{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2008:NW5G7POJ7AOFK2K5VVYL4NK5TD","short_pith_number":"pith:NW5G7POJ","canonical_record":{"source":{"id":"0807.3308","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2008-07-21T16:39:29Z","cross_cats_sorted":["math-ph","math.MP"],"title_canon_sha256":"1484c4b76c5aa212ffb006924e9aa09f6ccefd846bf9ee49641d84bae5d3a4d1","abstract_canon_sha256":"552e13708cf5922882d5018eb59a6695637423aa6cc019c5867cba80dda1bb79"},"schema_version":"1.0"},"canonical_sha256":"6dba6fbdc9f81c55695dad70be355d98c6cf52f899f9d049f8cbb14eefd6ac52","source":{"kind":"arxiv","id":"0807.3308","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0807.3308","created_at":"2026-07-04T15:13:03Z"},{"alias_kind":"arxiv_version","alias_value":"0807.3308v1","created_at":"2026-07-04T15:13:03Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0807.3308","created_at":"2026-07-04T15:13:03Z"},{"alias_kind":"pith_short_12","alias_value":"NW5G7POJ7AOF","created_at":"2026-07-04T15:13:03Z"},{"alias_kind":"pith_short_16","alias_value":"NW5G7POJ7AOFK2K5","created_at":"2026-07-04T15:13:03Z"},{"alias_kind":"pith_short_8","alias_value":"NW5G7POJ","created_at":"2026-07-04T15:13:03Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2008:NW5G7POJ7AOFK2K5VVYL4NK5TD","target":"record","payload":{"canonical_record":{"source":{"id":"0807.3308","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2008-07-21T16:39:29Z","cross_cats_sorted":["math-ph","math.MP"],"title_canon_sha256":"1484c4b76c5aa212ffb006924e9aa09f6ccefd846bf9ee49641d84bae5d3a4d1","abstract_canon_sha256":"552e13708cf5922882d5018eb59a6695637423aa6cc019c5867cba80dda1bb79"},"schema_version":"1.0"},"canonical_sha256":"6dba6fbdc9f81c55695dad70be355d98c6cf52f899f9d049f8cbb14eefd6ac52","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-04T15:13:03.793388Z","signature_b64":"d0LH8IlUdglzC9xi8ikIn1nKeGj3SNwHYljCizFlupN0XgV1L+puqwopcvBo63Rx2eSPtK37eGH2TbwtD9F+AQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6dba6fbdc9f81c55695dad70be355d98c6cf52f899f9d049f8cbb14eefd6ac52","last_reissued_at":"2026-07-04T15:13:03.792997Z","signature_status":"signed_v1","first_computed_at":"2026-07-04T15:13:03.792997Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"0807.3308","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":"2026-07-04T15:13:03Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"54i5tmhaIjdhoMHXZk5YDi5E74iwhAwu/J9oBlEX72Vt8RsKwDiiq5N8vNZ0wLOvA77TWkh9kTsqSzrzGvZqBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-11T09:45:35.829169Z"},"content_sha256":"2f58909bc7ca17749b24b732f9fa83c2d4bd3a2f17bb27e0cbe00cc3a531c7ae","schema_version":"1.0","event_id":"sha256:2f58909bc7ca17749b24b732f9fa83c2d4bd3a2f17bb27e0cbe00cc3a531c7ae"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2008:NW5G7POJ7AOFK2K5VVYL4NK5TD","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Visibility to infinity in the hyperbolic plane, despite obstacles","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.MP"],"primary_cat":"math.PR","authors_text":"Itai Benjamini, Johan Jonasson, Johan Tykesson, Oded Schramm","submitted_at":"2008-07-21T16:39:29Z","abstract_excerpt":"Suppose that $Z$ is a random closed subset of the hyperbolic plane $\\H^2$, whose law is invariant under isometries of $\\H^2$. We prove that if the probability that $Z$ contains a fixed ball of radius 1 is larger than some universal constant $p<1$, then there is positive probability that $Z$ contains (bi-infinite) lines.\n  We then consider a family of random sets in $\\H^2$ that satisfy some additional natural assumptions. An example of such a set is the covered region in the Poisson Boolean model. Let $f(r)$ be the probability that a line segment of length $r$ is contained in such a set $Z$. We"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0807.3308","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/0807.3308/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":"2026-07-04T15:13:03Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"QhXDI8yqr4Sa5Wq3l+RxagPfD/H68JDoTLMjv8Be8IZEPenzszF+7wYuZuvvAsBunUWQY+2cYlUu8RxavmJLAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-11T09:45:35.829832Z"},"content_sha256":"a73ad4cee8e8ae91bca64358cf320b8f7001d23f83dda93a7e3f0169dba93b03","schema_version":"1.0","event_id":"sha256:a73ad4cee8e8ae91bca64358cf320b8f7001d23f83dda93a7e3f0169dba93b03"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/NW5G7POJ7AOFK2K5VVYL4NK5TD/bundle.json","state_url":"https://pith.science/pith/NW5G7POJ7AOFK2K5VVYL4NK5TD/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/NW5G7POJ7AOFK2K5VVYL4NK5TD/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":"2026-08-11T09:45:35Z","links":{"resolver":"https://pith.science/pith/NW5G7POJ7AOFK2K5VVYL4NK5TD","bundle":"https://pith.science/pith/NW5G7POJ7AOFK2K5VVYL4NK5TD/bundle.json","state":"https://pith.science/pith/NW5G7POJ7AOFK2K5VVYL4NK5TD/state.json","well_known_bundle":"https://pith.science/.well-known/pith/NW5G7POJ7AOFK2K5VVYL4NK5TD/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2008:NW5G7POJ7AOFK2K5VVYL4NK5TD","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":"552e13708cf5922882d5018eb59a6695637423aa6cc019c5867cba80dda1bb79","cross_cats_sorted":["math-ph","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2008-07-21T16:39:29Z","title_canon_sha256":"1484c4b76c5aa212ffb006924e9aa09f6ccefd846bf9ee49641d84bae5d3a4d1"},"schema_version":"1.0","source":{"id":"0807.3308","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0807.3308","created_at":"2026-07-04T15:13:03Z"},{"alias_kind":"arxiv_version","alias_value":"0807.3308v1","created_at":"2026-07-04T15:13:03Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0807.3308","created_at":"2026-07-04T15:13:03Z"},{"alias_kind":"pith_short_12","alias_value":"NW5G7POJ7AOF","created_at":"2026-07-04T15:13:03Z"},{"alias_kind":"pith_short_16","alias_value":"NW5G7POJ7AOFK2K5","created_at":"2026-07-04T15:13:03Z"},{"alias_kind":"pith_short_8","alias_value":"NW5G7POJ","created_at":"2026-07-04T15:13:03Z"}],"graph_snapshots":[{"event_id":"sha256:a73ad4cee8e8ae91bca64358cf320b8f7001d23f83dda93a7e3f0169dba93b03","target":"graph","created_at":"2026-07-04T15:13:03Z","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/0807.3308/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Suppose that $Z$ is a random closed subset of the hyperbolic plane $\\H^2$, whose law is invariant under isometries of $\\H^2$. We prove that if the probability that $Z$ contains a fixed ball of radius 1 is larger than some universal constant $p<1$, then there is positive probability that $Z$ contains (bi-infinite) lines.\n  We then consider a family of random sets in $\\H^2$ that satisfy some additional natural assumptions. An example of such a set is the covered region in the Poisson Boolean model. Let $f(r)$ be the probability that a line segment of length $r$ is contained in such a set $Z$. We","authors_text":"Itai Benjamini, Johan Jonasson, Johan Tykesson, Oded Schramm","cross_cats":["math-ph","math.MP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2008-07-21T16:39:29Z","title":"Visibility to infinity in the hyperbolic plane, despite obstacles"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0807.3308","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":"sha256:2f58909bc7ca17749b24b732f9fa83c2d4bd3a2f17bb27e0cbe00cc3a531c7ae","target":"record","created_at":"2026-07-04T15:13:03Z","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":"552e13708cf5922882d5018eb59a6695637423aa6cc019c5867cba80dda1bb79","cross_cats_sorted":["math-ph","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2008-07-21T16:39:29Z","title_canon_sha256":"1484c4b76c5aa212ffb006924e9aa09f6ccefd846bf9ee49641d84bae5d3a4d1"},"schema_version":"1.0","source":{"id":"0807.3308","kind":"arxiv","version":1}},"canonical_sha256":"6dba6fbdc9f81c55695dad70be355d98c6cf52f899f9d049f8cbb14eefd6ac52","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"6dba6fbdc9f81c55695dad70be355d98c6cf52f899f9d049f8cbb14eefd6ac52","first_computed_at":"2026-07-04T15:13:03.792997Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-04T15:13:03.792997Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"d0LH8IlUdglzC9xi8ikIn1nKeGj3SNwHYljCizFlupN0XgV1L+puqwopcvBo63Rx2eSPtK37eGH2TbwtD9F+AQ==","signature_status":"signed_v1","signed_at":"2026-07-04T15:13:03.793388Z","signed_message":"canonical_sha256_bytes"},"source_id":"0807.3308","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:2f58909bc7ca17749b24b732f9fa83c2d4bd3a2f17bb27e0cbe00cc3a531c7ae","sha256:a73ad4cee8e8ae91bca64358cf320b8f7001d23f83dda93a7e3f0169dba93b03"],"state_sha256":"4f049406412c7b0ca96bf9233b810a5494cd49bcc53d1c323edd91c4fa05589c"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"/QIDLWByBVEun5DeLRDH7ofZqyq2jvMWKbbsmyvHapo0ziIKFe9RG6oQLlHvwVV48ULjP4Z79WsnPnQ+1Uh5AA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-11T09:45:35.834726Z","bundle_sha256":"31a9017172c02146263160fccbaa5729d3dc4396e0e03d5469358db9d8ba4d1b"}}