{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2021:Y4QR4UP5Q74BP6BRKYZKY2X4EE","short_pith_number":"pith:Y4QR4UP5","canonical_record":{"source":{"id":"2106.08012","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2021-06-15T09:46:13Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"90728192a43b4dc20d95cab95bbb9fc0b6cad31ed4122f6fe0ad732f7a38ccaa","abstract_canon_sha256":"10be381e08465d1f805bc1a046c85999b0096305a1eb9bf0bb2f63a7b0b394de"},"schema_version":"1.0"},"canonical_sha256":"c7211e51fd87f817f8315632ac6afc213327d949ddea53462d81c28706a6fb5a","source":{"kind":"arxiv","id":"2106.08012","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2106.08012","created_at":"2026-07-05T03:13:29Z"},{"alias_kind":"arxiv_version","alias_value":"2106.08012v2","created_at":"2026-07-05T03:13:29Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2106.08012","created_at":"2026-07-05T03:13:29Z"},{"alias_kind":"pith_short_12","alias_value":"Y4QR4UP5Q74B","created_at":"2026-07-05T03:13:29Z"},{"alias_kind":"pith_short_16","alias_value":"Y4QR4UP5Q74BP6BR","created_at":"2026-07-05T03:13:29Z"},{"alias_kind":"pith_short_8","alias_value":"Y4QR4UP5","created_at":"2026-07-05T03:13:29Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2021:Y4QR4UP5Q74BP6BRKYZKY2X4EE","target":"record","payload":{"canonical_record":{"source":{"id":"2106.08012","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2021-06-15T09:46:13Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"90728192a43b4dc20d95cab95bbb9fc0b6cad31ed4122f6fe0ad732f7a38ccaa","abstract_canon_sha256":"10be381e08465d1f805bc1a046c85999b0096305a1eb9bf0bb2f63a7b0b394de"},"schema_version":"1.0"},"canonical_sha256":"c7211e51fd87f817f8315632ac6afc213327d949ddea53462d81c28706a6fb5a","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T03:13:29.922639Z","signature_b64":"W5k6hdQF29POWPNQLwbHfJcKasByfiYxax+A/QvdJORFoSJ3eHNGBsfCLRSXM2RZ4dpSSIxpwx0upQkjDPt7Cw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c7211e51fd87f817f8315632ac6afc213327d949ddea53462d81c28706a6fb5a","last_reissued_at":"2026-07-05T03:13:29.922185Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T03:13:29.922185Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2106.08012","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":"2026-07-05T03:13:29Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"CXgn8eGxqKsSzCMI4Znoz7Q2mt71bRuQFsKpJE27lY+201L/N7iRNz3g7SmlNBywnBZOntkEiuiNBw7Kn1cGCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-04T05:16:23.587086Z"},"content_sha256":"725958d57c0ed852681fb526ddbd06eb3f9eeaaf0b68d75cdfb94a3263cd52d6","schema_version":"1.0","event_id":"sha256:725958d57c0ed852681fb526ddbd06eb3f9eeaaf0b68d75cdfb94a3263cd52d6"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2021:Y4QR4UP5Q74BP6BRKYZKY2X4EE","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Strong convexity in flip-graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.GT","authors_text":"Lionel Pournin, Zili Wang","submitted_at":"2021-06-15T09:46:13Z","abstract_excerpt":"The triangulations of a surface $\\Sigma$ with a prescribed set of vertices can be endowed with a graph structure $\\mathcal{F}(\\Sigma)$. Its edges connect two triangulations that differ by a single arc. It is known that, when $\\Sigma$ is a convex polygon or a topological surface, the subgraph $\\mathcal{F}_\\varepsilon(\\Sigma)$ induced in $\\mathcal{F}(\\Sigma)$ by the triangulations that contain a given arc $\\varepsilon$ is strongly convex in the sense that all the geodesic paths between two such triangulations remain in that subgraph. Here, we provide a related result that involves a triangle ins"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2106.08012","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2106.08012/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":"2026-07-05T03:13:29Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"MDGLuftF3GDticAarMO7A7a6sC8IjlCujOD9xPYQVX2jiN+iXh6RUpLSW1272SdZWRsa4niY4dTwCvXoxrioAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-04T05:16:23.587743Z"},"content_sha256":"8265c0470186da13a80439d648f2f06c14f93bd941b887947a4343f9d5cfe7d8","schema_version":"1.0","event_id":"sha256:8265c0470186da13a80439d648f2f06c14f93bd941b887947a4343f9d5cfe7d8"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/Y4QR4UP5Q74BP6BRKYZKY2X4EE/bundle.json","state_url":"https://pith.science/pith/Y4QR4UP5Q74BP6BRKYZKY2X4EE/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/Y4QR4UP5Q74BP6BRKYZKY2X4EE/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":"2026-08-04T05:16:23Z","links":{"resolver":"https://pith.science/pith/Y4QR4UP5Q74BP6BRKYZKY2X4EE","bundle":"https://pith.science/pith/Y4QR4UP5Q74BP6BRKYZKY2X4EE/bundle.json","state":"https://pith.science/pith/Y4QR4UP5Q74BP6BRKYZKY2X4EE/state.json","well_known_bundle":"https://pith.science/.well-known/pith/Y4QR4UP5Q74BP6BRKYZKY2X4EE/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2021:Y4QR4UP5Q74BP6BRKYZKY2X4EE","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":"10be381e08465d1f805bc1a046c85999b0096305a1eb9bf0bb2f63a7b0b394de","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2021-06-15T09:46:13Z","title_canon_sha256":"90728192a43b4dc20d95cab95bbb9fc0b6cad31ed4122f6fe0ad732f7a38ccaa"},"schema_version":"1.0","source":{"id":"2106.08012","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2106.08012","created_at":"2026-07-05T03:13:29Z"},{"alias_kind":"arxiv_version","alias_value":"2106.08012v2","created_at":"2026-07-05T03:13:29Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2106.08012","created_at":"2026-07-05T03:13:29Z"},{"alias_kind":"pith_short_12","alias_value":"Y4QR4UP5Q74B","created_at":"2026-07-05T03:13:29Z"},{"alias_kind":"pith_short_16","alias_value":"Y4QR4UP5Q74BP6BR","created_at":"2026-07-05T03:13:29Z"},{"alias_kind":"pith_short_8","alias_value":"Y4QR4UP5","created_at":"2026-07-05T03:13:29Z"}],"graph_snapshots":[{"event_id":"sha256:8265c0470186da13a80439d648f2f06c14f93bd941b887947a4343f9d5cfe7d8","target":"graph","created_at":"2026-07-05T03:13:29Z","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/2106.08012/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"The triangulations of a surface $\\Sigma$ with a prescribed set of vertices can be endowed with a graph structure $\\mathcal{F}(\\Sigma)$. Its edges connect two triangulations that differ by a single arc. It is known that, when $\\Sigma$ is a convex polygon or a topological surface, the subgraph $\\mathcal{F}_\\varepsilon(\\Sigma)$ induced in $\\mathcal{F}(\\Sigma)$ by the triangulations that contain a given arc $\\varepsilon$ is strongly convex in the sense that all the geodesic paths between two such triangulations remain in that subgraph. Here, we provide a related result that involves a triangle ins","authors_text":"Lionel Pournin, Zili Wang","cross_cats":["math.CO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2021-06-15T09:46:13Z","title":"Strong convexity in flip-graphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2106.08012","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":"sha256:725958d57c0ed852681fb526ddbd06eb3f9eeaaf0b68d75cdfb94a3263cd52d6","target":"record","created_at":"2026-07-05T03:13:29Z","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":"10be381e08465d1f805bc1a046c85999b0096305a1eb9bf0bb2f63a7b0b394de","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2021-06-15T09:46:13Z","title_canon_sha256":"90728192a43b4dc20d95cab95bbb9fc0b6cad31ed4122f6fe0ad732f7a38ccaa"},"schema_version":"1.0","source":{"id":"2106.08012","kind":"arxiv","version":2}},"canonical_sha256":"c7211e51fd87f817f8315632ac6afc213327d949ddea53462d81c28706a6fb5a","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"c7211e51fd87f817f8315632ac6afc213327d949ddea53462d81c28706a6fb5a","first_computed_at":"2026-07-05T03:13:29.922185Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T03:13:29.922185Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"W5k6hdQF29POWPNQLwbHfJcKasByfiYxax+A/QvdJORFoSJ3eHNGBsfCLRSXM2RZ4dpSSIxpwx0upQkjDPt7Cw==","signature_status":"signed_v1","signed_at":"2026-07-05T03:13:29.922639Z","signed_message":"canonical_sha256_bytes"},"source_id":"2106.08012","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:725958d57c0ed852681fb526ddbd06eb3f9eeaaf0b68d75cdfb94a3263cd52d6","sha256:8265c0470186da13a80439d648f2f06c14f93bd941b887947a4343f9d5cfe7d8"],"state_sha256":"351f8bcdcbb97e871316e9ad973213a1dae198d22cd92571d1b759b74c881adc"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"tyDS8lsP/GtJiF2VFJYAAyeQmFCBxV79arWApBOgxQq/ec7Rn6BhHQ72L5boSRgqistaQizyTsVoGwxklQPjAA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-04T05:16:23.593945Z","bundle_sha256":"223035ab44cf08e0c997d068aad96d9eed2767a269c3a44547eb75598bd9114f"}}