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Our theorem is for piecewise linear minimal valid functions that have only rational breakpoints (in $1/q\\,\\mathbb{Z}$ for some $q\\in \\mathbb{N}$) and that take rational values at the breakpoints. In contrast to Basu et al.'s construction, our construction preserves all function values on $1/q\\,\\mathbb{Z}$. As a corollary, we obtain"},"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":"1807.09758","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2018-07-24T18:18:30Z","cross_cats_sorted":["math.OC"],"title_canon_sha256":"8116144965caf16d369300e3f14c186b5aef12d26931ca4f2484ac2cf1c00369","abstract_canon_sha256":"678cd8641b5bc3200fdcf797b7900dbc6e7b44c09ca0d2e33784bb24c13e7dfd"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:48:22.542519Z","signature_b64":"0jhlIarl1+A36IVbo6cy2FrpwFH2zT5jg60fHgLkDutGA3ZLxMB5qHw6H9NRPK6z9n0VPgmZJzwqF0sU2DKRDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"bce97db42a2118e10272cd2e84e036b2e5c7b0293da3191c88fcb274f77cb035","last_reissued_at":"2026-05-17T23:48:22.541887Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:48:22.541887Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"All Cyclic Group Facets Inject","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.OC"],"primary_cat":"math.GR","authors_text":"Matthias K\\\"oppe, Yuan Zhou","submitted_at":"2018-07-24T18:18:30Z","abstract_excerpt":"We give a variant of Basu-Hildebrand-Molinaro's approximation theorem for continuous minimal valid functions for Gomory-Johnson's infinite group problem by piecewise linear two-slope extreme functions [Minimal cut-generating functions are nearly extreme, IPCO 2016]. Our theorem is for piecewise linear minimal valid functions that have only rational breakpoints (in $1/q\\,\\mathbb{Z}$ for some $q\\in \\mathbb{N}$) and that take rational values at the breakpoints. In contrast to Basu et al.'s construction, our construction preserves all function values on $1/q\\,\\mathbb{Z}$. 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