{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:KZTU2CJJ57M2FO5X2FIPBRSVQ4","short_pith_number":"pith:KZTU2CJJ","canonical_record":{"source":{"id":"2606.06768","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2026-06-04T23:14:19Z","cross_cats_sorted":[],"title_canon_sha256":"a334bcd74d7401d020675be304bc174a1d63f8b0740000a18fe8446f29c1bb26","abstract_canon_sha256":"d1cc5850bf57aaba18161a204b70d18e8b8a1f1da892802b9f8be94b9e11a26d"},"schema_version":"1.0"},"canonical_sha256":"56674d0929efd9a2bbb7d150f0c655871c197537f4cc9fb5600d3e32e59dd485","source":{"kind":"arxiv","id":"2606.06768","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2606.06768","created_at":"2026-06-08T01:04:27Z"},{"alias_kind":"arxiv_version","alias_value":"2606.06768v1","created_at":"2026-06-08T01:04:27Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.06768","created_at":"2026-06-08T01:04:27Z"},{"alias_kind":"pith_short_12","alias_value":"KZTU2CJJ57M2","created_at":"2026-06-08T01:04:27Z"},{"alias_kind":"pith_short_16","alias_value":"KZTU2CJJ57M2FO5X","created_at":"2026-06-08T01:04:27Z"},{"alias_kind":"pith_short_8","alias_value":"KZTU2CJJ","created_at":"2026-06-08T01:04:27Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:KZTU2CJJ57M2FO5X2FIPBRSVQ4","target":"record","payload":{"canonical_record":{"source":{"id":"2606.06768","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2026-06-04T23:14:19Z","cross_cats_sorted":[],"title_canon_sha256":"a334bcd74d7401d020675be304bc174a1d63f8b0740000a18fe8446f29c1bb26","abstract_canon_sha256":"d1cc5850bf57aaba18161a204b70d18e8b8a1f1da892802b9f8be94b9e11a26d"},"schema_version":"1.0"},"canonical_sha256":"56674d0929efd9a2bbb7d150f0c655871c197537f4cc9fb5600d3e32e59dd485","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-08T01:04:27.361074Z","signature_b64":"fF3FQhita0kn9tWJ34SvJszNqMEOgNmc4oOstTnxFzvoRo+uRXUO4c+zXVWFbVHhpmkvWminigPXW7MxvBNzBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"56674d0929efd9a2bbb7d150f0c655871c197537f4cc9fb5600d3e32e59dd485","last_reissued_at":"2026-06-08T01:04:27.360378Z","signature_status":"signed_v1","first_computed_at":"2026-06-08T01:04:27.360378Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2606.06768","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":"2026-06-08T01:04:27Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"i//A0NlURJJpMcvnezqoHd3CxTtBEgk25MTu8BOrdBDpYnhawtQl/2eCl/3iF9r52RIuQpVxu7SYHYm3WicABg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-03T20:10:34.585557Z"},"content_sha256":"347fa40a4df422b310d6fe1ffd1d8c7e3baa0b8f15777b946c3f024850da0d8e","schema_version":"1.0","event_id":"sha256:347fa40a4df422b310d6fe1ffd1d8c7e3baa0b8f15777b946c3f024850da0d8e"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:KZTU2CJJ57M2FO5X2FIPBRSVQ4","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"2-cell embeddings of cubic graphs I. The unstable dual","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Bojan Mohar, MacKenzie Carr","submitted_at":"2026-06-04T23:14:19Z","abstract_excerpt":"In this paper, the first of a two-part series, we explore 2-cell embeddings of cubic graphs, particularly those with small genus. Using local rotations, we introduce a new way of describing the space of 2-cell embeddings and their mutual relationship for any fixed (cubic) graph. We introduce the unstable dual of an embedding of a cubic graph, a subgraph of the dual graph, and describe how the genus of the corresponding embedding can be recovered from properties of the unstable dual. Finally, we characterize the unstable duals of embeddings with genus at most 2 of cubic cyclically 5-edge connec"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.06768","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/2606.06768/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":"2026-06-08T01:04:27Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"kFWwJR2F/n9wRlbi9YTvSPlZMNz48npqsyZ9Sx7bjBHJalgAtPldlJ4frvo3t1RFTxRgEOsPSamDwzVMzPEADQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-03T20:10:34.586060Z"},"content_sha256":"692e1d3ca45960b33a8d704ef7c7eb01175a893526ad66c0f3d7a808b50a3eac","schema_version":"1.0","event_id":"sha256:692e1d3ca45960b33a8d704ef7c7eb01175a893526ad66c0f3d7a808b50a3eac"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/KZTU2CJJ57M2FO5X2FIPBRSVQ4/bundle.json","state_url":"https://pith.science/pith/KZTU2CJJ57M2FO5X2FIPBRSVQ4/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/KZTU2CJJ57M2FO5X2FIPBRSVQ4/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":"2026-08-03T20:10:34Z","links":{"resolver":"https://pith.science/pith/KZTU2CJJ57M2FO5X2FIPBRSVQ4","bundle":"https://pith.science/pith/KZTU2CJJ57M2FO5X2FIPBRSVQ4/bundle.json","state":"https://pith.science/pith/KZTU2CJJ57M2FO5X2FIPBRSVQ4/state.json","well_known_bundle":"https://pith.science/.well-known/pith/KZTU2CJJ57M2FO5X2FIPBRSVQ4/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:KZTU2CJJ57M2FO5X2FIPBRSVQ4","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":"d1cc5850bf57aaba18161a204b70d18e8b8a1f1da892802b9f8be94b9e11a26d","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2026-06-04T23:14:19Z","title_canon_sha256":"a334bcd74d7401d020675be304bc174a1d63f8b0740000a18fe8446f29c1bb26"},"schema_version":"1.0","source":{"id":"2606.06768","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2606.06768","created_at":"2026-06-08T01:04:27Z"},{"alias_kind":"arxiv_version","alias_value":"2606.06768v1","created_at":"2026-06-08T01:04:27Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.06768","created_at":"2026-06-08T01:04:27Z"},{"alias_kind":"pith_short_12","alias_value":"KZTU2CJJ57M2","created_at":"2026-06-08T01:04:27Z"},{"alias_kind":"pith_short_16","alias_value":"KZTU2CJJ57M2FO5X","created_at":"2026-06-08T01:04:27Z"},{"alias_kind":"pith_short_8","alias_value":"KZTU2CJJ","created_at":"2026-06-08T01:04:27Z"}],"graph_snapshots":[{"event_id":"sha256:692e1d3ca45960b33a8d704ef7c7eb01175a893526ad66c0f3d7a808b50a3eac","target":"graph","created_at":"2026-06-08T01:04:27Z","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/2606.06768/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"In this paper, the first of a two-part series, we explore 2-cell embeddings of cubic graphs, particularly those with small genus. Using local rotations, we introduce a new way of describing the space of 2-cell embeddings and their mutual relationship for any fixed (cubic) graph. We introduce the unstable dual of an embedding of a cubic graph, a subgraph of the dual graph, and describe how the genus of the corresponding embedding can be recovered from properties of the unstable dual. Finally, we characterize the unstable duals of embeddings with genus at most 2 of cubic cyclically 5-edge connec","authors_text":"Bojan Mohar, MacKenzie Carr","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2026-06-04T23:14:19Z","title":"2-cell embeddings of cubic graphs I. The unstable dual"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.06768","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":"sha256:347fa40a4df422b310d6fe1ffd1d8c7e3baa0b8f15777b946c3f024850da0d8e","target":"record","created_at":"2026-06-08T01:04:27Z","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":"d1cc5850bf57aaba18161a204b70d18e8b8a1f1da892802b9f8be94b9e11a26d","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2026-06-04T23:14:19Z","title_canon_sha256":"a334bcd74d7401d020675be304bc174a1d63f8b0740000a18fe8446f29c1bb26"},"schema_version":"1.0","source":{"id":"2606.06768","kind":"arxiv","version":1}},"canonical_sha256":"56674d0929efd9a2bbb7d150f0c655871c197537f4cc9fb5600d3e32e59dd485","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"56674d0929efd9a2bbb7d150f0c655871c197537f4cc9fb5600d3e32e59dd485","first_computed_at":"2026-06-08T01:04:27.360378Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-06-08T01:04:27.360378Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"fF3FQhita0kn9tWJ34SvJszNqMEOgNmc4oOstTnxFzvoRo+uRXUO4c+zXVWFbVHhpmkvWminigPXW7MxvBNzBQ==","signature_status":"signed_v1","signed_at":"2026-06-08T01:04:27.361074Z","signed_message":"canonical_sha256_bytes"},"source_id":"2606.06768","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:347fa40a4df422b310d6fe1ffd1d8c7e3baa0b8f15777b946c3f024850da0d8e","sha256:692e1d3ca45960b33a8d704ef7c7eb01175a893526ad66c0f3d7a808b50a3eac"],"state_sha256":"9385c33a902dec3d322fecf4ebdb48bd3cec37f69e44fc8ef3322e8affc28636"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"VxVwh7rQxJyEgS5BCnIQsVGXXLrMT7wKqCIXxvkD7bDh5dFU2TozBkKEDJsns8wDlhPPmD7Sn2k8hzb5c+MmDA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-03T20:10:34.589687Z","bundle_sha256":"388542f03ca3388fdcd71f0ba041854a8cbfd5786a207dbece4b11110d51e851"}}