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The case we consider is the first case where the regularity was unknown, following earlier computations (by several authors) of the regularity when G is a tree, cycle, complete graph or complete bipartite graph, but specially in light of the reduction of the computation of th"},"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":"1606.08621","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2016-06-28T09:20:25Z","cross_cats_sorted":["math.AG"],"title_canon_sha256":"16aff109b9af247275741b80d5e8f6b33dbb4e6d6a9456b326e674ce099a30a6","abstract_canon_sha256":"9ff0bcdfa4472d0db5b2cf7628d99db7eec25e31a355ce0ee1b2c21887e0ae3d"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:11:47.377101Z","signature_b64":"jbmpsMsspd5F4afHci4icyGJ+9+v19BrWwMCxUzbsqXNb88827dxnH1vQlnlK/Qde94xFRPgyBJ0CvORjgRfBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"56908e90b01a336f5c200c48cd37b2c80a87bd85418b02a932cd982c5693d889","last_reissued_at":"2026-05-18T01:11:47.376731Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:11:47.376731Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Regularity of the vanishing ideal over a parallel composition of paths","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AG"],"primary_cat":"math.AC","authors_text":"Antonio Macchia, Jorge Neves, Maria Vaz Pinto, Rafael H. 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