ISHBESBGHOEDNQ6JJHVQ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"W*-correlations of II$_1$ factors and rigidity of tensor products and graph products","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GR"],"primary_cat":"math.OA","authors_text":"Daniel Drimbe, Stefaan Vaes","submitted_at":"2025-07-07T06:13:56Z","abstract_excerpt":"A variant of Gromov's notion of measure equivalence for groups has been introduced for II$_1$ factors under different names. We propose the terminology of W*-correlated II$_1$ factors. We prove rigidity results up to W*-correlations for tensor products and graph products of II$_1$ factors. As a consequence, we construct the first uncountable family of discrete groups $\\Gamma$ that are not von Neumann equivalent, which means that their group von Neumann algebras $L(\\Gamma)$ are not W*-correlated, and which implies that these groups are neither measure equivalent, nor have isomorphic or virtuall"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2507.04691","kind":"arxiv","version":2},"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/2507.04691/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":"2026-07-05T11:52:22Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ICBp9lT+3yDIBoQ4JLbnJE/HWm1Py4pGmYYmoU38vDtZBJ08kgrOYoFHd3LMuSaOmxtzgKanz9Uipm05JKzxCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-15T19:01:31.172122Z"},"content_sha256":"3f6fe850fb4e3b0ce7f573be8f7d3746c1a6a8f41d1768cd71a86917d22313b8","schema_version":"1.0","event_id":"sha256:3f6fe850fb4e3b0ce7f573be8f7d3746c1a6a8f41d1768cd71a86917d22313b8"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/5TBHA3ISHBESBGHOEDNQ6JJHVQ/bundle.json","state_url":"https://pith.science/pith/5TBHA3ISHBESBGHOEDNQ6JJHVQ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/5TBHA3ISHBESBGHOEDNQ6JJHVQ/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":"2026-08-15T19:01:31Z","links":{"resolver":"https://pith.science/pith/5TBHA3ISHBESBGHOEDNQ6JJHVQ","bundle":"https://pith.science/pith/5TBHA3ISHBESBGHOEDNQ6JJHVQ/bundle.json","state":"https://pith.science/pith/5TBHA3ISHBESBGHOEDNQ6JJHVQ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/5TBHA3ISHBESBGHOEDNQ6JJHVQ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2025:5TBHA3ISHBESBGHOEDNQ6JJHVQ","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":"b9114dbfa2b4c05f347f249d45146cc36238ff851a069ac9af1db8e3febc14a8","cross_cats_sorted":["math.GR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OA","submitted_at":"2025-07-07T06:13:56Z","title_canon_sha256":"15ae8d25090af4ea8f61a75e617f395a9d429972e07f536a73c75f24741c2937"},"schema_version":"1.0","source":{"id":"2507.04691","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2507.04691","created_at":"2026-07-05T11:52:22Z"},{"alias_kind":"arxiv_version","alias_value":"2507.04691v2","created_at":"2026-07-05T11:52:22Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2507.04691","created_at":"2026-07-05T11:52:22Z"},{"alias_kind":"pith_short_12","alias_value":"5TBHA3ISHBES","created_at":"2026-07-05T11:52:22Z"},{"alias_kind":"pith_short_16","alias_value":"5TBHA3ISHBESBGHO","created_at":"2026-07-05T11:52:22Z"},{"alias_kind":"pith_short_8","alias_value":"5TBHA3IS","created_at":"2026-07-05T11:52:22Z"}],"graph_snapshots":[{"event_id":"sha256:3f6fe850fb4e3b0ce7f573be8f7d3746c1a6a8f41d1768cd71a86917d22313b8","target":"graph","created_at":"2026-07-05T11:52:22Z","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/2507.04691/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"A variant of Gromov's notion of measure equivalence for groups has been introduced for II$_1$ factors under different names. We propose the terminology of W*-correlated II$_1$ factors. We prove rigidity results up to W*-correlations for tensor products and graph products of II$_1$ factors. As a consequence, we construct the first uncountable family of discrete groups $\\Gamma$ that are not von Neumann equivalent, which means that their group von Neumann algebras $L(\\Gamma)$ are not W*-correlated, and which implies that these groups are neither measure equivalent, nor have isomorphic or virtuall","authors_text":"Daniel Drimbe, Stefaan Vaes","cross_cats":["math.GR"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OA","submitted_at":"2025-07-07T06:13:56Z","title":"W*-correlations of II$_1$ factors and rigidity of tensor products and graph products"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2507.04691","kind":"arxiv","version":2},"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":"sha256:2d570adb76252a6ecac59ab8f5096d80bca3e7558333bd8cab51a1ce8bfa038e","target":"record","created_at":"2026-07-05T11:52:22Z","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":"b9114dbfa2b4c05f347f249d45146cc36238ff851a069ac9af1db8e3febc14a8","cross_cats_sorted":["math.GR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OA","submitted_at":"2025-07-07T06:13:56Z","title_canon_sha256":"15ae8d25090af4ea8f61a75e617f395a9d429972e07f536a73c75f24741c2937"},"schema_version":"1.0","source":{"id":"2507.04691","kind":"arxiv","version":2}},"canonical_sha256":"ecc2706d1238492098ee20db0f2527ac3cf072793bdf1c03afbcee6ddb6956ed","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"ecc2706d1238492098ee20db0f2527ac3cf072793bdf1c03afbcee6ddb6956ed","first_computed_at":"2026-07-05T11:52:22.076065Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T11:52:22.076065Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"hwQauSKLixuagayKjnkB9mWd8/TBLjBE/lO+IX5UK9TyR5UsqtJMxBFNYqlnP9QQbSYwZcvRXn6G0FoxLb4WDg==","signature_status":"signed_v1","signed_at":"2026-07-05T11:52:22.076563Z","signed_message":"canonical_sha256_bytes"},"source_id":"2507.04691","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:2d570adb76252a6ecac59ab8f5096d80bca3e7558333bd8cab51a1ce8bfa038e","sha256:3f6fe850fb4e3b0ce7f573be8f7d3746c1a6a8f41d1768cd71a86917d22313b8"],"state_sha256":"396f2720fab88b206a23364cbb631d46958213a0899d4643a63cc82e8182a427"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"bnOsqLWlIb9YwADJ4TAjDxslgc7RturS/UcXH/azT4nCeHm+Tghl2Hwn1rBiXGizhAUOBYfZAzVKx3akFaKxAQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-15T19:01:31.176139Z","bundle_sha256":"1ba5d1c0715b12184fa4c852081f1f9642a869cc01c541a707aaf9f4bbf3d87d"}}