{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2008:W33DRXPED6LENSLHFNLS2WZFK2","short_pith_number":"pith:W33DRXPE","canonical_record":{"source":{"id":"0802.0033","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2008-01-31T23:25:01Z","cross_cats_sorted":["math.GT"],"title_canon_sha256":"be7312f911a28887a5d221423a47273f9a79fa9e8283b4e4bc49eb57750ec785","abstract_canon_sha256":"8d7e395c348eb6aaa49f61c0dd93201216b23e281600d74ffb57e772cf6f87e5"},"schema_version":"1.0"},"canonical_sha256":"b6f638dde41f9646c9672b572d5b2556b487c2e1d58ca135cdfffb7e1b0674b5","source":{"kind":"arxiv","id":"0802.0033","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0802.0033","created_at":"2026-05-18T02:41:51Z"},{"alias_kind":"arxiv_version","alias_value":"0802.0033v2","created_at":"2026-05-18T02:41:51Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0802.0033","created_at":"2026-05-18T02:41:51Z"},{"alias_kind":"pith_short_12","alias_value":"W33DRXPED6LE","created_at":"2026-05-18T12:25:58Z"},{"alias_kind":"pith_short_16","alias_value":"W33DRXPED6LENSLH","created_at":"2026-05-18T12:25:58Z"},{"alias_kind":"pith_short_8","alias_value":"W33DRXPE","created_at":"2026-05-18T12:25:58Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2008:W33DRXPED6LENSLHFNLS2WZFK2","target":"record","payload":{"canonical_record":{"source":{"id":"0802.0033","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2008-01-31T23:25:01Z","cross_cats_sorted":["math.GT"],"title_canon_sha256":"be7312f911a28887a5d221423a47273f9a79fa9e8283b4e4bc49eb57750ec785","abstract_canon_sha256":"8d7e395c348eb6aaa49f61c0dd93201216b23e281600d74ffb57e772cf6f87e5"},"schema_version":"1.0"},"canonical_sha256":"b6f638dde41f9646c9672b572d5b2556b487c2e1d58ca135cdfffb7e1b0674b5","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:41:51.809942Z","signature_b64":"jO+MQyiMeIy1uqeSHT4Bo8EN1hLbDeeaN+orHc6gvPHmea5VgfZfXhQZkyf95gTfZcgjNfvCawU9Ai33iHHtCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b6f638dde41f9646c9672b572d5b2556b487c2e1d58ca135cdfffb7e1b0674b5","last_reissued_at":"2026-05-18T02:41:51.809378Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:41:51.809378Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"0802.0033","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":"2026-05-18T02:41:51Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"wCzoUrf6fdxBPj6l9KB3E8wtY8FeZ/4xCHpABvonChc1VyfadRhu8l4x+au+2IXDaO/fg97BhibYLC/SDHkPBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-26T17:01:50.489490Z"},"content_sha256":"5ee801f8632c9a6da03bed4d02eabfeef37150391f6ab2bb6517ce06d95b5f7e","schema_version":"1.0","event_id":"sha256:5ee801f8632c9a6da03bed4d02eabfeef37150391f6ab2bb6517ce06d95b5f7e"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2008:W33DRXPED6LENSLHFNLS2WZFK2","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Intersections and joins of free groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GT"],"primary_cat":"math.GR","authors_text":"Richard P. Kent IV","submitted_at":"2008-01-31T23:25:01Z","abstract_excerpt":"Let H and K be subgroups of a free group of ranks h and k \\geq h. We prove the following strong form of Burns' inequality: rank(H \\cap K) - 1 \\leq 2(h-1)(k-1) - (h-1)(rank(H \\vee K) -1). A corollary of this, also obtained by L. Louder and D. B. McReynolds, has been used by M. Culler and P. Shalen to obtain information regarding the volumes of hyperbolic 3-manifolds. We also prove the following particular case of the Hanna Neumann Conjecture, which has also been obtained by Louder. If the join of H and K has rank at least (h + k + 1)/2, then the intersection of H and K has rank no more than (h-"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0802.0033","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":""},"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":"2026-05-18T02:41:51Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"07xT4DRnmu7UzcbMIoIhextct27JDdRABJ01Gw5TJS/d+fqSQQav2xRmoCk7Sqy6ZaAWFF/ovO6U3f0bZ61zCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-26T17:01:50.490173Z"},"content_sha256":"6d627d10ddbbc980f4c91332fbe9670e23fc89a726be9fb091a29d9c116c65cb","schema_version":"1.0","event_id":"sha256:6d627d10ddbbc980f4c91332fbe9670e23fc89a726be9fb091a29d9c116c65cb"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/W33DRXPED6LENSLHFNLS2WZFK2/bundle.json","state_url":"https://pith.science/pith/W33DRXPED6LENSLHFNLS2WZFK2/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/W33DRXPED6LENSLHFNLS2WZFK2/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":"2026-05-26T17:01:50Z","links":{"resolver":"https://pith.science/pith/W33DRXPED6LENSLHFNLS2WZFK2","bundle":"https://pith.science/pith/W33DRXPED6LENSLHFNLS2WZFK2/bundle.json","state":"https://pith.science/pith/W33DRXPED6LENSLHFNLS2WZFK2/state.json","well_known_bundle":"https://pith.science/.well-known/pith/W33DRXPED6LENSLHFNLS2WZFK2/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2008:W33DRXPED6LENSLHFNLS2WZFK2","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":"8d7e395c348eb6aaa49f61c0dd93201216b23e281600d74ffb57e772cf6f87e5","cross_cats_sorted":["math.GT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2008-01-31T23:25:01Z","title_canon_sha256":"be7312f911a28887a5d221423a47273f9a79fa9e8283b4e4bc49eb57750ec785"},"schema_version":"1.0","source":{"id":"0802.0033","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0802.0033","created_at":"2026-05-18T02:41:51Z"},{"alias_kind":"arxiv_version","alias_value":"0802.0033v2","created_at":"2026-05-18T02:41:51Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0802.0033","created_at":"2026-05-18T02:41:51Z"},{"alias_kind":"pith_short_12","alias_value":"W33DRXPED6LE","created_at":"2026-05-18T12:25:58Z"},{"alias_kind":"pith_short_16","alias_value":"W33DRXPED6LENSLH","created_at":"2026-05-18T12:25:58Z"},{"alias_kind":"pith_short_8","alias_value":"W33DRXPE","created_at":"2026-05-18T12:25:58Z"}],"graph_snapshots":[{"event_id":"sha256:6d627d10ddbbc980f4c91332fbe9670e23fc89a726be9fb091a29d9c116c65cb","target":"graph","created_at":"2026-05-18T02:41:51Z","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"},"paper":{"abstract_excerpt":"Let H and K be subgroups of a free group of ranks h and k \\geq h. We prove the following strong form of Burns' inequality: rank(H \\cap K) - 1 \\leq 2(h-1)(k-1) - (h-1)(rank(H \\vee K) -1). A corollary of this, also obtained by L. Louder and D. B. McReynolds, has been used by M. Culler and P. Shalen to obtain information regarding the volumes of hyperbolic 3-manifolds. We also prove the following particular case of the Hanna Neumann Conjecture, which has also been obtained by Louder. If the join of H and K has rank at least (h + k + 1)/2, then the intersection of H and K has rank no more than (h-","authors_text":"Richard P. Kent IV","cross_cats":["math.GT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2008-01-31T23:25:01Z","title":"Intersections and joins of free groups"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0802.0033","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":"sha256:5ee801f8632c9a6da03bed4d02eabfeef37150391f6ab2bb6517ce06d95b5f7e","target":"record","created_at":"2026-05-18T02:41:51Z","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":"8d7e395c348eb6aaa49f61c0dd93201216b23e281600d74ffb57e772cf6f87e5","cross_cats_sorted":["math.GT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2008-01-31T23:25:01Z","title_canon_sha256":"be7312f911a28887a5d221423a47273f9a79fa9e8283b4e4bc49eb57750ec785"},"schema_version":"1.0","source":{"id":"0802.0033","kind":"arxiv","version":2}},"canonical_sha256":"b6f638dde41f9646c9672b572d5b2556b487c2e1d58ca135cdfffb7e1b0674b5","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"b6f638dde41f9646c9672b572d5b2556b487c2e1d58ca135cdfffb7e1b0674b5","first_computed_at":"2026-05-18T02:41:51.809378Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:41:51.809378Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"jO+MQyiMeIy1uqeSHT4Bo8EN1hLbDeeaN+orHc6gvPHmea5VgfZfXhQZkyf95gTfZcgjNfvCawU9Ai33iHHtCA==","signature_status":"signed_v1","signed_at":"2026-05-18T02:41:51.809942Z","signed_message":"canonical_sha256_bytes"},"source_id":"0802.0033","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:5ee801f8632c9a6da03bed4d02eabfeef37150391f6ab2bb6517ce06d95b5f7e","sha256:6d627d10ddbbc980f4c91332fbe9670e23fc89a726be9fb091a29d9c116c65cb"],"state_sha256":"16012bed44f86e93cd9aa875c2e91c76ad0bf7a634ad8f48ac297585044389b5"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"yYBYsrYXGObY9olQ/38lemqyHhKOUoq74KQaNM30hyimKe2SgaZjvnNkjte+MSdq/hd5iAQ9ZUIb9eTVDY3EBg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-26T17:01:50.493782Z","bundle_sha256":"6700e7fe44d042c95f34cc25c34cb124197a10d603994903cfc4d70c084814c3"}}