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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 "},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"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"},"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"},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"2606.06817","created_at":"2026-06-08T01:04:29.650058+00:00"},{"alias_kind":"arxiv_version","alias_value":"2606.06817v1","created_at":"2026-06-08T01:04:29.650058+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.06817","created_at":"2026-06-08T01:04:29.650058+00:00"},{"alias_kind":"pith_short_12","alias_value":"5ZQEJKOV6XC7","created_at":"2026-06-08T01:04:29.650058+00:00"},{"alias_kind":"pith_short_16","alias_value":"5ZQEJKOV6XC7J7OM","created_at":"2026-06-08T01:04:29.650058+00:00"},{"alias_kind":"pith_short_8","alias_value":"5ZQEJKOV","created_at":"2026-06-08T01:04:29.650058+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/5ZQEJKOV6XC7J7OMXYBXZKPNDU","json":"https://pith.science/pith/5ZQEJKOV6XC7J7OMXYBXZKPNDU.json","graph_json":"https://pith.science/api/pith-number/5ZQEJKOV6XC7J7OMXYBXZKPNDU/graph.json","events_json":"https://pith.science/api/pith-number/5ZQEJKOV6XC7J7OMXYBXZKPNDU/events.json","paper":"https://pith.science/paper/5ZQEJKOV"},"agent_actions":{"view_html":"https://pith.science/pith/5ZQEJKOV6XC7J7OMXYBXZKPNDU","download_json":"https://pith.science/pith/5ZQEJKOV6XC7J7OMXYBXZKPNDU.json","view_paper":"https://pith.science/paper/5ZQEJKOV","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2606.06817&json=true","fetch_graph":"https://pith.science/api/pith-number/5ZQEJKOV6XC7J7OMXYBXZKPNDU/graph.json","fetch_events":"https://pith.science/api/pith-number/5ZQEJKOV6XC7J7OMXYBXZKPNDU/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/5ZQEJKOV6XC7J7OMXYBXZKPNDU/action/timestamp_anchor","attest_storage":"https://pith.science/pith/5ZQEJKOV6XC7J7OMXYBXZKPNDU/action/storage_attestation","attest_author":"https://pith.science/pith/5ZQEJKOV6XC7J7OMXYBXZKPNDU/action/author_attestation","sign_citation":"https://pith.science/pith/5ZQEJKOV6XC7J7OMXYBXZKPNDU/action/citation_signature","submit_replication":"https://pith.science/pith/5ZQEJKOV6XC7J7OMXYBXZKPNDU/action/replication_record"}},"created_at":"2026-06-08T01:04:29.650058+00:00","updated_at":"2026-06-08T01:04:29.650058+00:00"}