{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:5ZQEJKOV6XC7J7OMXYBXZKPNDU","short_pith_number":"pith:5ZQEJKOV","canonical_record":{"source":{"id":"2606.06817","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2026-06-05T01:40:16Z","cross_cats_sorted":[],"title_canon_sha256":"1440a9f6564e84106b8315364f24dab486222f50ea11a7761cd3b0316da1208b","abstract_canon_sha256":"251ed5e8c457d64f36871f3c50160eb63a0705368114886e9735057cce70deb5"},"schema_version":"1.0"},"canonical_sha256":"ee6044a9d5f5c5f4fdccbe037ca9ed1d11733945dc838e2636afca032dc9dd10","source":{"kind":"arxiv","id":"2606.06817","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2606.06817","created_at":"2026-06-08T01:04:29Z"},{"alias_kind":"arxiv_version","alias_value":"2606.06817v1","created_at":"2026-06-08T01:04:29Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.06817","created_at":"2026-06-08T01:04:29Z"},{"alias_kind":"pith_short_12","alias_value":"5ZQEJKOV6XC7","created_at":"2026-06-08T01:04:29Z"},{"alias_kind":"pith_short_16","alias_value":"5ZQEJKOV6XC7J7OM","created_at":"2026-06-08T01:04:29Z"},{"alias_kind":"pith_short_8","alias_value":"5ZQEJKOV","created_at":"2026-06-08T01:04:29Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:5ZQEJKOV6XC7J7OMXYBXZKPNDU","target":"record","payload":{"canonical_record":{"source":{"id":"2606.06817","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2026-06-05T01:40:16Z","cross_cats_sorted":[],"title_canon_sha256":"1440a9f6564e84106b8315364f24dab486222f50ea11a7761cd3b0316da1208b","abstract_canon_sha256":"251ed5e8c457d64f36871f3c50160eb63a0705368114886e9735057cce70deb5"},"schema_version":"1.0"},"canonical_sha256":"ee6044a9d5f5c5f4fdccbe037ca9ed1d11733945dc838e2636afca032dc9dd10","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-08T01:04:29.650516Z","signature_b64":"z3PzyRdNvzgTYyTTzAHijk3qRv6ffk8rCyxtKgcVCBdOq4sFenIQRyYmZngTaxjeZS8Lc7Eroqt8jWyJqt+qBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ee6044a9d5f5c5f4fdccbe037ca9ed1d11733945dc838e2636afca032dc9dd10","last_reissued_at":"2026-06-08T01:04:29.649983Z","signature_status":"signed_v1","first_computed_at":"2026-06-08T01:04:29.649983Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2606.06817","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":"2026-06-08T01:04:29Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Z2Sq1/UJWwlG++dMg1L97D7KYDlyeHvLRYqRCGN+CU0qQH60SrdlC5lkbhNJQb5h3b/nb9JK++SVkjT7pRDLCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-24T18:08:31.691787Z"},"content_sha256":"02efee17cbc943e7270fdd5a09e0e71e9af893e639301d28795713f655fb2acf","schema_version":"1.0","event_id":"sha256:02efee17cbc943e7270fdd5a09e0e71e9af893e639301d28795713f655fb2acf"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:5ZQEJKOV6XC7J7OMXYBXZKPNDU","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"C4-face-magic labeling on a 4x4 Klein bottle grid graph","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Stephen J. Curran, Timothy Myers","submitted_at":"2026-06-05T01:40:16Z","abstract_excerpt":"For a graph G = (V, E) embedded in the Klein bottle, let F(G) denote the set of faces of G. A C_4-face-magic Klein bottle labeling on G is a bijection f: V(G) to {1, 2,..., |V(G)|} such that for any F in F(G) with F isomorphic C_4, the sum of all the vertex labelings along C_4 is a constant. We say that a C_4-face-magic labeling X={x_{i,j} : 0< i,j< 5} on the 4x4 Klein bottle grid graph is horizontally (or vertically) pairwise balanced if x_{2i-1,j} + x_{2i,j}=17 for 0< i <3 and 0< j \\le <5 (or x_{i,2j-1} + x_{i2,j}=17 for 0< i <5 and 0< j <3). We show that the 4x4 Klein bottle grid graph has "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.06817","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/2606.06817/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":"2026-06-08T01:04:29Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"f3TeDQ380DXnelGkDUrV2e8p+tQeJMvw4vVS74xDYRGLx3vC13dABuTUMbgLj9hkyJ0yfghbG5DFz95BT7MHCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-24T18:08:31.692162Z"},"content_sha256":"72d1a36da855dfd488f9a22e1b5b6f5a1a1aea7917060999a9cb3bade828c6ad","schema_version":"1.0","event_id":"sha256:72d1a36da855dfd488f9a22e1b5b6f5a1a1aea7917060999a9cb3bade828c6ad"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/5ZQEJKOV6XC7J7OMXYBXZKPNDU/bundle.json","state_url":"https://pith.science/pith/5ZQEJKOV6XC7J7OMXYBXZKPNDU/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/5ZQEJKOV6XC7J7OMXYBXZKPNDU/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":"2026-06-24T18:08:31Z","links":{"resolver":"https://pith.science/pith/5ZQEJKOV6XC7J7OMXYBXZKPNDU","bundle":"https://pith.science/pith/5ZQEJKOV6XC7J7OMXYBXZKPNDU/bundle.json","state":"https://pith.science/pith/5ZQEJKOV6XC7J7OMXYBXZKPNDU/state.json","well_known_bundle":"https://pith.science/.well-known/pith/5ZQEJKOV6XC7J7OMXYBXZKPNDU/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:5ZQEJKOV6XC7J7OMXYBXZKPNDU","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":"251ed5e8c457d64f36871f3c50160eb63a0705368114886e9735057cce70deb5","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2026-06-05T01:40:16Z","title_canon_sha256":"1440a9f6564e84106b8315364f24dab486222f50ea11a7761cd3b0316da1208b"},"schema_version":"1.0","source":{"id":"2606.06817","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2606.06817","created_at":"2026-06-08T01:04:29Z"},{"alias_kind":"arxiv_version","alias_value":"2606.06817v1","created_at":"2026-06-08T01:04:29Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.06817","created_at":"2026-06-08T01:04:29Z"},{"alias_kind":"pith_short_12","alias_value":"5ZQEJKOV6XC7","created_at":"2026-06-08T01:04:29Z"},{"alias_kind":"pith_short_16","alias_value":"5ZQEJKOV6XC7J7OM","created_at":"2026-06-08T01:04:29Z"},{"alias_kind":"pith_short_8","alias_value":"5ZQEJKOV","created_at":"2026-06-08T01:04:29Z"}],"graph_snapshots":[{"event_id":"sha256:72d1a36da855dfd488f9a22e1b5b6f5a1a1aea7917060999a9cb3bade828c6ad","target":"graph","created_at":"2026-06-08T01:04:29Z","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/2606.06817/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"For a graph G = (V, E) embedded in the Klein bottle, let F(G) denote the set of faces of G. A C_4-face-magic Klein bottle labeling on G is a bijection f: V(G) to {1, 2,..., |V(G)|} such that for any F in F(G) with F isomorphic C_4, the sum of all the vertex labelings along C_4 is a constant. We say that a C_4-face-magic labeling X={x_{i,j} : 0< i,j< 5} on the 4x4 Klein bottle grid graph is horizontally (or vertically) pairwise balanced if x_{2i-1,j} + x_{2i,j}=17 for 0< i <3 and 0< j \\le <5 (or x_{i,2j-1} + x_{i2,j}=17 for 0< i <5 and 0< j <3). We show that the 4x4 Klein bottle grid graph has ","authors_text":"Stephen J. Curran, Timothy Myers","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2026-06-05T01:40:16Z","title":"C4-face-magic labeling on a 4x4 Klein bottle grid graph"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.06817","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":"sha256:02efee17cbc943e7270fdd5a09e0e71e9af893e639301d28795713f655fb2acf","target":"record","created_at":"2026-06-08T01:04:29Z","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":"251ed5e8c457d64f36871f3c50160eb63a0705368114886e9735057cce70deb5","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2026-06-05T01:40:16Z","title_canon_sha256":"1440a9f6564e84106b8315364f24dab486222f50ea11a7761cd3b0316da1208b"},"schema_version":"1.0","source":{"id":"2606.06817","kind":"arxiv","version":1}},"canonical_sha256":"ee6044a9d5f5c5f4fdccbe037ca9ed1d11733945dc838e2636afca032dc9dd10","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"ee6044a9d5f5c5f4fdccbe037ca9ed1d11733945dc838e2636afca032dc9dd10","first_computed_at":"2026-06-08T01:04:29.649983Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-06-08T01:04:29.649983Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"z3PzyRdNvzgTYyTTzAHijk3qRv6ffk8rCyxtKgcVCBdOq4sFenIQRyYmZngTaxjeZS8Lc7Eroqt8jWyJqt+qBQ==","signature_status":"signed_v1","signed_at":"2026-06-08T01:04:29.650516Z","signed_message":"canonical_sha256_bytes"},"source_id":"2606.06817","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:02efee17cbc943e7270fdd5a09e0e71e9af893e639301d28795713f655fb2acf","sha256:72d1a36da855dfd488f9a22e1b5b6f5a1a1aea7917060999a9cb3bade828c6ad"],"state_sha256":"9d5585384335ce0774984ab75ad3ee6179ede68a8d100d2a84a842c75618e034"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"qAznRTpFaVNPPk9N3PrBFpiuH6mW3jqzqxaolj8yx1FJHMDgBcYS5iqSP9dBkYrKc+huZikYvuELIFS8AbCDBQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-24T18:08:31.694175Z","bundle_sha256":"4ee95e529cd165d486c348da2df975b836209ed4afb7b5e1c2d2ae272358ff9a"}}