{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:76NZ3EA35XNE4PQ3PZBXV2GXVZ","short_pith_number":"pith:76NZ3EA3","canonical_record":{"source":{"id":"2606.13632","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2026-06-11T17:42:31Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"e3c259b59cfa4196e5db1180bed82823c5f49380008e1e8001e4808ec885a5d6","abstract_canon_sha256":"f8e72503892b15722e54a6572da87e35661e2539a0f9e883acfd7e4655786102"},"schema_version":"1.0"},"canonical_sha256":"ff9b9d901bedda4e3e1b7e437ae8d7ae556ed167135a0108d03b844b506f19ba","source":{"kind":"arxiv","id":"2606.13632","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2606.13632","created_at":"2026-06-12T01:10:20Z"},{"alias_kind":"arxiv_version","alias_value":"2606.13632v1","created_at":"2026-06-12T01:10:20Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.13632","created_at":"2026-06-12T01:10:20Z"},{"alias_kind":"pith_short_12","alias_value":"76NZ3EA35XNE","created_at":"2026-06-12T01:10:20Z"},{"alias_kind":"pith_short_16","alias_value":"76NZ3EA35XNE4PQ3","created_at":"2026-06-12T01:10:20Z"},{"alias_kind":"pith_short_8","alias_value":"76NZ3EA3","created_at":"2026-06-12T01:10:20Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:76NZ3EA35XNE4PQ3PZBXV2GXVZ","target":"record","payload":{"canonical_record":{"source":{"id":"2606.13632","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2026-06-11T17:42:31Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"e3c259b59cfa4196e5db1180bed82823c5f49380008e1e8001e4808ec885a5d6","abstract_canon_sha256":"f8e72503892b15722e54a6572da87e35661e2539a0f9e883acfd7e4655786102"},"schema_version":"1.0"},"canonical_sha256":"ff9b9d901bedda4e3e1b7e437ae8d7ae556ed167135a0108d03b844b506f19ba","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-12T01:10:20.111656Z","signature_b64":"hVcIkaK3sfDDHa4/AaM1m71a8jmL/xe4gwIybJng+Rk7wSNhCqWO2eCVc1cET2Kg/c5LMMSlG8ENuJtp+0MiAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ff9b9d901bedda4e3e1b7e437ae8d7ae556ed167135a0108d03b844b506f19ba","last_reissued_at":"2026-06-12T01:10:20.110671Z","signature_status":"signed_v1","first_computed_at":"2026-06-12T01:10:20.110671Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2606.13632","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":"2026-06-12T01:10:20Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"E/exoKfEvNAcD8z1hQwZNrgNaTQ1dIcGzzfPtW4/RVc9DDmIfsKDPKFgozj6qt2I2n4xo2pbHf4mR4eGbtL7Cg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-09T06:59:51.479747Z"},"content_sha256":"99f650a01993db48d47325a45519a7664fd76ac702800536705647b0af730a26","schema_version":"1.0","event_id":"sha256:99f650a01993db48d47325a45519a7664fd76ac702800536705647b0af730a26"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:76NZ3EA35XNE4PQ3PZBXV2GXVZ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Growth of Approximate Groups in Hyperbolic Groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.GR","authors_text":"Gal Yehuda, Michael Saks","submitted_at":"2026-06-11T17:42:31Z","abstract_excerpt":"We prove a growth dichotomy for infinite approximate groups, and more generally approximate semigroups, in hyperbolic groups. If \\(G\\) is a finitely generated hyperbolic group and \\(A\\subseteq G\\) is infinite with \\[\n  A^2\\subseteq AX \\] for some finite \\(X\\subseteq G\\), then either \\(\\langle A\\rangle\\) is virtually cyclic, or \\(A\\) has positive exponential growth in the ambient word metric.\n  We also introduce a product-growth criterion for the existence of growth rates of approximate semigroups. The criterion applies to hyperbolic groups: if \\(G\\) is hyperbolic with finite generating set \\(S"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.13632","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/2606.13632/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":"2026-06-12T01:10:20Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"uJ6jTQBBfyOpWih6x9vTai+gOpKGP9adbzk9l1lkdQewbirhaayDr0Ng9p0p2jffhCZJx5PuhsXEXBvISv+nDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-09T06:59:51.480255Z"},"content_sha256":"7bfb7a66160c29fa83da17baf15f99d9a48ac3d4c1ea0bbb7ada537aa671c8d5","schema_version":"1.0","event_id":"sha256:7bfb7a66160c29fa83da17baf15f99d9a48ac3d4c1ea0bbb7ada537aa671c8d5"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/76NZ3EA35XNE4PQ3PZBXV2GXVZ/bundle.json","state_url":"https://pith.science/pith/76NZ3EA35XNE4PQ3PZBXV2GXVZ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/76NZ3EA35XNE4PQ3PZBXV2GXVZ/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":"2026-08-09T06:59:51Z","links":{"resolver":"https://pith.science/pith/76NZ3EA35XNE4PQ3PZBXV2GXVZ","bundle":"https://pith.science/pith/76NZ3EA35XNE4PQ3PZBXV2GXVZ/bundle.json","state":"https://pith.science/pith/76NZ3EA35XNE4PQ3PZBXV2GXVZ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/76NZ3EA35XNE4PQ3PZBXV2GXVZ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:76NZ3EA35XNE4PQ3PZBXV2GXVZ","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":"f8e72503892b15722e54a6572da87e35661e2539a0f9e883acfd7e4655786102","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2026-06-11T17:42:31Z","title_canon_sha256":"e3c259b59cfa4196e5db1180bed82823c5f49380008e1e8001e4808ec885a5d6"},"schema_version":"1.0","source":{"id":"2606.13632","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2606.13632","created_at":"2026-06-12T01:10:20Z"},{"alias_kind":"arxiv_version","alias_value":"2606.13632v1","created_at":"2026-06-12T01:10:20Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.13632","created_at":"2026-06-12T01:10:20Z"},{"alias_kind":"pith_short_12","alias_value":"76NZ3EA35XNE","created_at":"2026-06-12T01:10:20Z"},{"alias_kind":"pith_short_16","alias_value":"76NZ3EA35XNE4PQ3","created_at":"2026-06-12T01:10:20Z"},{"alias_kind":"pith_short_8","alias_value":"76NZ3EA3","created_at":"2026-06-12T01:10:20Z"}],"graph_snapshots":[{"event_id":"sha256:7bfb7a66160c29fa83da17baf15f99d9a48ac3d4c1ea0bbb7ada537aa671c8d5","target":"graph","created_at":"2026-06-12T01:10:20Z","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/2606.13632/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We prove a growth dichotomy for infinite approximate groups, and more generally approximate semigroups, in hyperbolic groups. If \\(G\\) is a finitely generated hyperbolic group and \\(A\\subseteq G\\) is infinite with \\[\n  A^2\\subseteq AX \\] for some finite \\(X\\subseteq G\\), then either \\(\\langle A\\rangle\\) is virtually cyclic, or \\(A\\) has positive exponential growth in the ambient word metric.\n  We also introduce a product-growth criterion for the existence of growth rates of approximate semigroups. The criterion applies to hyperbolic groups: if \\(G\\) is hyperbolic with finite generating set \\(S","authors_text":"Gal Yehuda, Michael Saks","cross_cats":["math.CO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2026-06-11T17:42:31Z","title":"Growth of Approximate Groups in Hyperbolic Groups"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.13632","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":"sha256:99f650a01993db48d47325a45519a7664fd76ac702800536705647b0af730a26","target":"record","created_at":"2026-06-12T01:10:20Z","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":"f8e72503892b15722e54a6572da87e35661e2539a0f9e883acfd7e4655786102","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2026-06-11T17:42:31Z","title_canon_sha256":"e3c259b59cfa4196e5db1180bed82823c5f49380008e1e8001e4808ec885a5d6"},"schema_version":"1.0","source":{"id":"2606.13632","kind":"arxiv","version":1}},"canonical_sha256":"ff9b9d901bedda4e3e1b7e437ae8d7ae556ed167135a0108d03b844b506f19ba","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"ff9b9d901bedda4e3e1b7e437ae8d7ae556ed167135a0108d03b844b506f19ba","first_computed_at":"2026-06-12T01:10:20.110671Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-06-12T01:10:20.110671Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"hVcIkaK3sfDDHa4/AaM1m71a8jmL/xe4gwIybJng+Rk7wSNhCqWO2eCVc1cET2Kg/c5LMMSlG8ENuJtp+0MiAA==","signature_status":"signed_v1","signed_at":"2026-06-12T01:10:20.111656Z","signed_message":"canonical_sha256_bytes"},"source_id":"2606.13632","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:99f650a01993db48d47325a45519a7664fd76ac702800536705647b0af730a26","sha256:7bfb7a66160c29fa83da17baf15f99d9a48ac3d4c1ea0bbb7ada537aa671c8d5"],"state_sha256":"28cffccb50fb590bac9c5ab4616408757f190fc9aac5d04c5a8163a0b4c84286"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"7ZkV5LxA3O/z1C0hVlHApzB40LWZQU87D5J+LRlbeIMgzdH0vKc8oXiEgftNKEYFuXMV5IL0MXQhBASoZStdAw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-09T06:59:51.492898Z","bundle_sha256":"aa3a1e952dee89016099207531bba0766f90d906e49f0bccbebd7a099ce259f4"}}