{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:5O4BMWY477GMX4HTWWPATIHLG4","short_pith_number":"pith:5O4BMWY4","canonical_record":{"source":{"id":"2606.02284","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2026-06-01T14:06:56Z","cross_cats_sorted":["math.MG"],"title_canon_sha256":"2e4ea4b293c2ec29c632b0329b0704257cda83647b9f7719da60eda98ada36fc","abstract_canon_sha256":"3efda688e729ce7dd08abad06d0a7aeedd75917edd7782a7f727ce600b388d58"},"schema_version":"1.0"},"canonical_sha256":"ebb8165b1cffcccbf0f3b59e09a0eb373de48b86b8804362d292f2ac5b5bb63b","source":{"kind":"arxiv","id":"2606.02284","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2606.02284","created_at":"2026-06-02T03:04:55Z"},{"alias_kind":"arxiv_version","alias_value":"2606.02284v1","created_at":"2026-06-02T03:04:55Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.02284","created_at":"2026-06-02T03:04:55Z"},{"alias_kind":"pith_short_12","alias_value":"5O4BMWY477GM","created_at":"2026-06-02T03:04:55Z"},{"alias_kind":"pith_short_16","alias_value":"5O4BMWY477GMX4HT","created_at":"2026-06-02T03:04:55Z"},{"alias_kind":"pith_short_8","alias_value":"5O4BMWY4","created_at":"2026-06-02T03:04:55Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:5O4BMWY477GMX4HTWWPATIHLG4","target":"record","payload":{"canonical_record":{"source":{"id":"2606.02284","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2026-06-01T14:06:56Z","cross_cats_sorted":["math.MG"],"title_canon_sha256":"2e4ea4b293c2ec29c632b0329b0704257cda83647b9f7719da60eda98ada36fc","abstract_canon_sha256":"3efda688e729ce7dd08abad06d0a7aeedd75917edd7782a7f727ce600b388d58"},"schema_version":"1.0"},"canonical_sha256":"ebb8165b1cffcccbf0f3b59e09a0eb373de48b86b8804362d292f2ac5b5bb63b","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-02T03:04:55.063999Z","signature_b64":"RB2YaeBvvSnMtepHZPWLIyLHmkEeWBihYOj42IRcMeUPmpDUsNBru3n/lz+hpvmL6LownATkbyFBnSbUIRsnBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ebb8165b1cffcccbf0f3b59e09a0eb373de48b86b8804362d292f2ac5b5bb63b","last_reissued_at":"2026-06-02T03:04:55.063624Z","signature_status":"signed_v1","first_computed_at":"2026-06-02T03:04:55.063624Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2606.02284","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":"2026-06-02T03:04:55Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"mqdMyxhb78fABadflEpfFrGu2Yyqq8CDxwxC9QaIBihtMQVK3Xxjv2hLdT2PvIMzcX/zopVF5MxmgtgL/F/EBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-29T18:16:22.968214Z"},"content_sha256":"27f031f2935fed1335412bf9f89e4260be974a60dc8f5ed8ef9f660563bb9ebe","schema_version":"1.0","event_id":"sha256:27f031f2935fed1335412bf9f89e4260be974a60dc8f5ed8ef9f660563bb9ebe"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:5O4BMWY477GMX4HTWWPATIHLG4","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Coarse median property of virtually nilpotent groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MG"],"primary_cat":"math.GR","authors_text":"Hyeonggeun Kim","submitted_at":"2026-06-01T14:06:56Z","abstract_excerpt":"We show that virtually nilpotent groups are coarse median if and only if they are virtually abelian. The main idea is that the sub-Riemannian geometry of the asymptotic cone obstructs the existence of a locally convex Lipschitz median of finite rank. As an application, we deduce that non-compact lattices in the isometry group of a rank 1 symmetric space of non-compact type other than real hyperbolic space are not coarse median. This establishes the remaining case in the classification of lattices with the coarse median property initiated by Haettel. The same approach applies more generally to "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.02284","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/2606.02284/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":"2026-06-02T03:04:55Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"7+G+6VLNsZQX+inzymTEd8D8eXuccPTM3NRPcxxnuEu4wcEqKH07gP5qee6jPZyNR+GuSDSq2nej910LRV/MBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-29T18:16:22.968597Z"},"content_sha256":"fab1829c09a4d745ad01f7296358875106e7a351506abca98904482bdfd78fb7","schema_version":"1.0","event_id":"sha256:fab1829c09a4d745ad01f7296358875106e7a351506abca98904482bdfd78fb7"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/5O4BMWY477GMX4HTWWPATIHLG4/bundle.json","state_url":"https://pith.science/pith/5O4BMWY477GMX4HTWWPATIHLG4/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/5O4BMWY477GMX4HTWWPATIHLG4/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":"2026-06-29T18:16:22Z","links":{"resolver":"https://pith.science/pith/5O4BMWY477GMX4HTWWPATIHLG4","bundle":"https://pith.science/pith/5O4BMWY477GMX4HTWWPATIHLG4/bundle.json","state":"https://pith.science/pith/5O4BMWY477GMX4HTWWPATIHLG4/state.json","well_known_bundle":"https://pith.science/.well-known/pith/5O4BMWY477GMX4HTWWPATIHLG4/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:5O4BMWY477GMX4HTWWPATIHLG4","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":"3efda688e729ce7dd08abad06d0a7aeedd75917edd7782a7f727ce600b388d58","cross_cats_sorted":["math.MG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2026-06-01T14:06:56Z","title_canon_sha256":"2e4ea4b293c2ec29c632b0329b0704257cda83647b9f7719da60eda98ada36fc"},"schema_version":"1.0","source":{"id":"2606.02284","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2606.02284","created_at":"2026-06-02T03:04:55Z"},{"alias_kind":"arxiv_version","alias_value":"2606.02284v1","created_at":"2026-06-02T03:04:55Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.02284","created_at":"2026-06-02T03:04:55Z"},{"alias_kind":"pith_short_12","alias_value":"5O4BMWY477GM","created_at":"2026-06-02T03:04:55Z"},{"alias_kind":"pith_short_16","alias_value":"5O4BMWY477GMX4HT","created_at":"2026-06-02T03:04:55Z"},{"alias_kind":"pith_short_8","alias_value":"5O4BMWY4","created_at":"2026-06-02T03:04:55Z"}],"graph_snapshots":[{"event_id":"sha256:fab1829c09a4d745ad01f7296358875106e7a351506abca98904482bdfd78fb7","target":"graph","created_at":"2026-06-02T03:04:55Z","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/2606.02284/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We show that virtually nilpotent groups are coarse median if and only if they are virtually abelian. The main idea is that the sub-Riemannian geometry of the asymptotic cone obstructs the existence of a locally convex Lipschitz median of finite rank. As an application, we deduce that non-compact lattices in the isometry group of a rank 1 symmetric space of non-compact type other than real hyperbolic space are not coarse median. This establishes the remaining case in the classification of lattices with the coarse median property initiated by Haettel. The same approach applies more generally to ","authors_text":"Hyeonggeun Kim","cross_cats":["math.MG"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2026-06-01T14:06:56Z","title":"Coarse median property of virtually nilpotent groups"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.02284","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":"sha256:27f031f2935fed1335412bf9f89e4260be974a60dc8f5ed8ef9f660563bb9ebe","target":"record","created_at":"2026-06-02T03:04:55Z","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":"3efda688e729ce7dd08abad06d0a7aeedd75917edd7782a7f727ce600b388d58","cross_cats_sorted":["math.MG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2026-06-01T14:06:56Z","title_canon_sha256":"2e4ea4b293c2ec29c632b0329b0704257cda83647b9f7719da60eda98ada36fc"},"schema_version":"1.0","source":{"id":"2606.02284","kind":"arxiv","version":1}},"canonical_sha256":"ebb8165b1cffcccbf0f3b59e09a0eb373de48b86b8804362d292f2ac5b5bb63b","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"ebb8165b1cffcccbf0f3b59e09a0eb373de48b86b8804362d292f2ac5b5bb63b","first_computed_at":"2026-06-02T03:04:55.063624Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-06-02T03:04:55.063624Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"RB2YaeBvvSnMtepHZPWLIyLHmkEeWBihYOj42IRcMeUPmpDUsNBru3n/lz+hpvmL6LownATkbyFBnSbUIRsnBg==","signature_status":"signed_v1","signed_at":"2026-06-02T03:04:55.063999Z","signed_message":"canonical_sha256_bytes"},"source_id":"2606.02284","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:27f031f2935fed1335412bf9f89e4260be974a60dc8f5ed8ef9f660563bb9ebe","sha256:fab1829c09a4d745ad01f7296358875106e7a351506abca98904482bdfd78fb7"],"state_sha256":"58f4910db56002198eb530b7290a399cc28713a775ecde7c9e67ba1418ca08c2"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"XndcAK/bhFJuY3wJvFWBdWl50z+GViQSYF1T39IS2BkYN5+vgK5feiA7gdYP7yJj629Je9WGS0a7pyuc7uPlDg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-29T18:16:22.970497Z","bundle_sha256":"b9b7d32ad2100b01510c70fa7e1d68a74bb440c1040d0feacdb9b58810aa86a3"}}