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We extend this characterization in two directions.\n  First, we show that $\\FO(\\Z,+,\\leq)$-definable \\emph{functions} are exactly the piecewise linear functions (Theorem~\\ref{thm:integer}), and that $\\FO(\\R,+,\\leq)$-definable functions are also exactly the piecewise linear functions (Theorem~\\ref{thm:real}). The proofs are direct algebraic argume"},"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":"2607.05701","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.LO","submitted_at":"2026-07-06T23:47:19Z","cross_cats_sorted":["cs.FL"],"title_canon_sha256":"a1901dab3e98ca03a0ba859432fd95d1b2b5aede122da434c32844893e4af4a0","abstract_canon_sha256":"b2c627a29d0bb3c0b70d10a2ef73065bded7ed684aa9b7505d1f2ba711637675"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-08T01:18:41.892063Z","signature_b64":"X0CzxCAzk7uQzOMyh80aQAEtjDD7L60uzN1SjNof14lZ2nrRdzP5l10omcy8kR87bKIUEpBcVOpPjK39gA5JDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0566b0c7f99eee72ae172b256ed5c908182cdf92a6f7903e22e1a43f7c8366c2","last_reissued_at":"2026-07-08T01:18:41.891587Z","signature_status":"signed_v1","first_computed_at":"2026-07-08T01:18:41.891587Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Extending the Ginsburg-Spanier Theorem to Functions and Mixed Arithmetic","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["cs.FL"],"primary_cat":"cs.LO","authors_text":"Alain Finkel, J\\'er\\^ome Leroux","submitted_at":"2026-07-06T23:47:19Z","abstract_excerpt":"We study sets and functions definable in the three additive theories $\\FO(\\Z,+,\\leq)$, $\\FO(\\R,+,\\leq)$, and $\\FO(\\R,\\Z,+,\\leq)$.\n  The Ginsburg--Spanier theorem~\\cite{GS66} characterizes $\\FO(\\Z,+,\\leq)$-definable sets as exactly the semi-linear sets. We extend this characterization in two directions.\n  First, we show that $\\FO(\\Z,+,\\leq)$-definable \\emph{functions} are exactly the piecewise linear functions (Theorem~\\ref{thm:integer}), and that $\\FO(\\R,+,\\leq)$-definable functions are also exactly the piecewise linear functions (Theorem~\\ref{thm:real}). 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