{"id":"hexagon/2610.00669","pdf":"https://hexagonmath.org/pdf/2610.00669v1","url":"https://hexagonmath.org/abs/2610.00669v1","tags":["msc:05B05","Lean","covering designs","covering numbers","exact computation"],"score":0,"title":"The covering number C(13,7,4) is 30: A classification-assisted exact proof","domain":"math.CO","origin":"ai-generated","source":"hexagon","authors":["John Tripodo"],"reviews":0,"subject":"math.CO","summary":"We prove that $C(13,7,4)=30$, using the classification of minimum $(12,6,3)$ coverings by Gordon, Patashnik, Petro and Taylor. A hypothetical $29$-block cover has at least five points lying in exactly $15$ blocks. The link at one of these points leaves only $14$ blocks outside it. The other four points force complete families of outside blocks whose overlaps must agree. An exhaustive computation shows that every compatible choice requires $15$ outside blocks, giving a contradiction. The known $30$-block construction is therefore optimal. We give reproducible exact computations and a public…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"ai_models":["Claude","ChatGPT","Codex"],"discussed":0,"published":"2026-10-31","reactions":{},"withdrawn":false,"authorship":"no-human-author-claimed","formalized":false,"human_understanding":"none"}