{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2024:YH732K4XG4FOJYHO7TW7FQ37KS","short_pith_number":"pith:YH732K4X","canonical_record":{"source":{"id":"2409.04438","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.GR","submitted_at":"2024-09-06T17:57:56Z","cross_cats_sorted":["math.CV"],"title_canon_sha256":"9d783fa53e7fd00af95a3b2d975557cdfbe6fc652f36456073a6563e18523d19","abstract_canon_sha256":"5f55c7825aa2249977e23532bfd9e0883b1afb40c953bc507bbe394410a32d15"},"schema_version":"1.0"},"canonical_sha256":"c1ffbd2b97370ae4e0eefcedf2c37f54bd16c68de096b520f9ea095343d957a9","source":{"kind":"arxiv","id":"2409.04438","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2409.04438","created_at":"2026-07-05T09:04:05Z"},{"alias_kind":"arxiv_version","alias_value":"2409.04438v1","created_at":"2026-07-05T09:04:05Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2409.04438","created_at":"2026-07-05T09:04:05Z"},{"alias_kind":"pith_short_12","alias_value":"YH732K4XG4FO","created_at":"2026-07-05T09:04:05Z"},{"alias_kind":"pith_short_16","alias_value":"YH732K4XG4FOJYHO","created_at":"2026-07-05T09:04:05Z"},{"alias_kind":"pith_short_8","alias_value":"YH732K4X","created_at":"2026-07-05T09:04:05Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2024:YH732K4XG4FOJYHO7TW7FQ37KS","target":"record","payload":{"canonical_record":{"source":{"id":"2409.04438","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.GR","submitted_at":"2024-09-06T17:57:56Z","cross_cats_sorted":["math.CV"],"title_canon_sha256":"9d783fa53e7fd00af95a3b2d975557cdfbe6fc652f36456073a6563e18523d19","abstract_canon_sha256":"5f55c7825aa2249977e23532bfd9e0883b1afb40c953bc507bbe394410a32d15"},"schema_version":"1.0"},"canonical_sha256":"c1ffbd2b97370ae4e0eefcedf2c37f54bd16c68de096b520f9ea095343d957a9","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T09:04:05.676986Z","signature_b64":"gZp020K91hjCf1FklKDhek4VR/6qlC7JkB7phzTGjNSUGV+l6ebYdEwhMqPSzpJ+3wlLNP7Eanp06k5CyQ3nCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c1ffbd2b97370ae4e0eefcedf2c37f54bd16c68de096b520f9ea095343d957a9","last_reissued_at":"2026-07-05T09:04:05.676519Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T09:04:05.676519Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2409.04438","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":"2026-07-05T09:04:05Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"g8HJz/9DehicAhYcQnoNVAgD7QNVfUoZjAK4NZjAhJS0szcL05GaG5B2XqUhL5dm1uVPGKL5eTunZpdZIMdOAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-04T06:07:07.366326Z"},"content_sha256":"424fb6b75c00293d800b06c9d0e89880e0399eaca6809c3845db885e7f8db4b8","schema_version":"1.0","event_id":"sha256:424fb6b75c00293d800b06c9d0e89880e0399eaca6809c3845db885e7f8db4b8"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2024:YH732K4XG4FOJYHO7TW7FQ37KS","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On Thin Heckoid and Generalised Triangle Groups in $PSL(2,\\mathbb{C})$}","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.CV"],"primary_cat":"math.GR","authors_text":"Alex Elzenaar, Gaven Martin, Jeroen Schillewaert","submitted_at":"2024-09-06T17:57:56Z","abstract_excerpt":"We provide a brief overview of our upcoming work identifying all the thin Heckoid groups in $PSL(2,\\mathbb{C})$. Here we give a complete list of the $55$ thin generalised triangle groups of slope $1/2$. This work was presented at the conference Computational Aspects of Thin Groups, IMSS, Singapore and presents an application of joint work initiated with Colin Maclachlan"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2409.04438","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/2409.04438/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":"2026-07-05T09:04:05Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"wP4xSSoRl+tp1ByWkYtHQnueOdUeB8BtGfDNDDoy2+y30VGQidrifue4OIKe/CM8O4LPXTTIjvEiy9MTq/FUDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-04T06:07:07.366698Z"},"content_sha256":"5ac8945248c902759a39bbc2f2eae120baa90766d297a1142265403f935eb76a","schema_version":"1.0","event_id":"sha256:5ac8945248c902759a39bbc2f2eae120baa90766d297a1142265403f935eb76a"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/YH732K4XG4FOJYHO7TW7FQ37KS/bundle.json","state_url":"https://pith.science/pith/YH732K4XG4FOJYHO7TW7FQ37KS/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/YH732K4XG4FOJYHO7TW7FQ37KS/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":"2026-08-04T06:07:07Z","links":{"resolver":"https://pith.science/pith/YH732K4XG4FOJYHO7TW7FQ37KS","bundle":"https://pith.science/pith/YH732K4XG4FOJYHO7TW7FQ37KS/bundle.json","state":"https://pith.science/pith/YH732K4XG4FOJYHO7TW7FQ37KS/state.json","well_known_bundle":"https://pith.science/.well-known/pith/YH732K4XG4FOJYHO7TW7FQ37KS/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2024:YH732K4XG4FOJYHO7TW7FQ37KS","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":"5f55c7825aa2249977e23532bfd9e0883b1afb40c953bc507bbe394410a32d15","cross_cats_sorted":["math.CV"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.GR","submitted_at":"2024-09-06T17:57:56Z","title_canon_sha256":"9d783fa53e7fd00af95a3b2d975557cdfbe6fc652f36456073a6563e18523d19"},"schema_version":"1.0","source":{"id":"2409.04438","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2409.04438","created_at":"2026-07-05T09:04:05Z"},{"alias_kind":"arxiv_version","alias_value":"2409.04438v1","created_at":"2026-07-05T09:04:05Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2409.04438","created_at":"2026-07-05T09:04:05Z"},{"alias_kind":"pith_short_12","alias_value":"YH732K4XG4FO","created_at":"2026-07-05T09:04:05Z"},{"alias_kind":"pith_short_16","alias_value":"YH732K4XG4FOJYHO","created_at":"2026-07-05T09:04:05Z"},{"alias_kind":"pith_short_8","alias_value":"YH732K4X","created_at":"2026-07-05T09:04:05Z"}],"graph_snapshots":[{"event_id":"sha256:5ac8945248c902759a39bbc2f2eae120baa90766d297a1142265403f935eb76a","target":"graph","created_at":"2026-07-05T09:04:05Z","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/2409.04438/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We provide a brief overview of our upcoming work identifying all the thin Heckoid groups in $PSL(2,\\mathbb{C})$. Here we give a complete list of the $55$ thin generalised triangle groups of slope $1/2$. This work was presented at the conference Computational Aspects of Thin Groups, IMSS, Singapore and presents an application of joint work initiated with Colin Maclachlan","authors_text":"Alex Elzenaar, Gaven Martin, Jeroen Schillewaert","cross_cats":["math.CV"],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.GR","submitted_at":"2024-09-06T17:57:56Z","title":"On Thin Heckoid and Generalised Triangle Groups in $PSL(2,\\mathbb{C})$}"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2409.04438","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":"sha256:424fb6b75c00293d800b06c9d0e89880e0399eaca6809c3845db885e7f8db4b8","target":"record","created_at":"2026-07-05T09:04:05Z","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":"5f55c7825aa2249977e23532bfd9e0883b1afb40c953bc507bbe394410a32d15","cross_cats_sorted":["math.CV"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.GR","submitted_at":"2024-09-06T17:57:56Z","title_canon_sha256":"9d783fa53e7fd00af95a3b2d975557cdfbe6fc652f36456073a6563e18523d19"},"schema_version":"1.0","source":{"id":"2409.04438","kind":"arxiv","version":1}},"canonical_sha256":"c1ffbd2b97370ae4e0eefcedf2c37f54bd16c68de096b520f9ea095343d957a9","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"c1ffbd2b97370ae4e0eefcedf2c37f54bd16c68de096b520f9ea095343d957a9","first_computed_at":"2026-07-05T09:04:05.676519Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T09:04:05.676519Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"gZp020K91hjCf1FklKDhek4VR/6qlC7JkB7phzTGjNSUGV+l6ebYdEwhMqPSzpJ+3wlLNP7Eanp06k5CyQ3nCg==","signature_status":"signed_v1","signed_at":"2026-07-05T09:04:05.676986Z","signed_message":"canonical_sha256_bytes"},"source_id":"2409.04438","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:424fb6b75c00293d800b06c9d0e89880e0399eaca6809c3845db885e7f8db4b8","sha256:5ac8945248c902759a39bbc2f2eae120baa90766d297a1142265403f935eb76a"],"state_sha256":"5210c00867796e5d26e75fd643fb9264059bda880aad4d3e5981e572bec3a9c9"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"FUQGaMVRd06kemho4ttE+2koD66qvJ29PCjkJ8s5WmBuZ7tKhHuTjpywOcyhi+k/frwva/klLV58uaCixF6zAg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-04T06:07:07.368963Z","bundle_sha256":"3a4d17c0928accfe01ee8be9ea74f31e3b69719de3b900f5643994be2599fcce"}}