{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2013:OJGD6CJS36X23WIO675H4RN2Q3","short_pith_number":"pith:OJGD6CJS","canonical_record":{"source":{"id":"1306.6025","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2013-06-25T16:37:23Z","cross_cats_sorted":[],"title_canon_sha256":"2e819e0d750d90f67957b576d57eea740e821b61775cf8152ac623e22ebb7fb7","abstract_canon_sha256":"dfdea92b56d5568f732ab3bce78d07ff76210538d920971b21f49d231356e319"},"schema_version":"1.0"},"canonical_sha256":"724c3f0932dfafadd90ef7fa7e45ba86dbfe32d703e56ace24b663279a20a9df","source":{"kind":"arxiv","id":"1306.6025","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1306.6025","created_at":"2026-05-17T23:54:42Z"},{"alias_kind":"arxiv_version","alias_value":"1306.6025v3","created_at":"2026-05-17T23:54:42Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1306.6025","created_at":"2026-05-17T23:54:42Z"},{"alias_kind":"pith_short_12","alias_value":"OJGD6CJS36X2","created_at":"2026-05-18T12:27:54Z"},{"alias_kind":"pith_short_16","alias_value":"OJGD6CJS36X23WIO","created_at":"2026-05-18T12:27:54Z"},{"alias_kind":"pith_short_8","alias_value":"OJGD6CJS","created_at":"2026-05-18T12:27:54Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2013:OJGD6CJS36X23WIO675H4RN2Q3","target":"record","payload":{"canonical_record":{"source":{"id":"1306.6025","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2013-06-25T16:37:23Z","cross_cats_sorted":[],"title_canon_sha256":"2e819e0d750d90f67957b576d57eea740e821b61775cf8152ac623e22ebb7fb7","abstract_canon_sha256":"dfdea92b56d5568f732ab3bce78d07ff76210538d920971b21f49d231356e319"},"schema_version":"1.0"},"canonical_sha256":"724c3f0932dfafadd90ef7fa7e45ba86dbfe32d703e56ace24b663279a20a9df","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:54:42.117979Z","signature_b64":"z3ZJXri1bLlGWgaiLqSptmf+gmsuHMfYdbycMoA9kKThzDKt4di4TF5ewsY4Cvx6UIR4/uNezYuS6JsWSYYxAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"724c3f0932dfafadd90ef7fa7e45ba86dbfe32d703e56ace24b663279a20a9df","last_reissued_at":"2026-05-17T23:54:42.117393Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:54:42.117393Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1306.6025","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-17T23:54:42Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"S8MbqjbBOwCdg48NJRrh6UFhySdKQHLiJQb37I9oQQY5AqNwaWGlCg9Ljc7zcuzSxCJv8zocc6Q9uymECDcdDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T02:29:38.926149Z"},"content_sha256":"14cb3990c5da4beade8ab09b01b06fc4a9e7bb8f866d447b6348afbeea409c97","schema_version":"1.0","event_id":"sha256:14cb3990c5da4beade8ab09b01b06fc4a9e7bb8f866d447b6348afbeea409c97"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2013:OJGD6CJS36X23WIO675H4RN2Q3","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Coxeter groups, hyperbolic cubes, and acute triangulations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GT","authors_text":"Genevieve S. Walsh, Sang-hyun Kim","submitted_at":"2013-06-25T16:37:23Z","abstract_excerpt":"Let $C(L)$ be the right-angled Coxeter group defined by an abstract triangulation $L$ of $\\mathbb{S}^2$. We show that $C(L)$ is isomorphic to a hyperbolic right-angled reflection group if and only if $L$ can be realized as an acute triangulation. The proof relies on the theory of CAT(-1) spaces. A corollary is that an abstract triangulation of $\\mathbb{S}^2$ can be realized as an acute triangulation exactly when it satisfies a combinatorial condition called \"flag no-square\". We also study generalizations of this result to other angle bounds, other planar surfaces and other dimensions."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1306.6025","kind":"arxiv","version":3},"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":"2026-05-17T23:54:42Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"bPev4RGeJF4PAQYSAxAKDp/xemFclayeCCgxewhs83BkIcrZMwUZrXE2YJWhtuUcY4WPJSjXcsUuiYaaFO/NBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T02:29:38.926748Z"},"content_sha256":"4485c399a732cd003e97de772143af70ee818e1cbf8a38570b811033475cc243","schema_version":"1.0","event_id":"sha256:4485c399a732cd003e97de772143af70ee818e1cbf8a38570b811033475cc243"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/OJGD6CJS36X23WIO675H4RN2Q3/bundle.json","state_url":"https://pith.science/pith/OJGD6CJS36X23WIO675H4RN2Q3/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/OJGD6CJS36X23WIO675H4RN2Q3/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":"2026-05-27T02:29:38Z","links":{"resolver":"https://pith.science/pith/OJGD6CJS36X23WIO675H4RN2Q3","bundle":"https://pith.science/pith/OJGD6CJS36X23WIO675H4RN2Q3/bundle.json","state":"https://pith.science/pith/OJGD6CJS36X23WIO675H4RN2Q3/state.json","well_known_bundle":"https://pith.science/.well-known/pith/OJGD6CJS36X23WIO675H4RN2Q3/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:OJGD6CJS36X23WIO675H4RN2Q3","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":"dfdea92b56d5568f732ab3bce78d07ff76210538d920971b21f49d231356e319","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2013-06-25T16:37:23Z","title_canon_sha256":"2e819e0d750d90f67957b576d57eea740e821b61775cf8152ac623e22ebb7fb7"},"schema_version":"1.0","source":{"id":"1306.6025","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1306.6025","created_at":"2026-05-17T23:54:42Z"},{"alias_kind":"arxiv_version","alias_value":"1306.6025v3","created_at":"2026-05-17T23:54:42Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1306.6025","created_at":"2026-05-17T23:54:42Z"},{"alias_kind":"pith_short_12","alias_value":"OJGD6CJS36X2","created_at":"2026-05-18T12:27:54Z"},{"alias_kind":"pith_short_16","alias_value":"OJGD6CJS36X23WIO","created_at":"2026-05-18T12:27:54Z"},{"alias_kind":"pith_short_8","alias_value":"OJGD6CJS","created_at":"2026-05-18T12:27:54Z"}],"graph_snapshots":[{"event_id":"sha256:4485c399a732cd003e97de772143af70ee818e1cbf8a38570b811033475cc243","target":"graph","created_at":"2026-05-17T23:54:42Z","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 $C(L)$ be the right-angled Coxeter group defined by an abstract triangulation $L$ of $\\mathbb{S}^2$. We show that $C(L)$ is isomorphic to a hyperbolic right-angled reflection group if and only if $L$ can be realized as an acute triangulation. The proof relies on the theory of CAT(-1) spaces. A corollary is that an abstract triangulation of $\\mathbb{S}^2$ can be realized as an acute triangulation exactly when it satisfies a combinatorial condition called \"flag no-square\". We also study generalizations of this result to other angle bounds, other planar surfaces and other dimensions.","authors_text":"Genevieve S. Walsh, Sang-hyun Kim","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2013-06-25T16:37:23Z","title":"Coxeter groups, hyperbolic cubes, and acute triangulations"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1306.6025","kind":"arxiv","version":3},"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":"sha256:14cb3990c5da4beade8ab09b01b06fc4a9e7bb8f866d447b6348afbeea409c97","target":"record","created_at":"2026-05-17T23:54:42Z","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":"dfdea92b56d5568f732ab3bce78d07ff76210538d920971b21f49d231356e319","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2013-06-25T16:37:23Z","title_canon_sha256":"2e819e0d750d90f67957b576d57eea740e821b61775cf8152ac623e22ebb7fb7"},"schema_version":"1.0","source":{"id":"1306.6025","kind":"arxiv","version":3}},"canonical_sha256":"724c3f0932dfafadd90ef7fa7e45ba86dbfe32d703e56ace24b663279a20a9df","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"724c3f0932dfafadd90ef7fa7e45ba86dbfe32d703e56ace24b663279a20a9df","first_computed_at":"2026-05-17T23:54:42.117393Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:54:42.117393Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"z3ZJXri1bLlGWgaiLqSptmf+gmsuHMfYdbycMoA9kKThzDKt4di4TF5ewsY4Cvx6UIR4/uNezYuS6JsWSYYxAw==","signature_status":"signed_v1","signed_at":"2026-05-17T23:54:42.117979Z","signed_message":"canonical_sha256_bytes"},"source_id":"1306.6025","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:14cb3990c5da4beade8ab09b01b06fc4a9e7bb8f866d447b6348afbeea409c97","sha256:4485c399a732cd003e97de772143af70ee818e1cbf8a38570b811033475cc243"],"state_sha256":"e5cab0818a69756150aee1b799eff6f09dc5e63545c0113c21c467394fbc7764"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"VU9kbWTmcrNFJV6J90sRVLgYfC/8I/IP0yu4TgJ9ep/OgJlbFpqcvdUs+X14RYkfGtfegOe3MMTXqL6NgYGhDA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-27T02:29:38.930141Z","bundle_sha256":"7ca718ca0166cb6872d77674c631e9e67c5e4a709366f5a998d4a30856d5595e"}}