{"paper":{"title":"An exact small-$n$ computation of the minimum 2-coloring discrepancy of $K_n^{(3)}$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"For n ≡ 1 or 3 mod 6 up to 21, the minimum 2-coloring discrepancy of Steiner triple systems equals min over x of |x(n-x)/2 - n(n-1)/12|.","cross_cats":[],"primary_cat":"math.CO","authors_text":"Tong Niu","submitted_at":"2026-05-01T08:02:38Z","abstract_excerpt":"For an integer $r \\ge 2$ and an order $n \\equiv 1, 3 \\pmod{6}$, write $\\delta_r(n)$ for the minimum, over all $r$-colourings $\\chi : \\binom{[n]}{3} \\to [r]$, of $\\max_{\\mathcal{S}} \\mathrm{disc}(\\mathcal{S}, \\chi)$, where the maximum is over labelled Steiner triple systems $\\mathcal{S}$ of order $n$ and $\\mathrm{disc}(\\mathcal{S}, \\chi) = \\max_c |\\#\\{T \\in \\mathcal{S} : \\chi(T) = c\\} - |\\mathcal{S}|/r|$. Following Gishboliner, Glock, and Sgueglia \\cite{GishbolinerGlockSgueglia2025}, the bulk of the recent work on this quantity has been on lower bounds for $r \\ge 3$ (proving $\\delta_r(n) = \\Ome"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"An exact value of δ₂(n) for each such n, matching the formula δ₂(n) = min_{x ∈ [0,n] ∩ Z} |x(n-x)/2 - n(n-1)/12| obtained by optimising the GGS Example 1.1 family.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"That simulated annealing finds the global minimum for n=13,15,19,21 and that the conjectural formula continues to hold for all larger n ≡1,3 mod 6.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Exact values of δ₂(n) for n=7,9,13,15,19,21 match the min over x of |x(n-x)/2 - n(n-1)/12|, with a conjecture that this holds for all n ≡1,3 mod 6.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"For n ≡ 1 or 3 mod 6 up to 21, the minimum 2-coloring discrepancy of Steiner triple systems equals min over x of |x(n-x)/2 - n(n-1)/12|.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"718a725207b3431946609269afc2afff8944646124494f4c7ed0089ca3e50440"},"source":{"id":"2605.00492","kind":"arxiv","version":2},"verdict":{"id":"adfaeb91-4963-4dc4-a73d-7f2cfa9b1ce4","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-09T19:21:58.843835Z","strongest_claim":"An exact value of δ₂(n) for each such n, matching the formula δ₂(n) = min_{x ∈ [0,n] ∩ Z} |x(n-x)/2 - n(n-1)/12| obtained by optimising the GGS Example 1.1 family.","one_line_summary":"Exact values of δ₂(n) for n=7,9,13,15,19,21 match the min over x of |x(n-x)/2 - n(n-1)/12|, with a conjecture that this holds for all n ≡1,3 mod 6.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"That simulated annealing finds the global minimum for n=13,15,19,21 and that the conjectural formula continues to hold for all larger n ≡1,3 mod 6.","pith_extraction_headline":"For n ≡ 1 or 3 mod 6 up to 21, the minimum 2-coloring discrepancy of Steiner triple systems equals min over x of |x(n-x)/2 - n(n-1)/12|."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.00492/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"ai_meta_artifact","ran_at":"2026-05-20T19:41:07.173604Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_compliance","ran_at":"2026-05-19T18:06:42.766307Z","status":"completed","version":"1.0.0","findings_count":0}],"snapshot_sha256":"5572b9473d6e574d11377b1a56327d4df0f1bab3915b3b24e3f679abde72d59c"},"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"}