{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:VNOMTCNX7UUFGJJIESC5OLGKYV","short_pith_number":"pith:VNOMTCNX","canonical_record":{"source":{"id":"2607.17764","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2026-07-20T09:59:39Z","cross_cats_sorted":[],"title_canon_sha256":"b2a2a42a823dd711b642884eb01d0b62935c30e6e9dd55b0cf788486f99dc175","abstract_canon_sha256":"791b431ce0da1175e104731de5cf7e386de9c1b399ad8047c25e1fe7c9eca658"},"schema_version":"1.0"},"canonical_sha256":"ab5cc989b7fd285325282485d72ccac55b6fbd6c86978869c648da1ca78102f9","source":{"kind":"arxiv","id":"2607.17764","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2607.17764","created_at":"2026-07-21T02:21:58Z"},{"alias_kind":"arxiv_version","alias_value":"2607.17764v1","created_at":"2026-07-21T02:21:58Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.17764","created_at":"2026-07-21T02:21:58Z"},{"alias_kind":"pith_short_12","alias_value":"VNOMTCNX7UUF","created_at":"2026-07-21T02:21:58Z"},{"alias_kind":"pith_short_16","alias_value":"VNOMTCNX7UUFGJJI","created_at":"2026-07-21T02:21:58Z"},{"alias_kind":"pith_short_8","alias_value":"VNOMTCNX","created_at":"2026-07-21T02:21:58Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:VNOMTCNX7UUFGJJIESC5OLGKYV","target":"record","payload":{"canonical_record":{"source":{"id":"2607.17764","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2026-07-20T09:59:39Z","cross_cats_sorted":[],"title_canon_sha256":"b2a2a42a823dd711b642884eb01d0b62935c30e6e9dd55b0cf788486f99dc175","abstract_canon_sha256":"791b431ce0da1175e104731de5cf7e386de9c1b399ad8047c25e1fe7c9eca658"},"schema_version":"1.0"},"canonical_sha256":"ab5cc989b7fd285325282485d72ccac55b6fbd6c86978869c648da1ca78102f9","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-21T02:21:58.978417Z","signature_b64":"J8HWGfpEhRVRQGh4d6k+6njeQaigjWUtognIDd9FuA4m2lpQMAeveZDPaS+hm9d9OKg7ZqWDY6Cv2Gy0/DByAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ab5cc989b7fd285325282485d72ccac55b6fbd6c86978869c648da1ca78102f9","last_reissued_at":"2026-07-21T02:21:58.977617Z","signature_status":"signed_v1","first_computed_at":"2026-07-21T02:21:58.977617Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2607.17764","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":"2026-07-21T02:21:58Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"UkxT3KNGRRX6qMIueSnmXZwQSHxE49nZN8nqNuC73Ry6iElSJny4YX28t8SRsPb5bNEiB7Zmd1YmMfkySOrtAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-12T05:54:02.705988Z"},"content_sha256":"49f848dec104335c53e730e08d6f0e31378d20abb9412192290a269fc86f85b2","schema_version":"1.0","event_id":"sha256:49f848dec104335c53e730e08d6f0e31378d20abb9412192290a269fc86f85b2"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:VNOMTCNX7UUFGJJIESC5OLGKYV","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Non-vanishing uniqueness threshold for hyperbolic Poisson-Voronoi percolation in dimension at least three","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Matthias Irlbeck, Tobias M\\\"uller","submitted_at":"2026-07-20T09:59:39Z","abstract_excerpt":"We study the threshold for the existence of exactly one unbounded cluster for Poisson-Voronoi percolation on the $d$-dimensional hyperbolic space $\\mathbb{H}^d$ for $d\\geq 3$. By recent results of Greb\\'ik and Recke and d'Achille et al., this \"uniqueness threshold\" $p_u(\\lambda)$ tends to zero as the intensity $\\lambda$ of the underlying Poisson point process tends to zero, for Poisson-Voronoi percolation defined on an ambient space from a family of geometric spaces that includes Cartesian products $\\mathbb{H}^{d_1}\\times\\dots\\times\\mathbb{H}^{d_k}$ with $k,d_1,\\dots,d_k\\geq 2$. In contrast, f"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.17764","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/2607.17764/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":"2026-07-21T02:21:58Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"SP+6xhJnsiHeYbd+wquYghdn7P9ShEuC75bLeL+e2sKt3JfXvls1Wc8lJR+iaMeA7iZPt3dJ+AIfCeptTU/9CQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-12T05:54:02.706913Z"},"content_sha256":"b4e8749591b90890cc40906aa87d982f62fb2d2bf796691b7725cf950dff9fdb","schema_version":"1.0","event_id":"sha256:b4e8749591b90890cc40906aa87d982f62fb2d2bf796691b7725cf950dff9fdb"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/VNOMTCNX7UUFGJJIESC5OLGKYV/bundle.json","state_url":"https://pith.science/pith/VNOMTCNX7UUFGJJIESC5OLGKYV/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/VNOMTCNX7UUFGJJIESC5OLGKYV/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":"2026-08-12T05:54:02Z","links":{"resolver":"https://pith.science/pith/VNOMTCNX7UUFGJJIESC5OLGKYV","bundle":"https://pith.science/pith/VNOMTCNX7UUFGJJIESC5OLGKYV/bundle.json","state":"https://pith.science/pith/VNOMTCNX7UUFGJJIESC5OLGKYV/state.json","well_known_bundle":"https://pith.science/.well-known/pith/VNOMTCNX7UUFGJJIESC5OLGKYV/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:VNOMTCNX7UUFGJJIESC5OLGKYV","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":"791b431ce0da1175e104731de5cf7e386de9c1b399ad8047c25e1fe7c9eca658","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2026-07-20T09:59:39Z","title_canon_sha256":"b2a2a42a823dd711b642884eb01d0b62935c30e6e9dd55b0cf788486f99dc175"},"schema_version":"1.0","source":{"id":"2607.17764","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2607.17764","created_at":"2026-07-21T02:21:58Z"},{"alias_kind":"arxiv_version","alias_value":"2607.17764v1","created_at":"2026-07-21T02:21:58Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.17764","created_at":"2026-07-21T02:21:58Z"},{"alias_kind":"pith_short_12","alias_value":"VNOMTCNX7UUF","created_at":"2026-07-21T02:21:58Z"},{"alias_kind":"pith_short_16","alias_value":"VNOMTCNX7UUFGJJI","created_at":"2026-07-21T02:21:58Z"},{"alias_kind":"pith_short_8","alias_value":"VNOMTCNX","created_at":"2026-07-21T02:21:58Z"}],"graph_snapshots":[{"event_id":"sha256:b4e8749591b90890cc40906aa87d982f62fb2d2bf796691b7725cf950dff9fdb","target":"graph","created_at":"2026-07-21T02:21:58Z","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/2607.17764/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We study the threshold for the existence of exactly one unbounded cluster for Poisson-Voronoi percolation on the $d$-dimensional hyperbolic space $\\mathbb{H}^d$ for $d\\geq 3$. By recent results of Greb\\'ik and Recke and d'Achille et al., this \"uniqueness threshold\" $p_u(\\lambda)$ tends to zero as the intensity $\\lambda$ of the underlying Poisson point process tends to zero, for Poisson-Voronoi percolation defined on an ambient space from a family of geometric spaces that includes Cartesian products $\\mathbb{H}^{d_1}\\times\\dots\\times\\mathbb{H}^{d_k}$ with $k,d_1,\\dots,d_k\\geq 2$. In contrast, f","authors_text":"Matthias Irlbeck, Tobias M\\\"uller","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2026-07-20T09:59:39Z","title":"Non-vanishing uniqueness threshold for hyperbolic Poisson-Voronoi percolation in dimension at least three"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.17764","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":"sha256:49f848dec104335c53e730e08d6f0e31378d20abb9412192290a269fc86f85b2","target":"record","created_at":"2026-07-21T02:21:58Z","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":"791b431ce0da1175e104731de5cf7e386de9c1b399ad8047c25e1fe7c9eca658","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2026-07-20T09:59:39Z","title_canon_sha256":"b2a2a42a823dd711b642884eb01d0b62935c30e6e9dd55b0cf788486f99dc175"},"schema_version":"1.0","source":{"id":"2607.17764","kind":"arxiv","version":1}},"canonical_sha256":"ab5cc989b7fd285325282485d72ccac55b6fbd6c86978869c648da1ca78102f9","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"ab5cc989b7fd285325282485d72ccac55b6fbd6c86978869c648da1ca78102f9","first_computed_at":"2026-07-21T02:21:58.977617Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-21T02:21:58.977617Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"J8HWGfpEhRVRQGh4d6k+6njeQaigjWUtognIDd9FuA4m2lpQMAeveZDPaS+hm9d9OKg7ZqWDY6Cv2Gy0/DByAg==","signature_status":"signed_v1","signed_at":"2026-07-21T02:21:58.978417Z","signed_message":"canonical_sha256_bytes"},"source_id":"2607.17764","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:49f848dec104335c53e730e08d6f0e31378d20abb9412192290a269fc86f85b2","sha256:b4e8749591b90890cc40906aa87d982f62fb2d2bf796691b7725cf950dff9fdb"],"state_sha256":"aa94912a5caef6a4dbd55226acb570c82b3052fad89a59555b1525ce82024897"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"1plyHMlOAe0MXxn3g7jPG2L44/cRam1/dQhfgTUx1LiWlLgJgCJp2QmT76ODCK41kqgJwN3mqcj3w5ARKQ9UAw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-12T05:54:02.712883Z","bundle_sha256":"8788d3dbbe631722c45e4c201bb487246864f44519e00cef9f4e9fabf01059c8"}}