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If $g=1$, these are also called $B_2$-sets, Sidon sets, and Babcock sets. We define $G(g,n)$ to represent the minimum diameter of a $g$-Golomb Ruler. In this paper, we prove that for all $b\\ge 1$, if $g \\ge \\frac{7}{4}\\left(b^{3/2} -b\\right)+1,$ then $G(g,g+b)=g+2b-2$. Sharper bounds are given for $b\\le 18$. The main technique is through an arithmetic property of the integers that are \\emph{not} in a $g$-Golomb ruler, leading us to intr"},"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":true,"formal_links_present":true},"canonical_record":{"source":{"id":"2605.14229","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2026-05-14T00:47:08Z","cross_cats_sorted":[],"title_canon_sha256":"2689b1dee4d64e1887b510125b0d3da6f0a696f29b5f7efef0953a6322d80f73","abstract_canon_sha256":"b093b283b622f632f26d67fd84263b15f81d089324ae56f6829260a15bfc5a78"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:39:10.752297Z","signature_b64":"vZSahXjopPgAB9qvOeG21PrKCZ/N7m8aoNUH7B/Pi7MOIO00rwnwynA/L1Em5p9zDWmBxPg3/lzKUzhkFMl7Dg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"48d719aea18fe4b38122392a96303b70ece6ef53ee8c478d83233ec66c8cfaad","last_reissued_at":"2026-05-17T23:39:10.751873Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:39:10.751873Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Optimal Diameters of High Multiplicity g-Golomb Rulers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"For g at least roughly 1.75 times b to the 3/2 power, the shortest g-Golomb ruler with g plus b marks has diameter exactly g plus 2b minus 2.","cross_cats":[],"primary_cat":"math.CO","authors_text":"Aditya Gupta, Kevin O'Bryant","submitted_at":"2026-05-14T00:47:08Z","abstract_excerpt":"A set $\\cG$ of integers is called a $g$-Golomb ruler of length $n$ if the difference between any two distinct elements of $\\cG $ is repeated at most $g$ times. If $g=1$, these are also called $B_2$-sets, Sidon sets, and Babcock sets. We define $G(g,n)$ to represent the minimum diameter of a $g$-Golomb Ruler. In this paper, we prove that for all $b\\ge 1$, if $g \\ge \\frac{7}{4}\\left(b^{3/2} -b\\right)+1,$ then $G(g,g+b)=g+2b-2$. Sharper bounds are given for $b\\le 18$. The main technique is through an arithmetic property of the integers that are \\emph{not} in a $g$-Golomb ruler, leading us to intr"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"for all b≥1, if g ≥ (7/4)(b^{3/2} - b) +1, then G(g,g+b)=g+2b-2","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The arithmetic property of integers not belonging to a g-Golomb ruler is strong enough to force the diameter lower bound via the newly defined LM rulers.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"G(g, g+b) equals g + 2b - 2 when g meets the threshold (7/4)(b^{3/2} - b) + 1, with new bounds sqrt(8/9)(n-1)^{3/2} to (7/4)((n+1)^{3/2} - (n+1)) for the minimal diameter L(n) of n-element LM rulers.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"For g at least roughly 1.75 times b to the 3/2 power, the shortest g-Golomb ruler with g plus b marks has diameter exactly g plus 2b minus 2.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"1e098c088e5ef5ed50896e738ad0163deca0d34ae666267804164a840bb45cd6"},"source":{"id":"2605.14229","kind":"arxiv","version":1},"verdict":{"id":"088489e5-43a2-43f4-9bf2-92fe9e1325e3","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-15T02:53:21.968905Z","strongest_claim":"for all b≥1, if g ≥ (7/4)(b^{3/2} - b) +1, then G(g,g+b)=g+2b-2","one_line_summary":"G(g, g+b) equals g + 2b - 2 when g meets the threshold (7/4)(b^{3/2} - b) + 1, with new bounds sqrt(8/9)(n-1)^{3/2} to (7/4)((n+1)^{3/2} - (n+1)) for the minimal diameter L(n) of n-element LM rulers.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The arithmetic property of integers not belonging to a g-Golomb ruler is strong enough to force the diameter lower bound via the newly defined LM rulers.","pith_extraction_headline":"For g at least roughly 1.75 times b to the 3/2 power, the shortest g-Golomb ruler with g plus b marks has diameter exactly g plus 2b minus 2."},"references":{"count":8,"sample":[{"doi":"","year":1984,"title":"M. 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